diff 763cb89..b34550f hard_sphere_boltzmann_fluctuations_v2.txt
diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index 57fae9c..1e7629c 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -2,17 +2,17 @@
A gas of billiard balls. A large box holds identical hard spheres,
like a three-dimensional break of pool: they fly in straight lines and
-bounce elastically on contact. The diameter $\varepsilon$ is tiny and the density
-is $\mu=\varepsilon^{-2}$ spheres per unit volume, so the gas is dilute yet the
+bounce elastically on contact. The diameter $ε$ is tiny and the density
+is $\mu=ε^{-2}$ spheres per unit volume, so the gas is dilute yet the
number of spheres is enormous.
The average. The expected number of spheres per unit volume of
position-velocity space, the one-particle density, solves the Boltzmann
-equation as $\varepsilon\downarrow0$. That is Lanford's kinetic limit.
+equation as $ε↓0$. That is Lanford's kinetic limit.
The wobble. The random gas equals this deterministic density plus
a small error. Scaling the error by $\sqrt\mu$ gives the fluctuation field
-$\zeta^\varepsilon_t$; the question is its law as $\varepsilon\downarrow0$.
+$\zeta^ε_t$; the question is its law as $ε↓0$.
The answer. The scaled error is asymptotically Gaussian, with a
covariance governed by the fluctuating Boltzmann equation: the linearized
@@ -57,12 +57,12 @@ equilibrium.
Two new difficulties. First, size: the $h$-th factorial cumulant,
the connected $h$-point correlation, must be pinned at the sharp scale
$\mu^{1-h}$, far below a crude bound. A change of relative translations of
-collision clusters extracts one $\varepsilon^2$ per merging collision and does it.
+collision clusters extracts one $ε^2$ per merging collision and does it.
Second, centering: the gas is centered by its true microscopic mean, which
-differs from the Boltzmann mean already at order $\varepsilon$, since hard spheres
-cannot overlap; at the scale $\sqrt\mu=\varepsilon^{-1}$ that difference survives,
+differs from the Boltzmann mean already at order $ε$, since hard spheres
+cannot overlap; at the scale $\sqrt\mu=ε^{-1}$ that difference survives,
so the model dynamics must match the gas with probability better than
-$O(\varepsilon)$.
+$O(ε)$.
Notation is fixed once and used without variation.
@@ -81,7 +81,7 @@ $\Delta\phi$ of a test across one collision.
$A_t$, $C_t$ : linearized Boltzmann drift and collision-noise covariance:
the two coefficients of the fluctuating equation.
-$\zeta_t^\varepsilon$, $\zeta_t$ : the centered, $\sqrt\mu$-scaled fluctuation
+$\zeta_t^ε$, $\zeta_t$ : the centered, $\sqrt\mu$-scaled fluctuation
(microscopic) and its Gaussian limit.
$W$ : the Gaussian noise driving $\zeta_t$, with independent increments.
@@ -89,27 +89,27 @@ $W$ : the Gaussian noise driving $\zeta_t$, with independent increments.
$g_i$, $f_{i\tau}$ : one-root mean at interface $i$ (the independent input)
and the Boltzmann density $f_{i\tau}$ at the same time.
-$H_i,\Lambda_i,\Gamma$ : collision bookkeeping of the Deng--Hani--Ma
+$H_i,Λ_i,\Gamma$ : collision bookkeeping of the Deng--Hani--Ma
machinery: root count, particle-size cap, and cluster-excess cap.
$L, \tau=T/L$, $K$, $\rho$ : number of slabs, slab length, the complexity
cutoff, and the complexity of a history.
-$c_h^\varepsilon$ : the $h$-th factorial cumulant density; its target size is the
+$c_h^ε$ : the $h$-th factorial cumulant density; its target size is the
sharp scale $\mu^{1-h}$.
-Why one collision costs $\varepsilon^2$. Two groups of spheres that
+Why one collision costs $ε^2$. Two groups of spheres that
evolved independently can join one connected history only if, at the
-contact, their relative position lands exactly on a sphere of radius $\varepsilon$
+contact, their relative position lands exactly on a sphere of radius $ε$
around the other group. Their relative motion is a straight line, and a line
meets that tiny sphere in a window whose cross section is a disk of area of
-order $\varepsilon^2$, hit during a sliding time interval. This is the source of the
-factor $\varepsilon^2$ per merging collision, the merger Jacobian
-$\varepsilon^2[(v-v_*)\cdot\omega] d t d \omega$. The gas carries activity
-$\mu=\varepsilon^{-2}$ per particle, so a connected diagram with $h$ marked
-particles and $m$ lines weighs $\mu^{m-h}$, and each merger pays one $\varepsilon^2$:
-the two powers of $\varepsilon$ cancel line by line. The initial excluded-volume link
-is a ball, not a sphere, so it costs $\varepsilon^3$, the only place where the
+order $ε^2$, hit during a sliding time interval. This is the source of the
+factor $ε^2$ per merging collision, the merger Jacobian
+$ε^2[(v-v_*)\cdot\omega] d t d \omega$. The gas carries activity
+$\mu=ε^{-2}$ per particle, so a connected diagram with $h$ marked
+particles and $m$ lines weighs $\mu^{m-h}$, and each merger pays one $ε^2$:
+the two powers of $ε$ cancel line by line. The initial excluded-volume link
+is a ball, not a sphere, so it costs $ε^3$, the only place where the
static problem is even smaller than the dynamical one.
Where does the Gaussian randomness come from? A test sees a sum of very many
@@ -140,40 +140,40 @@ v_*' =v_*+[(v-v_*)\cdot\omega]\omega$$
with $[a]_+=\max\{a,0\}$ and $ d \omega$ the unnormalized surface measure.
The Boltzmann equation reads
-$$(\partial_t+v\cdot\nabla_x)f=Q(f,f),
+$$(∂_t+v\cdot\nabla_x)f=Q(f,f),
Q(f,f)=\int_{R^3\times\mathbb S^2}B(f'f_*'-ff_*) d v_* d \omega$$
all densities evaluated at the same $x$.
-Fix the sphere diameter $\varepsilon>0$ and set the mean density
-$\mu=\varepsilon^{-2}$. The grand-canonical initial law, including $N=0$, is
+Fix the sphere diameter $ε>0$ and set the mean density
+$\mu=ε^{-2}$. The grand-canonical initial law, including $N=0$, is
-$$\mathbf{P}_\varepsilon(N, d Z_N)
-=\mathcal Z_\varepsilon^{-1}\frac{\mu^N}{N!}
-1 _{\mathcal D_N^\varepsilon}(Z_N)\prod_{i=1}^N f_0(z_i) d z_i$$
+$$P_ε(N, d Z_N)
+=\mathcal Z_ε^{-1}(\mu^N)/(N!)
+1 _{\mathcal D_N^ε}(Z_N)\prod_{i=1}^N f_0(z_i) d z_i$$
with the non-overlap set
-$\mathcal D_N^\varepsilon=\{(z_i):|x_i-x_j|>\varepsilon, i\ne j\}$ and
-$1\le\mathcal Z_\varepsilon\le e^\mu$. Given the configuration, spheres move by
+$\mathcal D_N^ε=\{(z_i):|x_i-x_j|>ε, i\ne j\}$ and
+$1\le\mathcal Z_ε\le e^\mu$. Given the configuration, spheres move by
free flight and deterministic elastic collisions, defined almost surely
by Deng--Hani--Ma, Proposition 1.2. There is no random collision rule and no
resampling: the only randomness is the initial configuration.
Assumption. $f_0\ge0$ is a smooth probability density with
-$\|f_0\|_{Bol,2\beta}+\|\nabla_xf_0\|_{Bol,2\beta}<\infty$ for some
+$\|f_0\|_{Bol,2\beta}+\|\nabla_xf_0\|_{Bol,2\beta}<∞$ for some
$\beta>0$. There is a nonnegative classical solution $f$ of
the Boltzmann equation on a prescribed interval $[0,T]$ with
-$$M:=\sup_{0\le t\le T}\sup_{x,v}e^{2\beta|v|^2}f(t,x,v)<\infty$$
+$$M:=\sup_{0\le t\le T}\sup_{x,v}e^{2\beta|v|^2}f(t,x,v)<∞$$
Throughout, $[0,T]$ is one fixed regular kinetic interval; nothing
below asserts that $T$ can be taken infinite.
-For real $\phi\in C_c^\infty(R^6)$ put
+For real $\phi\in C_c^∞(R^6)$ put
-$$\pi_t^\varepsilon(\phi)=\mu^{-1}\sum_{i=1}^N\phi(z_i(t)),
+$$\pi_t^ε(\phi)=\mu^{-1}\sum_{i=1}^N\phi(z_i(t)),
-\zeta_t^\varepsilon(\phi)=\sqrt\mu [\pi_t^\varepsilon(\phi)-E_\varepsilon\pi_t^\varepsilon(\phi)]$$
+\zeta_t^ε(\phi)=\sqrt\mu [\pi_t^ε(\phi)-E_ε\pi_t^ε(\phi)]$$
The expectation is the exact microscopic one. This is essential; see
the centering remark.
@@ -210,23 +210,23 @@ $$E[\zeta_0(\phi)\zeta_0(\psi)] =\int_{R^6}f_0 \phi\psi d z$$
$$E[W_t(\phi)W_s(\psi)] =\int_0^{t\wedge s}C_r(\phi,\psi) d r$$
with $W$ of independent increments. For every $m\ge1$, all $t_1,...,t_m
-\in[0,T]$, and real $\phi_1,...,\phi_m\in C_c^\infty(R^6)$,
+\in[0,T]$, and real $\phi_1,...,\phi_m\in C_c^∞(R^6)$,
-$$(\zeta_{t_1}^\varepsilon(\phi_1),...,\zeta_{t_m}^\varepsilon(\phi_m))
-\Longrightarrow\
+$$(\zeta_{t_1}^ε(\phi_1),...,\zeta_{t_m}^ε(\phi_m))
+⇒\
(\zeta_{t_1}(\phi_1),...,\zeta_{t_m}(\phi_m))
-(\varepsilon\downarrow0)$$
+(ε↓0)$$
Remark (The initial covariance). The initial covariance has no subtraction of
$(\int f_0\phi)(\int f_0\psi)$: the particle number is random. More
importantly, microscopic centering is not the same as centering by the
-Boltzmann mean. With $\varrho_0(x)=\int f_0(x,v) d v$,
+Boltzmann mean. With $ρ_0(x)=\int f_0(x,v) d v$,
$$
-\lim_{\varepsilon\to0}\sqrt\mu(E\pi_0^\varepsilon(\phi)-\int f_0\phi)
-=-\frac{4\pi}{3}\int f_0(x,v)\phi(x,v)\varrho_0(x) d x d v .
+\lim_{ε\to0}\sqrt\mu(E\pi_0^ε(\phi)-\int f_0\phi)
+=-(4\pi)/(3)\int f_0(x,v)\phi(x,v)ρ_0(x) d x d v .
$$
-So exclusion shifts the mean by order $\varepsilon$, which survives the factor
-$\sqrt\mu=\varepsilon^{-1}$. Centering by the Boltzmann density alone would
+So exclusion shifts the mean by order $ε$, which survives the factor
+$\sqrt\mu=ε^{-1}$. Centering by the Boltzmann density alone would
leave a nonzero deterministic drift already at time zero. This is why
the proof must compare exact expectations, not merely laws.
@@ -252,7 +252,7 @@ the sharp connected bounds). The $h$-th cumulant is $O(\mu^{1-h})$,
with the leading $h=1,2$ terms identified as signed sums of trees
weighted by $f_{i\tau}$, contact flux, and $\pm1$ signs. This is proved
by chronological merger coordinates: each collision that merges two
-clusters contributes $\varepsilon^2[(v-v_*)\cdot\omega]_+$, and summing all
+clusters contributes $ε^2[(v-v_*)\cdot\omega]_+$, and summing all
clusters costs $CT/\sqrt L<1$.
4. Gaussian limit for the model dynamics
@@ -263,7 +263,7 @@ vanish, giving the Gaussian law with the correct covariance.
5. Accuracy of the model dynamics (the cutoff comparison, below,
the pasted accuracy bound). The pasted and true dynamics agree with
-probability $O(\varepsilon^{1+\eta})$, $\eta=1/20$. This is a positive
+probability $O(ε^{1+\eta})$, $\eta=1/20$. This is a positive
geometric estimate on terminal witnesses, carried out with local
gains, a common birth-energy bound, and a causal time-order sum free of
per-vertex logarithms.
@@ -279,85 +279,85 @@ grand-canonical hard-sphere measure), and construct the Gaussian field
that the fluctuation limit theorem claims to be the limit. It also records the initial
correlation bounds that later parts use.
-Let $F_{h,0}^{\varepsilon}$ be the density of the ordered factorial measure
+Let $F_{h,0}^{ε}$ be the density of the ordered factorial measure
divided by $\mu^h$:
-$$\int F_{h,0}^{\varepsilon}(Z_h)\Phi(Z_h) d Z_h
+$$\int F_{h,0}^{ε}(Z_h)\Phi(Z_h) d Z_h
=\mu^{-h}E\sum_{i_1,...,i_h distinct}\Phi(z_{i_1},...,z_{i_h})$$
Write $[h]=\{1,...,h\}$, $Z_B=(z_j)_{j\in B}$, and let $\mathcal P(V)$
be the set of set-partitions of $V$. The factorial cumulant
-(connected) densities $c_{h,0}^{\varepsilon}$ are defined by the cumulant
+(connected) densities $c_{h,0}^{ε}$ are defined by the cumulant
inversion of the factorial hierarchy,
-$$F_{h,0}^{\varepsilon}(Z_h)
-=\sum_{\pi\in\mathcal P([h])}\prod_{B\in\pi}c_{|B|,0}^{\varepsilon}(Z_B)$$
+$$F_{h,0}^{ε}(Z_h)
+=\sum_{\pi\in\mathcal P([h])}\prod_{B\in\pi}c_{|B|,0}^{ε}(Z_B)$$
-The one-point function is $g_0=c_{1,0}^{\varepsilon}=F_{1,0}^{\varepsilon}$. Set
+The one-point function is $g_0=c_{1,0}^{ε}=F_{1,0}^{ε}$. Set
$$
-\varrho_0(x)=\int f_0(x,v) d v,
-b_\varepsilon=\tfrac{4\pi}{3}\|\varrho_0\|_\infty\varepsilon^3,
-d_\varepsilon=\mu b_\varepsilon=O(\varepsilon),
+ρ_0(x)=\int f_0(x,v) d v,
+b_ε=(4\pi)/(3)\|ρ_0\|_∞ε^3,
+d_ε=\mu b_ε=O(ε),
$$
and let $(N)_p=N...(N-p+1)$ denote the falling factorial.
The key geometric fact, before the initial CLT can even be stated, is that
-connected correlations are order $\varepsilon^3$ per connected pair.
+connected correlations are order $ε^3$ per connected pair.
This is the static shadow of the dynamical statement that each merging
-collision costs $\varepsilon^2$; excluded volume makes it $\varepsilon^3$.
+collision costs $ε^2$; excluded volume makes it $ε^3$.
-Proposition (Initial ensemble). For all sufficiently small $\varepsilon$:
+Proposition (Initial ensemble). For all sufficiently small $ε$:
-(i) $0\le F_{h,0}^{\varepsilon}\le\prod_{j=1}^hf_0(z_j)$ and
-$(1-C\varepsilon)f_0\le g_0\le f_0$. Hence
-$\|g_0-f_0\|_{Bol,2\beta}=O(\varepsilon)$ and, for every fixed $p\ge1$,
+(i) $0\le F_{h,0}^{ε}\le\prod_{j=1}^hf_0(z_j)$ and
+$(1-Cε)f_0\le g_0\le f_0$. Hence
+$\|g_0-f_0\|_{Bol,2\beta}=O(ε)$ and, for every fixed $p\ge1$,
-$$E(N)_p\le\mu^p, \sup_{0<\varepsilon\le\varepsilon_0}E(N/\mu)^p<\infty$$
+$$E(N)_p\le\mu^p, \sup_{0<ε\leε_0}E(N/\mu)^p<∞$$
-(ii) With $\upsilon=1/81$, for every $h\ge2$ and a constant $C$ independent
+(ii) With $υ=1/81$, for every $h\ge2$ and a constant $C$ independent
of $h$,
-$$|c_{h,0}^{\varepsilon}(Z_h)|\le(Ch)^{Ch}\prod_{j=1}^hf_0(z_j)
+$$|c_{h,0}^{ε}(Z_h)|\le(Ch)^{Ch}\prod_{j=1}^hf_0(z_j)
\sum_{\mathcal T}\prod_{\{i,j\}\in\mathcal T}
-( 1 _{|x_i-x_j|\le\varepsilon}+\varepsilon^\upsilon)$$
+( 1 _{|x_i-x_j|\leε}+ε^υ)$$
the sum over trees $\mathcal T$ on $[h]$. For fixed $h\ge2$ and any
$D>0$ there is $C_{D,h}$ with
-$$|c_{h,0}^{\varepsilon}(Z_h)|\le C_{D,h}\prod_{j=1}^hf_0(z_j)
+$$|c_{h,0}^{ε}(Z_h)|\le C_{D,h}\prod_{j=1}^hf_0(z_j)
\sum_{\mathcal T}\prod_{\{i,j\}\in\mathcal T}
-( 1 _{|x_i-x_j|\le C_{D,h}\varepsilon}+\varepsilon^D)$$
+( 1 _{|x_i-x_j|\le C_{D,h}ε}+ε^D)$$
and
-$$\|c_{h,0}^{\varepsilon}\|_{L^1}\le C_h\varepsilon^{3(h-1)} (h\ge2)$$
+$$\|c_{h,0}^{ε}\|_{L^1}\le C_hε^{3(h-1)} (h\ge2)$$
(iii) For every finite list of bounded real tests, the exact-centered
initial fluctuations converge jointly to a centered Gaussian vector with
covariance $\int f_0\phi\psi$. For smooth compactly supported real
$\phi$,
-$$\lim_{\varepsilon\to0}\sqrt\mu(E\pi_0^\varepsilon(\phi)-\int f_0\phi)
-=-\frac{4\pi}{3}\int_{R^6}f_0\phi \varrho_0 d z$$
+$$\lim_{ε\to0}\sqrt\mu(E\pi_0^ε(\phi)-\int f_0\phi)
+=-(4\pi)/(3)\int_{R^6}f_0\phi ρ_0 d z$$
Proof.
(i) Factorial bound and moments. For fixed $Z_h$, let
-$\mathcal Z_\varepsilon[Z_h]$ be the partition function of the remaining
+$\mathcal Z_ε[Z_h]$ be the partition function of the remaining
particles, still mutually excluding and additionally avoiding the balls
-$B(x_j,\varepsilon)$. Directly from the grand-canonical law,
+$B(x_j,ε)$. Directly from the grand-canonical law,
-$$F_{h,0}^{\varepsilon}(Z_h)= 1 _{\mathcal D_h^\varepsilon}(Z_h)\prod_{j=1}^hf_0(z_j)
-\frac{\mathcal Z_\varepsilon[Z_h]}{\mathcal Z_\varepsilon}$$
+$$F_{h,0}^{ε}(Z_h)= 1 _{\mathcal D_h^ε}(Z_h)\prod_{j=1}^hf_0(z_j)
+(\mathcal Z_ε[Z_h])/(\mathcal Z_ε)$$
The ratio is $\le1$, giving the factorial upper bound. For $h=1$ it is
-the probability that $B(x_1,\varepsilon)$ is empty of particle centers; by the
+the probability that $B(x_1,ε)$ is empty of particle centers; by the
factorial upper bound with one particle and the union bound this is
-$\ge1-\mu b_\varepsilon=1-O(\varepsilon)$, giving $(1-C\varepsilon)f_0\le g_0\le f_0$.
+$\ge1-\mu b_ε=1-O(ε)$, giving $(1-Cε)f_0\le g_0\le f_0$.
Integrating the factorial bound gives $E(N)_p\le\mu^p$; writing $N^p$
as a positive combination of $(N)_j$ gives the moment bounds.
(ii) The Penrose tree bound. Write $\chi_{ij}= 1 _{|x_i-x_j|\le
-\varepsilon}$ and set
+ε}$ and set
$$
\mathcal U_1=1,
\mathcal U_n(Z_n)=\sum_{H conn. on [n]}\prod_{\{i,j\}\in H}(-\chi_{ij}) (n\ge2).
@@ -370,12 +370,12 @@ This is the Penrose forest scheme. Root a connected labelled graph at
vertex $1$, assign graph-distance levels, and let the parent of each
vertex be its least-labelled neighbour in the preceding level. For a
fixed resulting tree $\mathcal T$, the graphs giving it are exactly
-$\mathcal T\subseteq H\subseteq R(\mathcal T)$, where $R(\mathcal T)$
+$\mathcal T⊆ H⊆ R(\mathcal T)$, where $R(\mathcal T)$
adds all same-level edges and every edge to a preceding-level vertex with
label $\ge$ that of the chosen parent. These intervals partition all
connected graphs, and summing one interval gives
$$
-\prod_{e\in\mathcal T}(-\chi_e)\prod_{e\in R(\mathcal T)\setminus\mathcal T}(1-\chi_e),
+\prod_{e\in\mathcal T}(-\chi_e)\prod_{e\in R(\mathcal T)∖\mathcal T}(1-\chi_e),
$$
whose second factor lies in $[0,1]$, proving the tree bound.
@@ -387,63 +387,63 @@ classical systems, in Statistical Mechanics (Benjamin, 1967).
Integrating a tree from its leaves, keeping one free vertex, gives
-$$\int f_0^{\otimes n}|\mathcal U_n|\le n^{n-2}b_\varepsilon^{n-1} (n\ge2)$$
+$$\int f_0^{ ⊗ n}|\mathcal U_n|\le n^{n-2}b_ε^{n-1} (n\ge2)$$
Hence the connected-graph (Mayer) series is absolutely convergent when
-$e\mu b_\varepsilon\|a\|_\infty<1$, and by exponentiation/analytic continuation
+$e\mu b_ε\|a\|_∞<1$, and by exponentiation/analytic continuation
-$$\log\mathcal Z_\varepsilon[a]=\sum_{n\ge1}\frac{\mu^n}{n!}\int\mathcal U_n(Z_n)
+$$\log\mathcal Z_ε[a]=\sum_{n\ge1}(\mu^n)/(n!)\int\mathcal U_n(Z_n)
\prod_{j=1}^nf_0(z_j)a(z_j) d Z_n$$
From the partition function to the cumulants. Apply
the logarithmic partition formula with $a=1+u$. The log of
-$E\prod_i(1+u(z_i))$ has coefficients $\mu^hc_{h,0}^{\varepsilon}/h!$, so
+$E\prod_i(1+u(z_i))$ has coefficients $\mu^hc_{h,0}^{ε}/h!$, so
differentiation gives the rooted Mayer series
-$$c_{h,0}^{\varepsilon}(Z_h)=\prod_{j=1}^hf_0(z_j)
-\sum_{n\ge0}\frac{\mu^n}{n!}\int f_0^{\otimes n}(Z_{h+1:h+n})
+$$c_{h,0}^{ε}(Z_h)=\prod_{j=1}^hf_0(z_j)
+\sum_{n\ge0}(\mu^n)/(n!)\int f_0^{ ⊗ n}(Z_{h+1:h+n})
\mathcal U_{h+n}(Z_{h+n}) d Z_{h+1:h+n}$$
For bounds uniform in $h$: delete $h-1$ edges of a tree on $h+n$ vertices
so each component contains one fixed root (possible by repeatedly cutting
an edge on a path between two roots). Dropping the deleted constraints and
-integrating the auxiliary leaves costs at most $b_\varepsilon^n$. With
+integrating the auxiliary leaves costs at most $b_ε^n$. With
$n!\ge(n/e)^n$, $(1+h/n)^n\le e^h$, $(h+n)^h\le h^he^n$,
$$\frac{(h+n)^{h+n-2}}{n!}\le(eh)^he^{2n}$$
and the $n=0$ count is $\le(Ch)^{Ch}$. Thus the rooted Mayer series
-converges absolutely for $\varepsilon\le\varepsilon_0$ with $\varepsilon_0$ independent of
+converges absolutely for $ε\leε_0$ with $ε_0$ independent of
$h$.
Coarse link bound. Let $q$ be the number of connected components
-of the graph on the $h$ roots with edges $|x_i-x_j|\le\varepsilon$. If $q\ge2$,
+of the graph on the $h$ roots with edges $|x_i-x_j|\leε$. If $q\ge2$,
every contributing tree must attach each such component to an auxiliary
vertex. One auxiliary vertex touches at most $27$ root components: pick
-one adjacent root from each; their mutual distances exceed $\varepsilon$, and
-their disjoint balls of radius $\varepsilon/2$ lie in a ball of radius
-$3\varepsilon/2$. Hence $q\le27n$ for a contributing tree with $n$ auxiliary
-vertices. Using the counting inequality and $d_\varepsilon=O(\varepsilon)$,
+one adjacent root from each; their mutual distances exceed $ε$, and
+their disjoint balls of radius $ε/2$ lie in a ball of radius
+$3ε/2$. Hence $q\le27n$ for a contributing tree with $n$ auxiliary
+vertices. Using the counting inequality and $d_ε=O(ε)$,
$$
-\frac{|c_{h,0}^{\varepsilon}(Z_h)|}{\prod_jf_0(z_j)}
-\le(Ch)^{Ch}\varepsilon^{q/27}\le(Ch)^{Ch}\varepsilon^{(q-1)/81} (q\ge2),
+\frac{|c_{h,0}^{ε}(Z_h)|}{\prod_jf_0(z_j)}
+\le(Ch)^{Ch}ε^{q/27}\le(Ch)^{Ch}ε^{(q-1)/81} (q\ge2),
$$
and for $q=1$ the same argument without the restriction on $n$ gives
$(Ch)^{Ch}$. A tree of the roots using intra-component and $q-1$
-inter-component edges has weight at least $\varepsilon^{(q-1)/81}$ in
+inter-component edges has weight at least $ε^{(q-1)/81}$ in
the factorial bound. This proves the factorial lower bound.
Sharp link bound. For fixed $h,D$, truncate
the rooted Mayer series at $n_0+1\ge D(h-1)$. The tail is
-$C_{D,h}\varepsilon^{D(h-1)}\prod_jf_0(z_j)$. In any term with $n\le n_0$, all
-roots are connected by paths of $\le h+n_0-1$ links of length $\le\varepsilon$,
-so their diameter is $\le(h+n_0)\varepsilon$; on that set the sharp link bound
+$C_{D,h}ε^{D(h-1)}\prod_jf_0(z_j)$. In any term with $n\le n_0$, all
+roots are connected by paths of $\le h+n_0-1$ links of length $\leε$,
+so their diameter is $\le(h+n_0)ε$; on that set the sharp link bound
holds with $C_{D,h}$ chosen large. This proves the sharp link bound.
Integrating the rooted Mayer series and using the rooted tree integral bound
instead gives
$$
-\|c_{h,0}^{\varepsilon}\|_1\le b_\varepsilon^{h-1}\sum_{n\ge0}\frac{(h+n)^{h+n-2}}{n!}d_\varepsilon^n
-\le C_h\varepsilon^{3(h-1)},
+\|c_{h,0}^{ε}\|_1\le b_ε^{h-1}\sum_{n\ge0}\frac{(h+n)^{h+n-2}}{n!}d_ε^n
+\le C_hε^{3(h-1)},
$$
which is the L1 bound.
@@ -452,32 +452,32 @@ $S_\phi=\sum_i\phi(z_i)$ and let $\kappa_k$ denote ordinary cumulants.
Substituting $a=e^{u\phi}$ in the logarithmic partition formula and
differentiating $k$ times at $u=0$,
$$
-\kappa_k(S_\phi)=\sum_{n\ge1}\frac{\mu^n}{n!}\int f_0^{\otimes n}\mathcal U_n
+\kappa_k(S_\phi)=\sum_{n\ge1}(\mu^n)/(n!)\int f_0^{ ⊗ n}\mathcal U_n
(\sum_{j=1}^n\phi(z_j))^k,
$$
and the rooted tree integral bound gives $\kappa_k(S_\phi)=\mu\int f_0\phi^k+
-O_{k,\phi}(\mu\varepsilon)$. After exact centering and division by $\sqrt\mu$,
+O_{k,\phi}(\muε)$. After exact centering and division by $\sqrt\mu$,
the second cumulant tends to $\int f_0\phi^2$ and every higher one to
zero. The moment-cumulant identity then gives every Gaussian moment; for
scalar convergence compare characteristic functions of the normalized
variable and the Gaussian of variance $\sigma^2=\int f_0\phi^2$ using
Taylor polynomials of degree $2n-1$, whose combined remainder after
-$\limsup$ is at most $2|u|^{2n}\sigma^{2n}/(2^n n!)\to0$. Joint
+$limsup$ is at most $2|u|^{2n}\sigma^{2n}/(2^n n!)\to0$. Joint
convergence follows on each linear combination. This includes zero
variance.
For the shift, the $n=1$ term of the rooted Mayer series with $h=1$
gives, uniformly in $z$,
$$
-g_0(z)=f_0(z)[1-\mu\int_{|x-x_*|\le\varepsilon}\varrho_0(x_*) d x_*+O(\varepsilon^2)].
+g_0(z)=f_0(z)[1-\mu\int_{|x-x_*|\leε}ρ_0(x_*) d x_*+O(ε^2)].
$$
-The bounded gradient of $\varrho_0$ gives
-$\int_{|x-x_*|\le\varepsilon}\varrho_0=\tfrac{4\pi}{3}\varepsilon^3\varrho_0(x)+O(\varepsilon^4)$,
-and $\sqrt\mu=\varepsilon^{-1}$ converts this into the centering shift.
+The bounded gradient of $ρ_0$ gives
+$\int_{|x-x_*|\leε}ρ_0=(4\pi)/(3)ε^3ρ_0(x)+O(ε^4)$,
+and $\sqrt\mu=ε^{-1}$ converts this into the centering shift.
Remark (The centering shift). The shift is generally nonzero, so exact
microscopic centering is essential already at time zero. Meanwhile
-$\mu\|c_{2,0}^{\varepsilon}\|_1=O(\varepsilon)$, so the limiting initial covariance has
+$\mu\|c_{2,0}^{ε}\|_1=O(ε)$, so the limiting initial covariance has
only the same-particle term $\int f_0\phi\psi$.
Write $M=\sup_{t\le T,x,v}e^{2\beta|v|^2}f_t(x,v)$. The noise form needs global velocity integrability, which does not
@@ -485,7 +485,7 @@ follow from a spatially uniform bound alone.
Lemma (Global weighted integrability). For every $0<q<2\beta$,
-$$\sup_{0\le t\le T}\int_{R^6}f_t(x,v)e^{q|v|^2} d x d v<\infty$$
+$$\sup_{0\le t\le T}\int_{R^6}f_t(x,v)e^{q|v|^2} d x d v<∞$$
Proof.
Let $Q^+$ be the gain part of $Q$ and $P_tg=Q^+(g,f_t)$ with the second
@@ -502,7 +502,7 @@ $$
Distributing the exponent from $\beta$ to $\beta/2$ over $n$ factors and
using that the time simplex has volume $T^n/n!$,
$\sup_{t\le T}\|(\mathcal V^nf)_t\|_{Y_{\beta/2}}\le M(CT)^n/n!(1+
-\sqrt{2n/\beta})^n\to0$, so $f$ is dominated pointwise by the positive
+√(2n/\beta))^n\to0$, so $f$ is dominated pointwise by the positive
gain series.
Let $X_q=L^1(e^{q|v|^2} d x d v)$. The pre/post substitution with
@@ -528,7 +528,7 @@ propagator $U(r,t):E_a^0\to E_b^0$ solving
$$U(r,t)h=\mathsf S_{t-r}h+\int_r^t\mathsf S_{s-r}K_sU(s,t)h d s$$
with the composition law at intermediate exponents and uniqueness as a
-mild propagator in these scales. For $\phi\in C_c^\infty$,
+mild propagator in these scales. For $\phi\in C_c^∞$,
$$U(r,t)\phi-\phi=\int_r^tU(r,s)A_s\phi d s$$
@@ -545,7 +545,7 @@ with $\eta_h(x)=\sup_ve^{-a|v|^2}|h|$, and $K_th$ is continuous in
$E_b^0$. Iterating the backward mild equation bounds the $n$-th
time-ordered term by
-$$\frac{(CM(t-r))^n}{n!}(1+\sqrt{n/(b-a)})^n\|h\|_a$$
+$$((CM(t-r))^n)/(n!)(1+√(n/(b-a)))^n\|h\|_a$$
so the series converges for every finite $T$. Splitting a time simplex at
an intermediate time gives composition, and the same estimate on an
@@ -553,7 +553,7 @@ iterated remainder gives uniqueness with a positive exponent allowance.
For compactly supported smooth $\phi$, differentiability of the series at
its upper end (its final transport gives $U(r,t)(v\cdot\nabla_x\phi)$;
the simplex faces give $U(r,t)K_t\phi$) converges with the Dyson bound,
-giving $\partial_tU(r,t)\phi=U(r,t)A_t\phi$ and hence the upper-end identity.
+giving $∂_tU(r,t)\phi=U(r,t)A_t\phi$ and hence the upper-end identity.
Proposition (Gaussian target). The fluctuating equation has a centered Gaussian solution on each
test scale $E_a^0$, $0\le a<\beta$, mapping continuously and linearly into
@@ -593,7 +593,7 @@ They are jointly Gaussian and linear with covariance the target covariance;
orthogonality of disjoint time intervals in $L^2(\nu)$ gives independent
increments of $W$, and $I_0\perp I$ gives initial independence.
-Weak equation. For $\phi\in C_c^\infty$, using the upper-end identity
+Weak equation. For $\phi\in C_c^∞$, using the upper-end identity
and boundedness of the isonormal maps against Bochner integrals,
$\int_0^t\zeta_s(A_s\phi) d s=I_0(U(0,t)\phi-\phi)+
I( 1 _{r\le t}\Delta_{U(r,t)\phi-\phi})$, which together with
@@ -606,12 +606,12 @@ $D_t$ has zero initial field and noise, mean-square continuity, and the
transport mild identity $D_t(\phi)=\int_0^tD_s(K_s\mathsf S_{t-s}\phi) d s$
(finite-support Riemann sums with $\mathsf S_{t-s}\phi$). Iterating $n$
times with $0\le a<b<\beta$ bounds its $L^2$ norm by
-$C_b\|\phi\|_a(CT)^n/n!(1+\sqrt{n/(b-a)})^n\to0$, proving uniqueness in
+$C_b\|\phi\|_a(CT)^n/n!(1+√(n/(b-a)))^n\to0$, proving uniqueness in
the class and uniqueness in law.
This is a Gaussian generalized field in the mean-square sense of
the field bound; it does not claim samplewise membership in the
-dual of $E_a^0$. Its restriction to $C_c^\infty$ does have a random
+dual of $E_a^0$. Its restriction to $C_c^∞$ does have a random
distribution version: expanding locally in Fourier functions times a
compact cutoff, variances bounded by the field bound with $a=0$
make the expected local $H^{-s}$ norm finite for $s>3$, and a cube
@@ -625,27 +625,27 @@ bound. Everything is either a definition that is needed or a
statement quoted from Deng--Hani--Ma with its source location. This is the only
place where an unproved external input enters.
-Fix $\upsilon=3^{-4}=1/81$. Choose the excess bound $\Gamma$ as a large
+Fix $υ=3^{-4}=1/81$. Choose the excess bound $\Gamma$ as a large
dimensional constant before the number of slabs $L$. Put
$\tau=T/L$ and, with harmless integer roundings,
-$$A_L=|\log\varepsilon|, \Lambda_i=A_i^{30}, A_{i-1}=\Lambda_i^{30}$$
+$$A_L=|\logε|, Λ_i=A_i^{30}, A_{i-1}=Λ_i^{30}$$
and for the analytic induction
-$$\beta_i=(1-\frac{i}{10L})\frac\beta2,
+$$\beta_i=(1-(i)/(10L))\frac\beta2,
\widetilde\beta_i=\frac{\beta_{i-1}+\beta_i}{2},
-\theta_i=(1-\frac{i}{10L})3^{-11}$$
+\theta_i=(1-(i)/(10L))3^{-11}$$
Constants written $C$ inside an exponential factor $C^b$ are independent
of $L$; constants $C_*$ and $c_*>0$ may depend on $L$ and the kinetic
-bounds but not on $\varepsilon$. Crucially, no moment order enters the choice of
+bounds but not on $ε$. Crucially, no moment order enters the choice of
$\Gamma$ or $L$.
Restarted truncated dynamics. Work slab by slab. Within slab $i$,
restart the collision prehistory. At an incoming contact, accept the
elastic collision iff adding it to the current collision graph of
-that component keeps the component's particle count $\le\Lambda_i$ and its
+that component keeps the component's particle count $\leΛ_i$ and its
excess $\le\Gamma$; otherwise both particles continue straight. The
excess of a graph with $b$ edges, $n$ vertices, $k$ components is
$r=b-n+k$ (repeated edges counted with multiplicity); an internal contact
@@ -659,7 +659,7 @@ the cutoff comparison, below.
Lemma (Kinetic envelopes). Under the kinetic assumption,
-$$\sup_{0\le t\le T}(\|f_t\|_{Bol,\beta}+\|\nabla_xf_t\|_{Bol,\beta})<\infty$$
+$$\sup_{0\le t\le T}(\|f_t\|_{Bol,\beta}+\|\nabla_xf_t\|_{Bol,\beta})<∞$$
Proof.
Apply Deng--Hani--Ma, Proposition A.1 with decreasing exponent
@@ -689,12 +689,12 @@ designated roots if $i$ is the top slab), and $H_0$ the lines incident to
an initial link, where initial links join pairs of lines born at zero with
weight
-$$\ell_\varepsilon(x,x')= 1 _{|x-x'|\le\varepsilon}+\varepsilon^\upsilon 1 _{|x-x'|>\varepsilon}$$
+$$\ell_ε(x,x')= 1 _{|x-x'|\leε}+ε^υ 1 _{|x-x'|>ε}$$
An admissible dependent history satisfies:
-(i) $|H_i|\le A_i$; every C-cluster has $\le\Lambda_i$ lines and excess
-$\le\Gamma$; every layer component has $<\Lambda_i$ C-clusters.
+(i) $|H_i|\le A_i$; every C-cluster has $\leΛ_i$ lines and excess
+$\le\Gamma$; every layer component has $<Λ_i$ C-clusters.
(ii) The cluster graph in each slab is a forest and every layer component
contains a root of its slab; every unrooted line is connected in its slab
@@ -719,16 +719,16 @@ atoms.
Integral convention. Give each bond $e$ a vector $z_e=(x_e,v_e)$
with position $X_e(t)=x_e+tv_e$; give atom $a$ a time $t_a$. Its incoming
edges are $1,2$ and serial outgoing edges $1',2'$. On the contact surface
-put $u=v_1-v_2$, $\omega=(X_1(t_a)-X_2(t_a))/\varepsilon$, and
+put $u=v_1-v_2$, $\omega=(X_1(t_a)-X_2(t_a))/ε$, and
$\mathfrak b=[-u\cdot\omega]_+$. Define
$$\Delta_a^{\mathrm C}
-=\delta(|X_1-X_2|-\varepsilon)\mathfrak b
+=\delta(|X_1-X_2|-ε)\mathfrak b
\prod_{j=1}^2\delta(X_{j'}(t_a)-X_j(t_a))
\delta(v_{1'}-v_1+(u\cdot\omega)\omega)\delta(v_{2'}-v_2-(u\cdot\omega)\omega)$$
$$\Delta_a^{\mathrm O}
-=\delta(|X_1-X_2|-\varepsilon)\mathfrak b
+=\delta(|X_1-X_2|-ε)\mathfrak b
\prod_{j=1}^2\delta(x_{j'}-x_j)\delta(v_{j'}-v_j)$$
Reverse $\omega$ to match the scattering convention; surface measure is
@@ -746,41 +746,41 @@ roots the normalized integral is
$$\mathcal J_M(Q)=\mu^{-h}I_M(Q):$$
each birth carries activity $\mu$, each root a normalization $\mu^{-1}$,
-each contact surface a factor $\varepsilon^2$ Deng--Hani--Ma, Proposition 7.5, Eq. (7.7).
+each contact surface a factor $ε^2$ Deng--Hani--Ma, Proposition 7.5, Eq. (7.7).
To keep a root endpoint $(x,v)$ at time $t$ free, make its top end a fixed
end evaluated at $(x-tv,v)$, removing the root activity.
The one-slab class $\mathcal T_i$ consists of connected histories with one
top root, conditions (i)--(ii) only, no continuation or initial-link
-requirement; it may have up to $\sim\Lambda_i^2$ lines.
+requirement; it may have up to $\simΛ_i^2$ lines.
$\mathcal T_i^{err}$ inserts one extra O-atom into a member of
$\mathcal T_i$. The one-root signed tree sum contains every one-root
-particle-line tree of size $<\Lambda_i/3$, with C/O choices and sign
+particle-line tree of size $<Λ_i/3$, with C/O choices and sign
$(-1)^{\#\mathrm O}$.
Theorem (Deng--Hani--Ma analytic package). Assume the kinetic assumption, the cutoff hierarchy, the exponent hierarchy, and a common Gaussian $Bol$ bound on all birth
-inputs. For $L$ sufficiently large (fixed) and $\varepsilon$ sufficiently small:
+inputs. For $L$ sufficiently large (fixed) and $ε$ sufficiently small:
(a) For admissible dependent histories with prescribed labeled roots,
fixed $b$, and complexity $\rho$, the normalized positive integral sums
to at most
-$$C^b\tau^{b/10}\varepsilon^{c_*\rho}|\log\varepsilon|^{C_*\rho}$$
+$$C^b\tau^{b/10}ε^{c_*\rho}|\logε|^{C_*\rho}$$
The sum includes graph enumeration and admissibility restrictions;
bounded observable multipliers cost their supremum. Birth inputs need
only satisfy the common Gaussian $Bol$ envelope.
(b) If the independent input $g$ in slab $i$ has
-$\|g-f_{(i-1)\tau}\|_{Bol,\beta_{i-1}}\le\varepsilon^{\theta_{i-1}}$, then the
-signed tree sum $\mathsf T_i^\varepsilon[g]$ satisfies
+$\|g-f_{(i-1)\tau}\|_{Bol,\beta_{i-1}}\leε^{\theta_{i-1}}$, then the
+signed tree sum $\mathsf T_i^ε[g]$ satisfies
-$$\|\mathsf T_i^\varepsilon[g]-f_{i\tau}\|_{Bol,\beta_i}\le\tfrac12\varepsilon^{\theta_i}$$
+$$\|\mathsf T_i^ε[g]-f_{i\tau}\|_{Bol,\beta_i}\le\tfrac12ε^{\theta_i}$$
In the fixed-top-endpoint norm $Bol,\widetilde\beta_i$, the summed
absolute contributions of cyclic $\mathcal T_i$, of
-$\mathcal T_i^{err}$, and of $\mathcal T_i$ with $\ge\Lambda_i/3$
-lines are $o(\varepsilon^{\theta_i})$. Fixed cutoff powers and a fixed
+$\mathcal T_i^{err}$, and of $\mathcal T_i$ with $\geΛ_i/3$
+lines are $o(ε^{\theta_i})$. Fixed cutoff powers and a fixed
exponential-atom factor can be included, the latter absorbed by large
$L$.
@@ -789,20 +789,20 @@ with target $f_{(i-1)\tau+\sigma}$; at $\sigma=0$ the map is the identity.
Proof (Source and envelope adaptations).
For (a): Deng--Hani--Ma, Proposition 7.2 counts root-labeled unlabeled histories by
-$C^b|\log\varepsilon|^{C_*\rho}$; Deng--Hani--Ma, Proposition 9.7 and Eq. (9.51) retain a
+$C^b|\logε|^{C_*\rho}$; Deng--Hani--Ma, Proposition 9.7 and Eq. (9.51) retain a
positive gain $\propto\rho$; after regular cutting, Deng--Hani--Ma, Eq. (9.53)
bounds each full edge integral by
-$\tau^{b/9}\varepsilon^{-2h+2\upsilon/5+c_*\rho}|\log\varepsilon|^{C_*\rho}$.
-Multiplication by $\mu^{-h}=\varepsilon^{2h}$ cancels the root power; weakening
+$\tau^{b/9}ε^{-2h+2υ/5+c_*\rho}|\logε|^{C_*\rho}$.
+Multiplication by $\mu^{-h}=ε^{2h}$ cancels the root power; weakening
$1/9\to1/10$ and discarding the extra positive power gives
the coarse bound. These are the estimates inside the proof of Deng--Hani--Ma, Proposition 6.2, Section 9.2, not only its displayed conclusion.
The only use of a birth density is the envelope decomposition Deng--Hani--Ma, Eqs. (9.58)--(9.63): from (1) choose a summable
sequence $(m_a)$ bounding the weighted supremum on unit balls; bounded
-overlap and $\sum_am_a<\infty$ dominate $|g|$ by
+overlap and $\sum_am_a<∞$ dominate $|g|$ by
$C\sum_am_a 1 _{|x-a|\le1}e^{-\beta'|v|^2}$, and the positive source
estimates applied to each envelope product sum by
-$\sum_am_a<\infty$. This extends (a) to arbitrary bounded inputs,
+$\sum_am_a<∞$. This extends (a) to arbitrary bounded inputs,
including absolute values of signed inputs. Multipliers in $[0,1]$ are
kept or dropped only when taking a positive upper bound.
@@ -812,15 +812,15 @@ noncausal trees; the causal remainder satisfies the Gaussian $Bol$
estimate Deng--Hani--Ma, Proposition 14.3, Eq. (14.8); comparison to the short-time
Boltzmann expansion, using the kinetic envelope bound, is Deng--Hani--Ma, Proposition 6.1, Section 14. To pass from $f_{(i-1)\tau}$ to $g$, expand each birth factor as
$f_{(i-1)\tau}+(g-f_{(i-1)\tau})$; each residual costs
-$\varepsilon^{\theta_{i-1}}$ by hypothesis, and the multilinear tree bound sums
+$ε^{\theta_{i-1}}$ by hypothesis, and the multilinear tree bound sums
all choices; since $\theta_{i-1}>\theta_i$ the residuals are
-$o(\varepsilon^{\theta_i})$. For the positive subleading terms the bound is
+$o(ε^{\theta_i})$. For the positive subleading terms the bound is
Deng--Hani--Ma, Eq. (14.28) in $Bol,\widetilde\beta_i$ with a time power per atom
-and gain $\varepsilon^{\upsilon/3+c_*\rho}$ in cyclic and extra-encounter
+and gain $ε^{υ/3+c_*\rho}$ in cyclic and extra-encounter
classes; Section 14.3 Points 1--2 keep the Gaussian weight at the fixed root and
the spatial $Bol$ sum, Point 4 regularizes an O-atom adjacent to the root
without changing its integral. The tree-tail branch of Point 5 uses only
-that the atom number is of cutoff order, so $\Lambda_i\to\Lambda_i/3$
+that the atom number is of cutoff order, so $Λ_i\toΛ_i/3$
leaves an exponentially small tail. Graph enumeration, the bounded
selected-edge choices, and fixed exponential factors are absorbed by
taking $L$ large. For the partial slab, restrict contact times to
@@ -834,14 +834,14 @@ $\theta_{i-1}>\theta_i$ gives the bound.
Lemma (Coarse boundary modifications). The positive coarse bound tolerates:
(i) dropping continuation condition (iv) in the top slab with $h$ roots,
-at the cost of a loss $\varepsilon^{-C_*h}$;
+at the cost of a loss $ε^{-C_*h}$;
-(ii) a connected pure-C terminal component with $\le3\Lambda_i$ lines and
+(ii) a connected pure-C terminal component with $\le3Λ_i$ lines and
excess $\le2\Gamma$, meeting a lower continuation but with no designated
root, at cost $\mu$ (add one artificial root);
(iii) supplying initial-link continuations to a fixed number $q$ of
-specified lines, at cost $\varepsilon^{-q\upsilon}$ (add $q$ dummy straight
+specified lines, at cost $ε^{-qυ}$ (add $q$ dummy straight
root lines), provided the root cutoffs have the corresponding slack.
Bounded records carried by these lines are kept.
@@ -853,19 +853,19 @@ into a bounded-$\rho$ case, where the UP algorithm makes $\le h$ free-
component ``bad pieces'' paid by the root deficit, and a large-$\rho$ case
using the dependent cutting argument with the continuation condition of a
strictly lower layer still available; the resulting positive integration
-gives the loss $\varepsilon^{-C_*h}$ of Deng--Hani--Ma, Eq. (9.82). Terminal size is used
+gives the loss $ε^{-C_*h}$ of Deng--Hani--Ma, Eq. (9.82). Terminal size is used
only later and absorbed by the time power.
(ii): take one line of the terminal component as root; its normalization
costs $\mu^{-1}$ and multiplication by $\mu$ restores the integral.
-Splitting into $\le\Lambda_i$ lines and $\le\Gamma$ excess (ordinary
+Splitting into $\leΛ_i$ lines and $\le\Gamma$ excess (ordinary
admissible) versus the enlarged terminal class of Deng--Hani--Ma, Definition 3.20(4), the same root operation is Deng--Hani--Ma, Eq. (15.4); summing over the
choice of line costs a cutoff polynomial.
(iii): fix nonnegative $q_0$ with $\int q_0=1$ and finite Gaussian $Bol$
norm; for every $x$,
$$
-\int q_0(x',v')\ell_\varepsilon(x,x') d x' d v'\ge\varepsilon^\upsilon .
+\int q_0(x',v')\ell_ε(x,x') d x' d v'\geε^υ .
$$
Insert once
per continuation: the dummy activity $\mu$ cancels the extra root factor
@@ -887,10 +887,10 @@ process, not a factorization assumption.
Put
-$$M_i=\lfloor\Lambda_i/2\rfloor$$
+$$M_i=\lfloorΛ_i/2\rfloor$$
the expansion cluster cap. It differs from the dynamics caps
-$\Lambda_i,\Gamma$ and belongs only to the expansion; take $\varepsilon$ small so
+$Λ_i,\Gamma$ and belongs only to the expansion; take $ε$ small so
$M_i\ge2$ for all $i$.
Idea. Run each subset of particles on its own, with the truncated
@@ -903,10 +903,10 @@ Fix a slab (suppress its index). Let $V$ be a finite labelled particle set.
A particle state may carry previously observed test values, or their
running sum, as extra coordinates called records; the acceptance rule uses
only positions, velocities, and the prehistory on this slab. For
-$C\subseteq V$, $\Psi_C$ is the autonomous truncated evolution of $C$
+$C⊆ V$, $\Psi_C$ is the autonomous truncated evolution of $C$
(including its observation updates), and $\kappa_C$ is the indicator that
its accepted collision graph is connected (singletons $\kappa_\bullet=1$;
-$|C|>\Lambda\Rightarrow\kappa_C=0$). For disjoint $C,D$, $a(C,D)$
+$|C|>Λ⇒\kappa_C=0$). For disjoint $C,D$, $a(C,D)$
indicates an incoming cross-contact that would be accepted with
their respective prefix prehistories, and $\chi(C,D)=1-a(C,D)$. These are
indicators, not counts. All identities are on the common full-measure set
@@ -915,12 +915,12 @@ where all finite subsystems are defined.
Lemma (Autonomous partition). For finite $V$,
$$\sum_{\mathcal C\in\mathcal P(V)}\prod_{C\in\mathcal C}\kappa_C
-\prod_{\{C,D\}\subseteq\mathcal C}\chi(C,D)=1$$
+\prod_{\{C,D\}⊆\mathcal C}\chi(C,D)=1$$
-For $S\subseteq V$ and a root test $H$,
+For $S⊆ V$ and a root test $H$,
$$H(\Psi_V|_S)=\sum_{\mathcal C\in\mathcal P(V)}
-\prod_{C\in\mathcal C}\kappa_C\prod_{\{C,D\}\subseteq\mathcal C}\chi(C,D)
+\prod_{C\in\mathcal C}\kappa_C\prod_{\{C,D\}⊆\mathcal C}\chi(C,D)
H(\Psi_{\mathcal C}|_S)$$
Proof.
@@ -935,7 +935,7 @@ $\mathcal C$.
The next lemma is finite algebra; its vertices will be autonomous clusters.
-Lemma (Rooted forest stopping). Let $W$ be finite, $R\subseteq W$ nonempty, $a_e\in\{0,1\}$ for each pair,
+Lemma (Rooted forest stopping). Let $W$ be finite, $R⊆ W$ nonempty, $a_e\in\{0,1\}$ for each pair,
$M\ge2$, and
$$u(B)=\sum_{\substack{G connected\\V(G)=B}}\prod_{e\in E(G)}(-a_e),
@@ -943,10 +943,10 @@ u(\{w\})=1$$
There is a finite signed remainder $\mathcal R$ with
-$$\prod_{e\subseteq W}(1-a_e)
-=\sum_{R\subseteq I\subseteq W}
-\sum_{\substack{\mathcal B\in\mathcal P(I)\ B\cap R\ne\varnothing, |B|<M (B\in\mathcal B)}}
-\prod_{B\in\mathcal B}u(B)\prod_{e\subseteq W\setminus I}(1-a_e)+\mathcal R$$
+$$\prod_{e⊆ W}(1-a_e)
+=\sum_{R⊆ I⊆ W}
+\sum_{\substack{\mathcal B\in\mathcal P(I)\ B\cap R\ne∅, |B|<M (B\in\mathcal B)}}
+\prod_{B\in\mathcal B}u(B)\prod_{e⊆ W∖ I}(1-a_e)+\mathcal R$$
$|\mathcal R|$ is bounded by a sum of rooted forests, each with exactly one
component of size in $[M,2M-2]$ and all other components smaller than $M$,
@@ -970,7 +970,7 @@ $u(B)$ with survivors keeping their compatibility product, giving the main
term; stopped branches give the forest bound. For the tree inequality, fix
a total edge order on $B$, assign each connected graph its minimum spanning
tree $T$, let $Q(T)$ be the edges later than the whole $T$-path; then
-connected graphs on $B$ are exactly $T\subseteq G\subseteq T\cup Q(T)$, so
+connected graphs on $B$ are exactly $T⊆ G⊆ T\cup Q(T)$, so
$u(B)=(-1)^{|B|-1}\sum_T\prod_{e\in T}a_e\prod_{e\in Q(T)}(1-a_e)$; since
the second product lies in $[0,1]$, dropping it proves
the autonomous partition.
@@ -985,20 +985,20 @@ allowed on a specified subset of the $s$ labels). Using symmetric Janossy
measures $p_n$ ($\sum_n p_n(E^n)/n!=1$, $p_n/n!$ the law restricted to
$N=n$),
-$$F_s=\mu^{-s}\sum_{n\ge s}\frac{1}{(n-s)!}(pr_{[s]})_\#p_n$$
+$$F_s=\mu^{-s}\sum_{n\ge s}(1)/((n-s)!)(pr_{[s]})_\#p_n$$
For $S=[s]$, $V=[s+j]$, define the signed root-output kernel
$$\mathcal K_{S,V}(X, d X'_S)
=\sum_{\mathcal C\in\mathcal P(V)}\prod_{C\in\mathcal C}\kappa_C(X)
\sum_{\substack{\mathcal B\in\mathcal P(\mathcal C)\\
-(\bigcup_{C\in B}C)\cap S\ne\varnothing, |B|<M}}
+(∪_{C\in B}C)\cap S\ne∅, |B|<M}}
\prod_{B\in\mathcal B}u(B;X) \delta_{\Psi_{\mathcal C}(X)|_S}( d X'_S)$$
and the retained operator
$$(\mathcal L_sF)( d X'_S)
-=\sum_{j=0}^{s(M-1)\Lambda-s}\frac{\mu^j}{j!}
+=\sum_{j=0}^{s(M-1)Λ-s}(\mu^j)/(j!)
\int F_{s+j}( d X_V) \mathcal K_{S,V}(X, d X'_S)$$
(An empty restriction is zero; at most $s$ retained blocks, so the bound on
@@ -1011,7 +1011,7 @@ $[M,2M-2]$, the rest smaller, no unrooted isolated cluster; keep
$\prod_C\kappa_C$ and the root-output map. Define $\mathcal E_sF$ by
the retained operator with this kernel and $j$ up to
-$$j\le(s+1)(M-1)\Lambda-s$$
+$$j\le(s+1)(M-1)Λ-s$$
Lemma (One-slab identity and variation remainder). For a positive input law with factorial hierarchy $F$, let $F'$ be its
hierarchy after the truncated slab. If factorial measures up to
@@ -1040,16 +1040,16 @@ summation and travel with the root-output map.
For a signed family $\Delta$ (uniformly in records and $s\le A_i/2$),
$$\|\mathcal L_{i,s}\Delta\|_{TV}+\|\mathcal E_{i,s}|\Delta|\|_{TV}
-\le e^{C_*A_i\Lambda_i^2\log(1/\varepsilon)}\max_{q\le A_{i-1}/2}\|\Delta_q\|_{TV}$$
+\le e^{C_*A_iΛ_i^2\log(1/ε)}\max_{q\le A_{i-1}/2}\|\Delta_q\|_{TV}$$
-Indeed relevant $q\le(s+1)\Lambda_i^2/2<A_{i-1}/2$, with at most $q^q$
+Indeed relevant $q\le(s+1)Λ_i^2/2<A_{i-1}/2$, with at most $q^q$
particle and $q^q$ cluster partitions, spanning/forest sum $\le q^{2q}$,
-activity $\le\mu^q$, and at most $A_i\Lambda_i^2$ values of $q$; take logs.
+activity $\le\mu^q$, and at most $A_iΛ_i^2$ values of $q$; take logs.
Define the unmarked one-root measures recursively by
$$g_i=\mathcal U_i(g_{i-1}),
-\mathcal U_i(g):=\mathcal L_{i,1}((g^{\otimes q})_{q\ge0})$$
+\mathcal U_i(g):=\mathcal L_{i,1}((g^{ ⊗ q})_{q\ge0})$$
a finite signed measure using the full retained one-root block. It
approximates the Boltzmann density:
@@ -1057,7 +1057,7 @@ approximates the Boltzmann density:
Proposition (Independent factors). $g_i$ has a Gaussian $Bol$-bounded density representative and, with the
exponents of the analytic package,
-$$\|g_i-f_{i\tau}\|_{Bol,\beta_i}\le\varepsilon^{\theta_i},
+$$\|g_i-f_{i\tau}\|_{Bol,\beta_i}\leε^{\theta_i},
\sup_{0\le i\le L}\|g_i\|_{Bol,\beta_i}\le C$$
with the same bounds for the one-root map run through any partial slab
@@ -1068,12 +1068,12 @@ following; it is where ``retained'' and ``DHM signed trees'' are shown to
differ only by error classes.
Lemma (Modified one-root bridge). The difference between $\mathcal U_i(g)$ and the signed unrestricted
-one-root particle-tree sum with $n<\Lambda_i/3$ is dominated, with inputs
+one-root particle-tree sum with $n<Λ_i/3$ is dominated, with inputs
replaced by $|g|$, by the positive integrals of
(i) retained one-root molecules with cycle excess $r\ge1$;
-(ii) retained one-root molecules with $n\ge\Lambda_i/3$;
+(ii) retained one-root molecules with $n\geΛ_i/3$;
(iii) small unrestricted trees with one extra incoming O-atom inserted
between two flight segments.
@@ -1094,8 +1094,8 @@ partition the encounter event by the first admissible segment pair, giving
disjoint encounter indicators with weights in $[0,1]$. Ordering these along
lines gives particle-line molecules. C-clusters plus the O forest determine
the connected block with sign $(-1)^{\#\mathrm O}$ and global excess equal
-to the sum of C-cluster excesses, so cyclic terms and $n\ge\Lambda_i/3$
-terms are exactly (i)--(ii), inside the DHM caps $\Lambda_i<\infty$,
+to the sum of C-cluster excesses, so cyclic terms and $n\geΛ_i/3$
+terms are exactly (i)--(ii), inside the DHM caps $Λ_i<∞$,
$\Gamma$. A remaining tree molecule with no other encounters has autonomous
C-clusters equal to the displayed ones, every O contact the unique selected
one, and unused factors equal to one; otherwise the first undesignated
@@ -1105,19 +1105,19 @@ cluster-pair contact) inserts an O-atom that changes no velocity, giving
is admitted with the displayed sign, so no symmetry factor is dropped.
Proof of the independent-factor construction.
-the initial ensemble gives $\|g_0-f_0\|_{Bol,\beta_0}\le C\varepsilon$, which is
+the initial ensemble gives $\|g_0-f_0\|_{Bol,\beta_0}\le Cε$, which is
the initial smallness for $\theta_0<1$. Assume the bound at the preceding
interface. the one-root bridge plus the fixed-endpoint part of
the analytic package bounds the three error families and compares the
signed small-tree sum with $f_{i\tau}$, under the preceding
-$Bol,\beta_{i-1}$ error, to $o(\varepsilon^{\theta_i})$, giving the induction and
+$Bol,\beta_{i-1}$ error, to $o(ε^{\theta_i})$, giving the induction and
the uniform $Bol$ bound. The estimate is a fixed-endpoint $Bol$ one, not
an integrated $L^1$ inference, so it also gives density representatives;
restricting contact times gives the uniform partial-slab version.
Fix finitely many observation pulses. Let $m$ bound both the number of
pulses and of record-carrying labels; record coordinates lie in a fixed
-bounded interval. For a root set $S$ and required records on $R\subseteq S$,
+bounded interval. For a root set $S$ and required records on $R⊆ S$,
let $F^R_{i,S}$ be the pasted factorial measure at interface $i$ with record
coordinates on $R$. The operators $\mathcal L_i$ append the slab
observations on the same labels; compatibility is record-independent. The
@@ -1129,7 +1129,7 @@ well defined order by order (finitely many previous orders per output).
Lemma (Resummation with records). There is an exact finite expansion
-$$\widehat F^R_{i,S}=\sum_{R\subseteq H\subseteq S}E^R_{i,H}\otimes g_i^{\otimes(S\setminus H)}$$
+$$\widehat F^R_{i,S}=\sum_{R⊆ H⊆ S}E^R_{i,H} ⊗ g_i^{ ⊗ (S∖ H)}$$
The $E^R_{i,H}$ are represented by layered diagrams: every retained block
in slab $i$ contains a root of the interface above; a block is retained if
@@ -1145,7 +1145,7 @@ into $g_0$ on the complement, marked singletons into $H$ with their
initial record. Insert the preceding expansion into
the retained operator; blocks with one new root and disjoint from the
dependent set $H_-$ have independent $g_{i-1}$ inputs and sum to
-the one-root definition (carrying no record since $R\subseteq H_-$);
+the one-root definition (carrying no record since $R⊆ H_-$);
keep the rest and call their new root set $H$. This gives
the dependent-subset expansion and the rule; record labels stay in $H_-$ at
every step, so their lines continue. Fresh labels on distinct blocks are
@@ -1158,13 +1158,13 @@ the initial updates, so it commutes with the composition.
Regard each nontrivial initial cumulant block as a connection. A block on
$k$ labels is dominated by $\prod f_0$, $(Ck)^{Ck}$, and a root tree with
edge kernel the edge kernel; the sharper kernel
-$ 1 _{|x-x'|\le C_{D,k}\varepsilon}+\varepsilon^D$ holds for bounded $k$. These dominate
+$ 1 _{|x-x'|\le C_{D,k}ε}+ε^D$ holds for bounded $k$. These dominate
the exact blocks; artificial links for marked lines are not connections.
Lemma (Labels and connected components). The retained expansion factors over global components. If $D^R_{i,B}$ is
the sum of diagrams connected on roots $B$, then
-$$\widehat F^R_{i,S}=\sum_{\pi\in\mathcal P(S)}\bigotimes_{B\in\pi}D^{R\cap B}_{i,B}$$
+$$\widehat F^R_{i,S}=\sum_{\pi\in\mathcal P(S)}⊗_{B\in\pi}D^{R\cap B}_{i,B}$$
Hence factorial-cumulant inversion keeps exactly the connected diagrams,
even after resumming into $g_i$. At fixed root order $h=|S|$ and bounded
@@ -1184,7 +1184,7 @@ Expand each $g_i$ to time zero (algebraically only). Weights are label-local;
size and excess caps are componentwise; every global component contains a
designated root (each slab block meets a root above). Distributing $j$ fresh
labels with sizes $j_1,...,j_q$ gives coefficient
-$\frac1{j!}\frac{j!}{j_1!... j_q!}=\prod_a\frac1{j_a!}$, so everything
+$\frac1{j!}(j!)/(j_1!... j_q!)=\prod_a\frac1{j_a!}$, so everything
factors; group components to get the component expansion, and finite
partition inversion gives the cumulant statement. A one-root resummation
contracts a history attached through one line and cannot split or merge
@@ -1200,7 +1200,7 @@ exponent controlled by crossings and initial links.
Lemma (Stopped variation errors). For $L$ large in terms of the fixed data and $\Gamma$, any fixed record
bound, and $|S|\le A_i/2$, there are $c_i>0$ with
-$$\|F^R_{i,S}-\widehat F^R_{i,S}\|_{TV}\le e^{-c_i\Lambda_i}$$
+$$\|F^R_{i,S}-\widehat F^R_{i,S}\|_{TV}\le e^{-c_iΛ_i}$$
uniformly for a partial final slab (no velocity weight claimed).
@@ -1213,26 +1213,26 @@ the resummation with records; refining autonomous histories into C graphs
and picking one incoming contact per incompatibility gives exactly the
positive C/O histories of the coarse package with previous-interface
continuations. The only top-slab difference, a stopped component with
-$M_i\le\cdot<2M_i-2<\Lambda_i$ clusters, is in the same coarse family with
+$M_i\le\cdot<2M_i-2<Λ_i$ clusters, is in the same coarse family with
atom count $\ge M_i-1$. Bookkeeping: the lower independent inputs are
already resummed into $g$; the top stopped forest is not resummed. Top
components failing continuation are paid by the coarse boundary modifications, part (i)
-($\varepsilon^{-C_*|S|}$), and bottom continuation of a record line with no
+($ε^{-C_*|S|}$), and bottom continuation of a record line with no
initial link uses the coarse boundary modifications, part (ii)--(iii) (add a unit-mass Gaussian
-Bol dummy and an initial link $\ge\varepsilon^\upsilon$, cost $\varepsilon^{-\upsilon}$,
+Bol dummy and an initial link $\geε^υ$, cost $ε^{-υ}$,
bounded many). Remaining label and initial-block factors are the cutoff
powers of the label bookkeeping. Choosing $L$ so
-$C\tau^{c_0}<1$ and using $\Lambda_i=A_i^{30}$, $A_i\ge|\log\varepsilon|$ makes the
+$C\tau^{c_0}<1$ and using $Λ_i=A_i^{30}$, $A_i\ge|\logε|$ makes the
top loss absorb into the exponential size gain,
$$\|\mathcal E_i\widehat F_{i-1}\|_{TV}\le
-\varepsilon^{-C_*|S|-C_m}\sum_{b\ge M_i-1}(C\tau^{c_0})^b
-\sum_{\rho\ge0}|\log\varepsilon|^{C_*\rho}\varepsilon^{c_*\rho}\le e^{-c\Lambda_i}$$
+ε^{-C_*|S|-C_m}\sum_{b\ge M_i-1}(C\tau^{c_0})^b
+\sum_{\rho\ge0}|\logε|^{C_*\rho}ε^{c_*\rho}\le e^{-cΛ_i}$$
-Input orders are at most $(|S|+1)\Lambda_i^2/2$, which is below
+Input orders are at most $(|S|+1)Λ_i^2/2$, which is below
$A_{i-1}/2$; the $\Delta_{i-1}$
terms are bounded by the stopped input bound, and
-$A_i\Lambda_i^2\log(1/\varepsilon)=o(\Lambda_{i-1})=\Lambda_i^{30}$ absorbs them.
+$A_iΛ_i^2\log(1/ε)=o(Λ_{i-1})=Λ_i^{30}$ absorbs them.
This is the induction; restricting the last slab preserves every bound.
Theorem (Marked component expansion). Choose $\Gamma$ then $L$ as above, independent of the moment order and the
@@ -1245,13 +1245,13 @@ of the independent-factor construction; label weights, continuations, and record
projections are those of the resummation with records, the label bookkeeping.
(ii) At fixed final order $h$, the actual normalized marked cumulant
-$c_h^\varepsilon$ is the sum of retained globally connected diagrams plus an
-error of total variation $\le C_he^{-c\Lambda_L}$. Nontrivial initial
+$c_h^ε$ is the sum of retained globally connected diagrams plus an
+error of total variation $\le C_he^{-cΛ_L}$. Nontrivial initial
blocks count as connections; dummy links do not.
(iii) For any accuracy power $P$, one can choose $K$ (after the record list
and $h$) so that all retained diagrams with $\rho>K$ have summed absolute
-variation $\le\varepsilon^P$; at $\rho\le K$ the crossing-factorial bound holds.
+variation $\leε^P$; at $\rho\le K$ the crossing-factorial bound holds.
Increasing $K$ or $h$ never changes the exponential base per ordinary atom.
Proof.
@@ -1261,14 +1261,14 @@ the retained one, and every term in the difference of the two finite
partition polynomials carries a factor $\Delta_{L,B}$. The actual
normalized factorial measures have mass $\le1$ (initially by the insertion
bound of the initial ensemble, and conserved since $N$ is preserved), and
-the retained ones have variation $\le1+e^{-c\Lambda_L}$ at fixed orders by
+the retained ones have variation $\le1+e^{-cΛ_L}$ at fixed orders by
the stopped variation bound. For (iii) apply the coarse bound to each
resummed core with the same fixed record and top-root losses; independent
-top singletons are bounded by the independent-factor construction; for small $\varepsilon$,
-$|\log\varepsilon|^{C_*\rho}\varepsilon^{c_*\rho}\le\varepsilon^{c_*\rho/2}$, so after summing
+top singletons are bounded by the independent-factor construction; for small $ε$,
+$|\logε|^{C_*\rho}ε^{c_*\rho}\leε^{c_*\rho/2}$, so after summing
the geometric atom factor the $\rho>K$ part is
-$\le C_{h,m}\varepsilon^{-C_*h-C_m}\varepsilon^{c_*K/2}$; raise $K$ to make it
-$\le\varepsilon^P$. Dummy lines change complexity by a fixed amount and are
+$\le C_{h,m}ε^{-C_*h-C_m}ε^{c_*K/2}$; raise $K$ to make it
+$\leε^P$. Dummy lines change complexity by a fixed amount and are
removed before low-complexity asymptotics. All root-order and observation
dependence sits in cutoff powers or fixed polynomial factors of
the crossing-factorial bound; the exponential atom base depends only on the
@@ -1280,27 +1280,27 @@ version with a connected final collision witness instead of observed
outputs, and without any bound on the cutoff probability.
Corollary (Terminal-witness expansion). In slab $i$ let a terminal witness be a connected true C history on
-$n\le3\Lambda_i$ incoming labels, excess $\le2\Gamma$, stopped at a contact
+$n\le3Λ_i$ incoming labels, excess $\le2\Gamma$, stopped at a contact
in that slab, with arbitrary further nonnegative realizability
restrictions. Its activity-normalized functional
-$$\sum_n\frac{\mu^n}{n!}\int F_{i-1,n}( d Z_n) 1 _{terminal witness}(Z_n)$$
+$$\sum_n(\mu^n)/(n!)\int F_{i-1,n}( d Z_n) 1 _{terminal witness}(Z_n)$$
has the retained layer expansion with independent $g$ inputs and initial
blocks. All lower C/O components attach through particle lines to the
witness. The preceding variation error is smaller than every fixed power of
-$\varepsilon$. After the artificial-top-root normalization $\mu$, the coarse part
+$ε$. After the artificial-top-root normalization $\mu$, the coarse part
with $\rho>K$ is also $<$ any assigned power for large $K$; the remaining
low-complexity witness integrals keep their realizability restrictions.
Proof.
-Since $n\le3\Lambda_i<A_{i-1}/2$, insert the preceding expansion from
+Since $n\le3Λ_i<A_{i-1}/2$, insert the preceding expansion from
the marked component expansion into the terminal-witness functional.
Every lower component has a root leading above and the final roots enter
the connected witness (attachment before discarding any initial link). The
variation error contributes at most
-$\sum_{n\le3\Lambda_i}\mu^n/n! e^{-c\Lambda_{i-1}}\le
-e^{C\Lambda_i\log(1/\varepsilon)-c\Lambda_{i-1}}$, which is super-polynomially
+$\sum_{n\le3Λ_i}\mu^n/n! e^{-cΛ_{i-1}}\le
+e^{CΛ_i\log(1/ε)-cΛ_{i-1}}$, which is super-polynomially
small. Designating an artificial top root places the dependent diagrams in
the terminal class at cost $\mu$; with no dependent continuation the
diagram lives in the terminal slab with independent inputs and complexity
@@ -1315,9 +1315,9 @@ survives at the sharp fluctuation scale $\mu^{1-h}$ for the $h$-th
factorial cumulant. The mechanism is a change of relative
translations: every collision that merges two clusters of particles
adjusts one relative position by the collision geometry, producing a
-factor $\varepsilon^2[(v-v_*)\cdot\omega]_+ d t d \omega$. After accounting
+factor $ε^2[(v-v_*)\cdot\omega]_+ d t d \omega$. After accounting
for the activity $\mu$ per birth line, a connected diagram with $p$
-physical initial links scales like $\mu^{1-h}\varepsilon^p$, which is exactly the
+physical initial links scales like $\mu^{1-h}ε^p$, which is exactly the
size sought; cycles and nontrivial initial correlations are smaller by a
positive power (proved via a null-set/dominated-convergence argument). The
sum over all cluster structures converges because each cluster slab has a
@@ -1333,17 +1333,17 @@ the measure forward under the corresponding coordinate projection, which
commutes with the finite hierarchy before regrouping by
the marked component expansion.
-Let $c_h^\varepsilon$ be the ordered normalized factorial cumulant measure of the
-pasted process, with records on its $h$ designated particles; thus $c_1^\varepsilon$
+Let $c_h^ε$ be the ordered normalized factorial cumulant measure of the
+pasted process, with records on its $h$ designated particles; thus $c_1^ε$
is the one-particle measure and the normalization at order $h$ is
$\mu^{-h}$. Constants $C_K$ may depend on a fixed complexity bound $K$, on
the fixed observations, and on $L$; the exponential base $C^n$ never does.
The independent factors of the independent-factor construction satisfy, for some $b>0$ and
-$B_0<\infty$,
+$B_0<∞$,
-$$\sup_{0<\varepsilon\le\varepsilon_0, i}\|g_i\|_{Bol,b}\le B_0,
-\|g_i-f_{i\tau}\|_{Bol,b}\longrightarrow0$$
+$$\sup_{0<ε\leε_0, i}\|g_i\|_{Bol,b}\le B_0,
+\|g_i-f_{i\tau}\|_{Bol,b}→0$$
The exponent $b$ is decreased a fixed number of times when reserving Gaussian weights;
this does not change the microscopic dynamics.
@@ -1353,9 +1353,9 @@ necessary by an amount depending only on the Boltzmann bounds and on $T$.
This single choice works for every fixed factorial order $h$ and every
fixed bounded observation list. For each fixed $h$,
-$$\|c_h^\varepsilon\|_{TV}\le C_h\mu^{1-h}$$
+$$\|c_h^ε\|_{TV}\le C_h\mu^{1-h}$$
-For $h=1,2$, the measures $\mu^{h-1}c_h^\varepsilon$ converge against bounded
+For $h=1,2$, the measures $\mu^{h-1}c_h^ε$ converge against bounded
continuous record tests to the signed sums of retained global trees with
trivial initial cumulants, with contact weight
$[(v-v_*)\cdot\omega]_+ d t d \omega$ (scattering $+1$,
@@ -1378,7 +1378,7 @@ Work with a term of complexity $\le K$. Replace each initial cumulant by
its bound from the initial ensemble,
$$C_K\sum_{\mathcal L forest}\prod_{(p,q)\in\mathcal L}
-( 1 _{|x_p(0)-x_q(0)|\le R_{K,D}\varepsilon}+\varepsilon^D)
+( 1 _{|x_p(0)-x_q(0)|\le R_{K,D}ε}+ε^D)
\prod_pf_0(z_p(0))$$
only for absolute bounds (not to sign an initial link). Expand the
@@ -1418,10 +1418,10 @@ Lemma (Chronological merger coordinates). Fix a finite line graph, the C/O choic
which contacts are redundant. Away from nontransversal and simultaneous
events, its history integral has coordinates: the birth velocities, one
absolute translation per global component, a vector in a ball of radius
-$R_{K,D}\varepsilon$ per physical initial merger, and $(t,\omega)\in R\times
+$R_{K,D}ε$ per physical initial merger, and $(t,\omega)\in R\times
S^2$ per dynamic merger. The positional Jacobian per dynamic merger is
-$$\varepsilon^2[(v-v_*)\cdot\omega]_+ d t d \omega$$
+$$ε^2[(v-v_*)\cdot\omega]_+ d t d \omega$$
A specified redundant contact has no free continuous parameter. The
coordinate map has multiplicity at most one on each incoming transversal
@@ -1436,20 +1436,20 @@ the forest toward one representative per group has determinant one.
Proceed in increasing contact time. Within each group the histories are
known up to a common translation. A merger of groups with translations
$X,X_*$ on the indicated straight segments satisfies
-$X-X_*=-d(t)-\varepsilon\omega$ with $d(t)$ the known relative displacement,
+$X-X_*=-d(t)-ε\omega$ with $d(t)$ the known relative displacement,
$\dot d=v-v_*$, and $\omega$ opposite the outward normal, so the incoming
condition is $(v-v_*)\cdot\omega>0$. The determinant of
$(t,\omega)\mapsto X-X_*$ on the sphere is
-$\varepsilon^2|(v-v_*)\cdot\omega|$, and the incoming convention selects the sign
+$ε^2|(v-v_*)\cdot\omega|$, and the incoming convention selects the sign
in the coordinate formula. A scattering reflection or O-continuation
updates velocities; the two groups now share one translation.
If the next specified contact is redundant, its relative displacement and
velocities are already fixed. A straight relative flight $r(t)=d+tu$ has
at most one incoming crossing, and
-$\int_I\delta(|r|-\varepsilon)[-u\cdot r/\varepsilon]_+ d t= 1 _{\{incoming
+$\int_I\delta(|r|-ε)[-u\cdot r/ε]_+ d t= 1 _{\{incoming
crossing in I\}}$, since the flux cancels the absolute derivative of
-$|r|-\varepsilon$; the outgoing deltas then make a deterministic substitution.
+$|r|-ε$; the outgoing deltas then make a deterministic substitution.
If the crossing is absent the branch has weight zero. Segments starting
inside the sphere have an exit, not an incoming crossing.
@@ -1467,16 +1467,16 @@ subsets of transversal branches and exhaust.
Multiplying the activity bookkeeping factor by the merger Jacobians and the
physical-link volumes,
-$$\mu^{m-h}\varepsilon^{2(m-c-p)}\varepsilon^{3p}=\mu^{c-h}\varepsilon^{p}$$
+$$\mu^{m-h}ε^{2(m-c-p)}ε^{3p}=\mu^{c-h}ε^{p}$$
All acceptance indicators can be carried through the change of variables;
for an upper bound they may be dropped, except that a redundant scattering
determines its branch velocities when it occurs (extend by zero where it
does not). For a connected diagram $c=1$, and a physical initial block has
-$p\ge1$ so gains a factor $\varepsilon$. A term of the sharp scaling
+$p\ge1$ so gains a factor $ε$. A term of the sharp scaling
using a tail factor: erase that link; erasing $\le K$ links raises $c$ by
-$\le K$ and costs $\le\varepsilon^{-2K}$ in the activity counting; choosing
-$D>2K+M$ makes such terms $O(\varepsilon^M\mu^{1-h})$, with $M$ arbitrary and
+$\le K$ and costs $\leε^{-2K}$ in the activity counting; choosing
+$D>2K+M$ makes such terms $O(ε^M\mu^{1-h})$, with $M$ arbitrary and
chosen after the moment order.
Lemma (Fresh-label factorials). For complexity $\le K$, the absolute sum with slab sizes $(n_1,...,n_L)$
@@ -1492,7 +1492,7 @@ Proof.
In slab $i$, $s_i$ labels arrive as roots from above (the final roots in
the top slab) and $j_i=n_i-s_i$ are fresh, integrated with coefficient
$1/j_i!$. Since $s_i\le K$,
-$\frac1{j_i!}=\frac{n_i!}{(n_i-s_i)!}\frac1{n_i!}\le
+$\frac1{j_i!}=(n_i!)/((n_i-s_i)!)\frac1{n_i!}\le
\frac{(1+n_i)^{s_i}}{n_i!}$. Continued labels, dependent subsets, and
initial-link labels are boundedly many, costing a fixed power of $1+n$.
The rooted compatibility expansion represents connected coefficients by
@@ -1593,17 +1593,17 @@ merger parameters:
observation times and are transversal;
(ii) before the first redundant contact, that contact's requirement is
-false for all small $\varepsilon$;
+false for all small $ε$;
(iii) on a full tree, any contact not in the prescribed list (including one
preceding the prescribed joining of its two groups) is absent for all
-small $\varepsilon$.
+small $ε$.
For (ii) the parameters are those of the forest prefix; the rest may be
appended. In (iii) an outgoing crossing after an O-contact is not counted.
Proof.
-Set $\varepsilon=0$. Segment velocities are polynomial in birth velocities and
+Set $ε=0$. Segment velocities are polynomial in birth velocities and
sphere coordinates, and displacements polynomial also in the times; the
parameter set is a connected real-analytic manifold, so a non-identically-
zero polynomial has a null zero set (grazing/ineffective reflections
@@ -1635,15 +1635,15 @@ Zero relative velocities and grazing contacts are null; at a merger the
incoming relative velocity is a nonzero linear function of independent
birth velocities and the normal is a free sphere parameter. Time
coincidences are null. A third line at a prescribed vertex has a nonneighbor
-path, killed above. Finally, formal tree velocities are $\varepsilon$-independent:
-$x_p^\varepsilon(t)=x_p^0(t)+\varepsilon d_p$ with $d_p$ a fixed sum of merger normals,
+path, killed above. Finally, formal tree velocities are $ε$-independent:
+$x_p^ε(t)=x_p^0(t)+ε d_p$ with $d_p$ a fixed sum of merger normals,
so convergence is uniform on each active interval. Away from small
neighborhoods of prescribed vertices every pair of segments has positive
separation or is the treated separating flight; near a vertex a different
pair needs an excluded zero-offset coincidence, and the incident pair has
no other velocity change in a small neighborhood, so it has only its one
incoming crossing. The same compactness handles interfaces and observation
-times, proving (ii)--(iii) including contacts whose times vary with $\varepsilon$.
+times, proving (ii)--(iii) including contacts whose times vary with $ε$.
For a diagram with a cycle, keep its first redundant-contact indicator. All
prior contacts form a forest, and the tested relative trajectory is
@@ -1659,8 +1659,8 @@ the Stirling bound. This order needs no size-uniform rate and proves
the asserted $o$ for cycles and nontrivial initial cumulants.
For a fixed tree all prescribed C-clusters and O-blocks fall below the caps
-once $\varepsilon$ is small; by the no-extra-contact lemma almost every merger parameter
-set has only prescribed incoming contacts for all small $\varepsilon$. Thus every
+once $ε$ is small; by the no-extra-contact lemma almost every merger parameter
+set has only prescribed incoming contacts for all small $ε$. Thus every
acceptance restriction holds, counting an encounter or its existence is the
same, and the connected compatibility coefficient equals its tree sign
$(-1)^{\#\mathrm O}$. Replacing $g_i$ by $f_{i\tau}$ in a fixed-size term is
@@ -1755,10 +1755,10 @@ Gaussian velocity growth.
Proof.
Temporarily expand the top-slab independent factors by their one-root
definitions, keep lower inputs canonical, and bound each limiting tree's
-line count by $N$; for small $\varepsilon$ every cap $\Lambda_i,M_i$ exceeds $N$,
+line count by $N$; for small $ε$ every cap $Λ_i,M_i$ exceeds $N$,
so the already-proved convergence gives each such tree its original
factorial weight. Do finite algebra on this limiting class (not a closure
-claim for finite-$\varepsilon$ capped classes). Deleting a tree edge makes two
+claim for finite-$ε$ capped classes). Deleting a tree edge makes two
components, each holding one incident line (now designated); a component
losing its old top root keeps the incident line or another original root,
so the rooted-block rule still holds, and lower attachments are put in
@@ -1770,7 +1770,7 @@ to $1/(j-1)!$, and
$\binom{j_1+j_2}{j_1}\frac1{(j_1+j_2)!}=\frac1{j_1!j_2!}$ splits the
remaining labels; a contact between two designated particles uses one
unordered pair, no fresh label; the deleted contact contributes its C/O
-sign, other signs multiply. Letting $N\to\infty$ passes through deletion
+sign, other signs multiply. Letting $N\to∞$ passes through deletion
and joining by the uniform summability bound, the spatial-trace construction, the uniform
conditional Gaussian bound controlling the joining flux with a fixed
exponent loss; records project because record forgetting commutes.
@@ -1787,19 +1787,19 @@ $|g_i(x,v)|e^{b|v|^2/4}$ (and $f_0$ likewise), which has a common
$Bol,3b/4$ bound. The positive coarse bound the analytic package(a)
needs only this envelope; apply it to the tilted inputs with the same
initial-link and record bounds. The geometric complexity sum makes the
-weighted retained tail smaller than any power of $\varepsilon$ after choosing $K$.
+weighted retained tail smaller than any power of $ε$ after choosing $K$.
The Gaussian fraction reserved is fixed, so its atom base is
independent of $J,h,K$.
Assembling the scale. Use the marked component expansion at accuracy
-$\varepsilon^M$, $M>2(h-1)$, with $K$ large enough for that accuracy and for
+$ε^M$, $M>2(h-1)$, with $K$ large enough for that accuracy and for
the tree-class completeness. The expansion error and the high-complexity
tail are negligible after multiplication by $\mu^{h-1}$ (they need only the
unweighted variation bound, since observation tests are bounded).
For the remaining connected diagrams, the activity bookkeeping and
the uniform summability bound prove the uniform bound at scale $\mu^{1-h}$.
-Physical initial links gain $\varepsilon$; nonphysical initial tails gain any
+Physical initial links gain $ε$; nonphysical initial tails gain any
power after choosing $D$; cycles are negligible by the no-extra-contact lemma plus
finite-size dominated convergence and the uniform tails; the survivors are
exactly the leading trees by the tree-class completeness. Their fluxes,
@@ -1823,20 +1823,20 @@ deferred to the next stage; exact centering is restored at the end.
Give every particle a scalar mark, initially $0$, and at time $t_j$ update
-$$(x,v,y)\longmapsto\Theta_j(x,v,y)=(x,v,y+\phi_j(x,v))$$
+$$(x,v,y)⟼Θ_j(x,v,y)=(x,v,y+\phi_j(x,v))$$
Collisions preserve the mark of each particle. Write
-$M_{obs}=\sum_j\|\phi_j\|_\infty$, $\xi=(x,v,y)$, $|y|\le
+$M_{obs}=\sum_j\|\phi_j\|_∞$, $\xi=(x,v,y)$, $|y|\le
M_{obs}$. The mark only records deterministic trajectories; it adds
no randomness. On either dynamics,
$$\sum_{i=1}^Ny_i(T)=\sum_j\sum_{i=1}^N\phi_j(z_i(t_j))$$
-Use a superscript $\varepsilon$ for the normalized factorial cumulants of the
+Use a superscript $ε$ for the normalized factorial cumulants of the
pasted marked process. At any observation-free time the sharp connected bounds
provides limits
-$$c_1^\varepsilon\longrightarrow F_t, \mu c_2^\varepsilon\longrightarrow J_t$$
+$$c_1^ε→ F_t, \mu c_2^ε→ J_t$$
against bounded continuous tests of the records; $F_t$ is a one-root
measure and $J_t$ a symmetric signed two-root measure, both with the
@@ -1845,7 +1845,7 @@ at time zero, the initial ensemble gives
$$F_0=f_0(x,v) d x d v \delta_0( d y), J_0=0$$
-(since $\mu\|c_{2,0}^\varepsilon\|_1=O(\varepsilon)$). Forgetting records in the finite
+(since $\mu\|c_{2,0}^ε\|_1=O(ε)$). Forgetting records in the finite
marked hierarchy commutes with the one-root regroupings; by
the marked component expansion, the independent-factor construction, the tree-class completeness, the
$(x,v)$ marginal of $F_t$ is $f_t(x,v) d x d v$ (also inside a slab, via
@@ -1882,16 +1882,16 @@ integral, with $ d x d \omega$.
Proposition (Evolution of the marked tree laws). Between updates,
-$$\partial_tF_t(\Phi) =F_t(D\Phi)+\tfrac12\int BF_tF_{t,*}\Delta\Phi$$
+$$∂_tF_t(\Phi) =F_t(D\Phi)+\tfrac12\int BF_tF_{t,*}\Delta\Phi$$
-$$\partial_tJ_t(\Phi\otimes\Psi) =J_t((D+H_t)\Phi\otimes\Psi+\Phi\otimes(D+H_t)\Psi)$$
+$$∂_tJ_t(\Phi ⊗ \Psi) =J_t((D+H_t)\Phi ⊗ \Psi+\Phi ⊗ (D+H_t)\Psi)$$
$$ +\int BF_tF_{t,*}(\Phi(\xi')\Psi(\xi_*')-\Phi(\xi)\Psi(\xi_*))$$
These hold in transport mild form, hence in weak differential form for
smooth tests for which the integrals exist. At an update,
-$$F_{t_j+}=(\Theta_j)_\#F_{t_j-}, J_{t_j+}=(\Theta_j\otimes\Theta_j)_\#J_{t_j-}$$
+$$F_{t_j+}=(Θ_j)_\#F_{t_j-}, J_{t_j+}=(Θ_j ⊗ Θ_j)_\#J_{t_j-}$$
Proof.
Derive the identities from the tree integrals after the limit. Start with
@@ -1918,8 +1918,8 @@ the last integral of the pushforward identity with no factor $1/2$.
(c) A contact between the first root and a new partner: the second original root
lies with the first (contribution $J$ on it, $F$ on the partner, change
$\Phi(\xi')-\Phi(\xi)$) or with the partner (interchange; the $J$ component
-carries $\Phi(\xi_*')-\Phi(\xi_*)$), summing to $J(H_t\Phi\otimes\Psi)$,
-and the same at the second root gives $J(\Phi\otimes H_t\Psi)$. Records
+carries $\Phi(\xi_*')-\Phi(\xi_*)$), summing to $J(H_t\Phi ⊗ \Psi)$,
+and the same at the second root gives $J(\Phi ⊗ H_t\Psi)$. Records
project when a partner has no record in the integrand. Free flight gives
$D$; a transport-mild version avoids differentiating a sum (freely
transport the test to the last-contact time $s$; no-contact term plus
@@ -1943,11 +1943,11 @@ has no extra $1/2$.
Restore the same-particle term by defining
-$$G_t(\Phi,\Psi)=J_t(\Phi\otimes\Psi)+F_t(\Phi\Psi)$$
+$$G_t(\Phi,\Psi)=J_t(\Phi ⊗ \Psi)+F_t(\Phi\Psi)$$
Then the tree-law evolution gives
-$$\partial_tG_t(\Phi,\Psi)=G_t((D+H_t)\Phi,\Psi)+G_t(\Phi,(D+H_t)\Psi)
+$$∂_tG_t(\Phi,\Psi)=G_t((D+H_t)\Phi,\Psi)+G_t(\Phi,(D+H_t)\Psi)
+\tfrac12\int BF_tF_{t,*}\Delta\Phi \Delta\Psi$$
Direct check of the source: with $P=\Phi+\Phi_*$, $R=\Psi+\Psi_*$, the
@@ -1987,7 +1987,7 @@ by local uniform approximation with a small Gaussian margin; the unmarked
collision operator is $K_t$, so the linearized operator estimate applies. Iterating
the bilinear mild identity chooses one of two slots per collision factor;
an $n$-fold remainder is bounded by
-$\frac{(CT)^n}{n!}(1+\sqrt{n/\delta})^n$ for fixed allowance
+$((CT)^n)/(n!)(1+√(n/\delta))^n$ for fixed allowance
$\delta=b-a>0$ (the $2^n$ slot choices absorbed in $C^n$), which vanishes.
Source iterations obey the same bound, justifying dual propagation by the
convergent mild series with no spatial/velocity differentiability of
@@ -2013,21 +2013,21 @@ identifies all multi-time and repeated-observation correlations.
Proposition (Gaussian limit on the pasted dynamics). For the single pasted dynamics fixed above and every fixed observation list,
$$
-\frac{\sum_iy_i(T)-E\sum_iy_i(T)}{\sqrt\mu} \Longrightarrow\
+(\sum_iy_i(T)-E\sum_iy_i(T))/(\sqrt\mu) ⇒\
\sum_j\zeta_{t_j}(\phi_j),
$$
and all centered moments converge to the Gaussian ones.
Proof.
-Put $S^\varepsilon=\sum_iy_i(T)$. Since $|S^\varepsilon|\le M_{obs}N$ and
-$\mathcal Z_\varepsilon\le e^\mu$, $S^\varepsilon$ has exponential moments for each fixed
-$\varepsilon$. The exact factorial-to-ordinary identity at the final marked time is
+Put $S^ε=\sum_iy_i(T)$. Since $|S^ε|\le M_{obs}N$ and
+$\mathcal Z_ε\le e^\mu$, $S^ε$ has exponential moments for each fixed
+$ε$. The exact factorial-to-ordinary identity at the final marked time is
-$$\kappa_k(S^\varepsilon)=\sum_{\pi\in\mathcal P([k])}\mu^{|\pi|}
-\int c_{|\pi|}^\varepsilon( d \xi_1... d \xi_{|\pi|})\prod_{B\in\pi}y_B^{|B|}$$
+$$\kappa_k(S^ε)=\sum_{\pi\in\mathcal P([k])}\mu^{|\pi|}
+\int c_{|\pi|}^ε( d \xi_1... d \xi_{|\pi|})\prod_{B\in\pi}y_B^{|B|}$$
Directly: write
-$\exp(\lambda S^\varepsilon)=\prod_i[1+(\exp(\lambda y_i)-1)]$, insert the factorial
+$\exp(\lambda S^ε)=\prod_i[1+(\exp(\lambda y_i)-1)]$, insert the factorial
series, and take the logarithm; each $\exp(\lambda y)-1$ must receive a
nonempty set of the $k$ derivatives, and distributing those sets gives
exactly the partitions of the factorial-to-ordinary identity, whose ordering
@@ -2037,40 +2037,40 @@ degree $k$.
All $y$-powers are bounded on the mark interval (extend to bounded
continuous functions on the mark line). By the sharp connected bounds every
term on the right is $O_k(\mu)$. For
-$X^\varepsilon=(S^\varepsilon-E S^\varepsilon)/\sqrt\mu$, the first cumulant vanishes and
+$X^ε=(S^ε-E S^ε)/\sqrt\mu$, the first cumulant vanishes and
$$
-\kappa_2(X^\varepsilon)=c_1^\varepsilon(y^2)+\mu c_2^\varepsilon(y\otimes y)\longrightarrow
-F_T(y^2)+J_T(y\otimes y)=G_T(y,y),
+\kappa_2(X^ε)=c_1^ε(y^2)+\mu c_2^ε(y ⊗ y)→
+F_T(y^2)+J_T(y ⊗ y)=G_T(y,y),
$$
-while for $k>2$, $\kappa_k(X^\varepsilon)=O_k(\mu^{1-k/2})\to0$. That these hold on
+while for $k>2$, $\kappa_k(X^ε)=O_k(\mu^{1-k/2})\to0$. That these hold on
one dynamics is essential: the bases fixing $\Gamma,L$ in
the marked component expansion, the sharp connected bounds are independent of the fixed
order; enlarging the auxiliary complexity cutoff or $D$ only changes the
-error representation, not the random variable $X^\varepsilon$.
+error representation, not the random variable $X^ε$.
The finite moment-cumulant identities give every centered Gaussian moment
with variance $\sigma^2=G_T(y,y)$. For scalar convergence, truncate the
-characteristic functions of $X^\varepsilon$ and of a centered Gaussian $Z$ at
+characteristic functions of $X^ε$ and of a centered Gaussian $Z$ at
degree $2n-1$; Taylor's integral remainder is bounded by
$|ux|^{2n}/(2n)!$, so
$$
-\limsup_{\varepsilon\to0}|E e^{iuX^\varepsilon}-E e^{iuZ}|
+limsup_{ε\to0}|E e^{iuX^ε}-E e^{iuZ}|
\le\frac{2|u|^{2n}E Z^{2n}}{(2n)!}
=\frac{2|u|^{2n}\sigma^{2n}}{2^nn!},
$$
-and $n\to\infty$ proves convergence for every real $u$, including
+and $n\to∞$ proves convergence for every real $u$, including
$\sigma=0$. This gives weak convergence with variance
the variance formula.
The pasted dynamics recolors every restarted slab, resets the collision
prehistory at each interface, and truncates components above the caps
-$\Lambda_i,\Gamma$. The true hard-sphere flow does none of this. It must be
+$Λ_i,\Gamma$. The true hard-sphere flow does none of this. It must be
shown that the two trajectories coincide up to time $T$ with probability
-$1-O(\varepsilon^{1+\eta})$, $\eta>0$, better than $O(\varepsilon)$. The reason:
-the fluctuation scale is $\sqrt\mu=\varepsilon^{-1}$, and the mean of the pasted
+$1-O(ε^{1+\eta})$, $\eta>0$, better than $O(ε)$. The reason:
+the fluctuation scale is $\sqrt\mu=ε^{-1}$, and the mean of the pasted
process differs from the exact microscopic mean by a deterministic
-$O(\varepsilon)$ amount; an error of probability $O(\varepsilon)$ alone would compete
-with that shift after multiplication by $\varepsilon^{-1}$, and exact centering
+$O(ε)$ amount; an error of probability $O(ε)$ alone would compete
+with that shift after multiplication by $ε^{-1}$, and exact centering
would fail. The sharper bound is proved.
Proposition (Accuracy of the pasted dynamics). There is a large dimensional integer $\Gamma_0$ with the following
@@ -2079,11 +2079,11 @@ $L_0=L_0(\Gamma,f_0,f,T,\beta)$ such that for every fixed $L\ge L_0$, the
true and pasted dynamics of the marked component expansion, coupled from the
same initial configuration, satisfy
-$$\mathbf{P}\{the two trajectories differ at some time in [0,T]\}\le
-C\varepsilon^{1+\eta}$$
+$$P\{the two trajectories differ at some time in [0,T]\}\le
+Cε^{1+\eta}$$
-for all small $\varepsilon$, with $\eta=1/20$. The size cutoffs are the
-$A_i,\Lambda_i$ of the analytic package; $C$ and the upper bound on $\varepsilon$
+for all small $ε$, with $\eta=1/20$. The size cutoffs are the
+$A_i,Λ_i$ of the analytic package; $C$ and the upper bound on $ε$
may depend on the displayed fixed data.
The rest of this argument proves it. The plan:
@@ -2093,12 +2093,12 @@ The rest of this argument proves it. The plan:
slab, of bounded excess, meeting a fresh label and stopped by a rejection.
2. Separate coarse from fine. Coarse witnesses (complexity $>K$, or
-terminal component larger than $\Lambda_i$) sum to less than any power of
-$\varepsilon$ by the Deng--Hani--Ma package---this fixes $K$ large.
+terminal component larger than $Λ_i$) sum to less than any power of
+$ε$ by the Deng--Hani--Ma package---this fixes $K$ large.
3. On the low-complexity fine regime, localize energy, then combine a
local gain at every good cutting piece with a global energy
-and causal time-order bound, gaining enough powers of $\varepsilon$ to beat
+and causal time-order bound, gaining enough powers of $ε$ to beat
$1+\eta$.
Distinguish the geometric cycle excess $r(M)$ of a graph from its
@@ -2127,13 +2127,13 @@ first disagreement, which therefore is a contact accepted by the true rule
but rejected by the pasted one (or vice versa). On agreement so far the
pasted state is the true state; the collision component in the true history
just before that contact has excess $\le2\Gamma$ (it was accepted), and the
-rejection is exactly ``component size $>\Lambda_i$ or excess added $>\Gamma$''
+rejection is exactly ``component size $>Λ_i$ or excess added $>\Gamma$''
within this block. Summing over possible blocks, each such rejection is
charged to a distinct terminal witness; summing charges gives the bound.
Insert the terminal-witness functional
and apply the terminal-witness expansion. Begin with the previous pasted
-factorial measure at interface $i-1$. Since $n\le3\Lambda_i<A_{i-1}/2$, the
+factorial measure at interface $i-1$. Since $n\le3Λ_i<A_{i-1}/2$, the
variation error is below every power. The connected terminal witness is
itself a block at the top of the pasted hierarchy and enters through its
positive layer expansion. Its lower dependent and rooted components attach
@@ -2150,22 +2150,22 @@ including continued empty lines).
Choose a complexity threshold $K$ after $L$ is fixed. By
the terminal-witness expansion, the analytic package, the sum of positive majorants
with $\rho>K$, including the activity cost of an artificial top root when
-required, is $O(\varepsilon^3)$ for $K$ large enough. The terms left to estimate
+required, is $O(ε^3)$ for $K$ large enough. The terms left to estimate
have $\rho\le K$. Their interface crossings, initial-link vertices,
within-slab cycles, and total geometric excess are bounded. Discard the
initial-link restrictions in these positive majorants; the trajectory
attachments to the terminal component persist.
Henceforth all graphs are connected low-complexity witnesses,
-summarized by a polylogarithmic exponent bound $B_\varepsilon$ on the total number
+summarized by a polylogarithmic exponent bound $B_ε$ on the total number
of atoms in the top-slab terminal component, with
-$B_\varepsilon\le C_*|\log\varepsilon|$ ($C_*$ may depend on $K,L$). A connected witness
+$B_ε\le C_*|\logε|$ ($C_*$ may depend on $K,L$). A connected witness
of low complexity has its top-slab component of bounded excess, and a
connected graph with $b$ atoms and excess $\le2\Gamma$ can be produced
without new crossings and within the same realization family by
$\le C^b$ choices of the initial labelled $M$ and its geometric data.
-The number of scattered velocities in any witness is at most $B_\varepsilon$, and
+The number of scattered velocities in any witness is at most $B_ε$, and
initial data carry only $\le2\beta$ Gaussian under the kinetic assumption; a
terminal witness has controllable total birth energy with Gaussian
cost $e^{-cP}$ for total energy $P$.
@@ -2174,16 +2174,16 @@ Lemma (Energy tail). Fix $\Gamma$ and then choose $L$ once, larger when needed.
with $n$ true input lines carry total birth energy $E_0=\frac12\sum_j|v_j|^2$,
put
-$$P_\varepsilon=(C_*B_\varepsilon)^{1/2}(1+B_\varepsilon|\log\varepsilon|^{1/2}),
-\gamma_\varepsilon=\varepsilon^{B_\varepsilon^2}$$
+$$P_ε=(C_*B_ε)^{1/2}(1+B_ε|\logε|^{1/2}),
+\gamma_ε=ε^{B_ε^2}$$
-For a witness with $n\le B_\varepsilon$ terminal lines, once energy is
-$>P_\varepsilon$ the
+For a witness with $n\le B_ε$ terminal lines, once energy is
+$>P_ε$ the
-$$integral\le C^{B_\varepsilon}e^{-cB_\varepsilon P_\varepsilon}\le\varepsilon^{cB_\varepsilon^2}$$
+$$integral\le C^{B_ε}e^{-cB_ε P_ε}\leε^{cB_ε^2}$$
-smaller than every fixed power. On the low-energy regime $E_0\le P_\varepsilon$
-the terminal terminals are localized to a ball of radius $R=C_*(1+B_\varepsilon P_\varepsilon^{1/2})$.
+smaller than every fixed power. On the low-energy regime $E_0\le P_ε$
+the terminal terminals are localized to a ball of radius $R=C_*(1+B_ε P_ε^{1/2})$.
Proof.
The forward bound below holds on the positive original measure, so it uses
@@ -2195,13 +2195,13 @@ an indicator; births at later interfaces are inserted at their times.
Dropping encounter restrictions and initial links, a graph with $n$ births
has mass at most
$$
-\mu^n\int\prod_{j=1}^nh_j(x_j,v_j) 1 _{\{\sum_j|v_j|^2>P_\varepsilon\}}
-\prod_j d x_j d v_j\le\mu^nC^ne^{-cP_\varepsilon},
+\mu^n\int\prod_{j=1}^nh_j(x_j,v_j) 1 _{\{\sum_j|v_j|^2>P_ε\}}
+\prod_j d x_j d v_j\le\mu^nC^ne^{-cP_ε},
$$
using half of the uniform birth Gaussian for the tail and the number of
-births $\le B_\varepsilon$; summing a finite count and $\mu^n=\varepsilon^{-2n}$ gives
-$e^{2B_\varepsilon|\log\varepsilon|-cP_\varepsilon}$ times a prefactor; choosing $P_\varepsilon$ a
-fixed large multiple of $B_\varepsilon|\log\varepsilon|$ makes this below any fixed
+births $\le B_ε$; summing a finite count and $\mu^n=ε^{-2n}$ gives
+$e^{2B_ε|\logε|-cP_ε}$ times a prefactor; choosing $P_ε$ a
+fixed large multiple of $B_ε|\logε|$ makes this below any fixed
power. With no exponential growth $M_{top}\to1$.
Lemma (Causal order and relative speeds). In a slab with $q_i$ ordinary atoms, retain the partial order induced by the
@@ -2218,21 +2218,21 @@ An antichain of ordinary vertices occupies disjoint incoming pairs on a
common causal slice: place all strict ancestors first in a total extension;
no two vertices share a line; energy conservation bounds the sum of their
pair energies by $P$, including interfaces via inactive coordinates. For
-$a_v=|v_v-v_{*v}|$, every antichain has $\sum_{A}a_v\le\sqrt{2q_iP}$.
+$a_v=|v_v-v_{*v}|$, every antichain has $\sum_{A}a_v\le√(2q_iP)$.
Summing the products of weights over the recursive enumeration of linear
extensions (available minimal vertices form an antichain) gives
$e_i\prod_va_v\le(2q_iP)^{q_i/2}$ for every feasible assignment, hence the
supremum.
Lemma (Free-parameter extraction). Consider a finite ordered composition of positive cut kernels. At an
-ordinary step the free variables are $\vartheta_j=(t_j,\omega_j,w_j)$ with
+ordinary step the free variables are $θ_j=(t_j,\omega_j,w_j)$ with
measure $a_j d t_j d \omega_j d w_j$, followed by a deterministic update.
Suppose every exceptional kernel $K_\ell(h, d z)$ has mass at most
$H_\ell$, uniformly in earlier-cut data (every restriction needed for that
bound included, zero outside the admissibility domain). Remove the $a_j$
from the ordinary measures to get a positive measure $\nu$. If $E$ is a set
of complete histories and $\mathcal D$ a measurable set containing its
-projection onto $\vartheta$, then
+projection onto $θ$, then
$$\nu(E)\le(\prod_\ell H_\ell)
\int_{\mathcal D}\prod_{j=1}^q d t_j d \omega_j d w_j$$
@@ -2242,13 +2242,13 @@ one.
Proof.
First put all ordinary variables in bounded boxes and fix their vector
-$\vartheta$. The exceptional kernels stay in place with $\vartheta$
+$θ$. The exceptional kernels stay in place with $θ$
exterior; integrating them backwards from the last exceptional step bounds
the surviving mass by $\prod_\ell H_\ell$, and intervening deterministic
substitutions have norm at most one on bounded functions. Nonnegative
-Fubini represents $\nu$ as the product in $\vartheta$ times these
+Fubini represents $\nu$ as the product in $θ$ times these
kernels, and majorizing the indicator of $E$ by
-$ 1 _{\mathcal D}(\vartheta)$ proves the extraction inequality.
+$ 1 _{\mathcal D}(θ)$ proves the extraction inequality.
Exhaustion removes the boxes. Equivalently, restrict ordinary relative
speeds away from zero and bound a rectangle by its product volume; cover a
compact containing set by products of rectangles and take the infimum;
@@ -2269,7 +2269,7 @@ bond; it is type A if one atom cuts as a type $\{3\}$ singleton and the
other becomes an allowed $\{2\}$ piece, type B otherwise. A $\{44\}$ piece
has two degree-four atoms joined by one bond
(Deng--Hani--Ma, Definitions 8.2, 8.11). The kernels compose
-$I_M=I_A\circ I_{M\setminus A}$ with repeated boundary variables identified
+$I_M=I_A\circ I_{M∖ A}$ with repeated boundary variables identified
(Deng--Hani--Ma, Proposition 8.14); this is how a later kernel is estimated
with earlier variables fixed. The uniform bound on the number of collisions in
semi-dispersing billiards behind these counting estimates is due to Burago,
@@ -2278,34 +2278,34 @@ Ferleger and Kononenko; Deng--Hani--Ma adapt it to the extended dynamics.
Lemma (Local kernels). Let $R$ bound velocities and localized radii; the time length is $\le T$.
For a singleton with two fixed ends in a free kernel:
-(i) If all incident free edges have relative speeds $\ge\varepsilon^\upsilon$, the
-mass is $\le C R^C\varepsilon^{2+\upsilon/3}$.
+(i) If all incident free edges have relative speeds $\geε^υ$, the
+mass is $\le C R^Cε^{2+υ/3}$.
(ii) If it is a scattering contact and the strong restrictions
-($|s-s_*|$, $|x-x_*|$ and exterior $x_*$ below $\varepsilon^{1-\upsilon}$ on both
-edges) hold, the mass is $\le\varepsilon^{2+\upsilon/3}$. The bound is uniform in
+($|s-s_*|$, $|x-x_*|$ and exterior $x_*$ below $ε^{1-υ}$ on both
+edges) hold, the mass is $\leε^{2+υ/3}$. The bound is uniform in
the exterior parameters $t_*,x_*$.
-(iii) For a type-B $\{33\}$ piece, both atoms C, with $|t_1-t_2|\ge\varepsilon^\upsilon$
-and every incident pair having $|v_e-v_{e'}|\ge\varepsilon^\upsilon$, the mass is
-$\le\varepsilon^\upsilon|\log\varepsilon|^{C_*}$ uniformly in the two fixed ends.
+(iii) For a type-B $\{33\}$ piece, both atoms C, with $|t_1-t_2|\geε^υ$
+and every incident pair having $|v_e-v_{e'}|\geε^υ$, the mass is
+$\leε^υ|\logε|^{C_*}$ uniformly in the two fixed ends.
Multiplying any such kernel by a bounded factor on its support costs its
supremum; no independent Gaussian distribution of a fixed-end velocity is
assumed. This is (Deng--Hani--Ma, Lemmas 8.15, 8.20, 8.23). A
uniformly good singleton or type-B pair is one of these restricted
-kernels with conditional mass $\varepsilon^\upsilon|\log\varepsilon|^{C_*}$; all
+kernels with conditional mass $ε^υ|\logε|^{C_*}$; all
gain-producing restrictions are part of the kernel.
Proposition (Pure-collision cutting alternatives). Let $M$ be a connected full realizable graph with only scattering contacts.
A transversal set $A\subset M$ satisfies
-$M\setminus A=A^-\sqcup A^+$ with no directed bond from $A^+$ to
+$M∖ A=A^-\sqcup A^+$ with no directed bond from $A^+$ to
$A\cup A^-$, nor from $A$ to $A^-$. Let $X_0(A)$ be vertices outside $A$
with two bonds to $A$, and $X(A)$ its closure obtained by repeatedly adding
vertices with two bonds to the grown set. Then:
(i) There is a chain of connected transversal sets $A_1,...,A_m=M$, $A_1$ a
-tree, $X_0(A_j)\ne\varnothing$ for $j<m$,
+tree, $X_0(A_j)\ne∅$ for $j<m$,
$A_j\cup X(A_j)\subset A_{j+1}$,
$r(A_{j+1})=r(A_j\cup X(A_j))$, $r(A_{j+1})-r(A_j)\le10|X(A_j)|$. The
intermediate growth adds one bond per new vertex. For $D$ transitions there
@@ -2328,7 +2328,7 @@ $\{44\}$, $\le S=a+z+y$ type $\{4\}$ singletons, all type-B pairs uniformly
good, and $\ge(m_\pm-10^5S)/10$ type-A or type-B pairs.
Neutral pieces keep their neutral kernels; partitions and operation
-sequences have $\le C^b|\log\varepsilon|^{C_*}$ subcases with $C$ independent of
+sequences have $\le C^b|\logε|^{C_*}$ subcases with $C$ independent of
$L$. These are (Deng--Hani--Ma, Definition 15.3; Propositions 15.1, 15.2, 15.4, 15.6, 15.10).
Lemma (Adjacent scattering contacts). Let $R\ge2$ bound velocities and localized radii, and length $\le T$. A pair
@@ -2337,23 +2337,23 @@ directed chain between them; a double bond between two consecutive
scattering contacts is impossible. For $e_1,e_2$ the prospective fixed
edges of a type-A pair with contact times $t_1,t_2$ and
$$
-strongly close: |t_2-t_1|\le\varepsilon^{9/10},
-|v_{e_2}-v_{e_1}|\le\varepsilon^{4/5},
+strongly close: |t_2-t_1|\leε^{9/10},
+|v_{e_2}-v_{e_1}|\leε^{4/5},
$$
(i) a primitive strongly close pair cut freely as one piece has mass at most
-$$CR^C\mu\varepsilon^{9/10+3\cdot4/5}=CR^C\varepsilon^{13/10};$$
+$$CR^C\muε^{9/10+3\cdot4/5}=CR^Cε^{13/10};$$
(ii) without any strongly close primitive pair, each type-A pair used by a
-sequence has conditional mass $\le CR^C\varepsilon^{1/20}$, or splits into a
+sequence has conditional mass $\le CR^Cε^{1/20}$, or splits into a
neutral type-two piece and a singleton with the same bound.
The bounds are uniform in previously exposed lines and times. This is
(Deng--Hani--Ma, Lemma 15.12).
-Fix $\delta=\tfrac12\min\{\upsilon,1/20\}>0$; every retained local gain is
-at least $\varepsilon^\delta$ up to a fixed polylogarithmic factor.
+Fix $\delta=\tfrac12\min\{υ,1/20\}>0$; every retained local gain is
+at least $ε^\delta$ up to a fixed polylogarithmic factor.
Lemma (Amplified pure-collision cut). For large dimensional $\Gamma_0$, every connected realizable full scattering
graph with $\Gamma<r(M)\le2\Gamma$, $\Gamma\ge\Gamma_0$, decomposes into cut
@@ -2363,7 +2363,7 @@ improved pieces $g$ and free singletons $s_4$ satisfy
$$\delta g-2s_4\ge6/5$$
-There are $\le C^b|\log\varepsilon|^{C_*}$ subcases; the number of non-ordinary
+There are $\le C^b|\logε|^{C_*}$ subcases; the number of non-ordinary
pieces is bounded in terms of $\Gamma$ and fixed choices. This is
(Deng--Hani--Ma, Proposition 15.11).
@@ -2377,14 +2377,14 @@ $s_2+p_{33}=r(M)+s_4+p_{44}-1$ on the complete graph (or its version with
boundedly many crossing ends fixed) bounds the extra type-two pieces. With
$b$ the total atoms and $q$ the ordinary ones, $q=b-O_K(1)$.
-Lemma (Total birth energy bounds the velocity domain). Fix a cutoff $E_{cut}=P_\varepsilon$ and, as in
+Lemma (Total birth energy bounds the velocity domain). Fix a cutoff $E_{cut}=P_ε$ and, as in
the energy tail, use the local kernels at the scopes where
the cutting alternatives and the amplified cut retain them. Then
for a witness with ordinary projection within the containing set $\mathcal D$
-(energy $\le P_\varepsilon$), there is a compact measurable set inside
+(energy $\le P_ε$), there is a compact measurable set inside
$\mathcal D$, of volume
-$$\le\prod_i(C\frac{L}{q_i})^{q_i}(1+P_\varepsilon/q_i)^{3q_i/2}$$
+$$\le\prod_i(C(L)/(q_i))^{q_i}(1+P_ε/q_i)^{3q_i/2}$$
that contains every admissible ordinary velocity vector.
@@ -2403,18 +2403,18 @@ $$W=A(\omega)u+d(\omega,a), \|A(\omega)\|_{HS}^2\le3q$$
Rows need not be orthogonal; only the sum of squared norms is used, and the
maps do not depend on contact times. Take a Euclidean net of $a$ of mesh
-$c_s(1+q)^{-1/2}$, cardinality $\le C_s\{(1+P_\varepsilon)(1+q)\}^{C_s}$; a change
+$c_s(1+q)^{-1/2}$, cardinality $\le C_s\{(1+P_ε)(1+q)\}^{C_s}$; a change
$\zeta$ in $a$ changes one chronological velocity by $\le\sqrt s\zeta$ and
-the observations by $\le C\sqrt{qs}\zeta$. Choose $c_s$ so simultaneous
-error is $\le1$. The allowed $W$ lie in $\bigcup_{a_0}
-(d(\omega,a_0)+A(\omega)B_{3n}(\sqrt{P_\varepsilon})+B_{3q}(1))$, and the Minkowski
-sum is in $\sqrt2(P_\varepsilon AA^{\mathsf T}+Id_m)^{1/2}B_m(1)$ for $m=3q$
-(support function: $\sqrt{P_\varepsilon}|A^{\mathsf T}\xi|+|\xi|\le
-\sqrt{2(P_\varepsilon|A^{\mathsf T}\xi|^2+|\xi|^2)}$). Combining the trace bound
+the observations by $\le C√(qs)\zeta$. Choose $c_s$ so simultaneous
+error is $\le1$. The allowed $W$ lie in $∪_{a_0}
+(d(\omega,a_0)+A(\omega)B_{3n}(√(P_ε))+B_{3q}(1))$, and the Minkowski
+sum is in $\sqrt2(P_ε AA^{\mathsf T}+Id_m)^{1/2}B_m(1)$ for $m=3q$
+(support function: $√(P_ε)|A^{\mathsf T}\xi|+|\xi|\le
+\sqrt{2(P_ε|A^{\mathsf T}\xi|^2+|\xi|^2)}$). Combining the trace bound
with AM-GM on eigenvalues and $vol(B_m(1))\le(C/\sqrt m)^m$,
$$
-2^{m/2}vol(B_m(1))\det(Id_m+P_\varepsilon AA^{\mathsf T})^{1/2}
-\le(C\sqrt{(1+P_\varepsilon)/q})^{3q}.
+2^{m/2}vol(B_m(1))\det(Id_m+P_ε AA^{\mathsf T})^{1/2}
+\le(C√((1+P_ε)/q))^{3q}.
$$
The identity enlargement handles small singular values without an inverse
Jacobian; a finite union of continuous images of compact parameter sets
@@ -2422,17 +2422,17 @@ gives the compact containing set for the extraction lemma.
Lemma (Summation with finitely many exceptional pieces). Suppose a cut sequence has $q=b-O_K(1)$ ordinary atoms and boundedly many
exceptional pieces whose conditional mass bounds have product
-$\varepsilon^\alpha|\log\varepsilon|^{C_*}$. After summing spatial cells and energy bins in
+$ε^\alpha|\logε|^{C_*}$. After summing spatial cells and energy bins in
a polylogarithmic energy region, its integral is at most
-$$\varepsilon^\alpha|\log\varepsilon|^{C_*}(C\tau\sqrt L)^q$$
+$$ε^\alpha|\logε|^{C_*}(C\tau\sqrt L)^q$$
The base $C$ depends on the fixed kinetic envelopes and dimension, not on
$K,L$. This is (Deng--Hani--Ma, Lemma 16.2).
Proof of the pasted accuracy bound.
Start from the first-disagreement bound, the terminal-witness expansion, and fix the
-complexity cutoff $K$ large enough that the coarse part is $<\varepsilon^3$
+complexity cutoff $K$ large enough that the coarse part is $<ε^3$
(the marked component expansion, part (iii) after the terminal-witness expansion):
$\rho>K$ is discarded. The remaining witnesses are connected, full, realizable
graphs with $\rho\le K$, hence bounded crossing/initial/cycle counts, and
@@ -2441,20 +2441,20 @@ polylogarithmic energy region the sum is smaller than any power.
On the low-energy region, apply the cutting alternatives, the amplified cut
to the top-slab graph. The freely cut primitive pair (if any) gives
-the gain balance, product $\varepsilon^{13/10}|\log\varepsilon|^{C_*}$. Otherwise
+the gain balance, product $ε^{13/10}|\logε|^{C_*}$. Otherwise
the summation bound holds: the retained improved pieces number $g$
-with a fixed bound except for the bounded seed, and gain $\varepsilon^{\delta g}$,
-while each free type-four seed costs $\mu=\varepsilon^{-2}$; so the exceptional
-product is at most $\varepsilon^{6/5}|\log\varepsilon|^{C_*}$ (using
-$\delta g-2s_4\ge6/5$ in both the $\varepsilon^{13/10}$ and the $\varepsilon^{6/5}$
+with a fixed bound except for the bounded seed, and gain $ε^{\delta g}$,
+while each free type-four seed costs $\mu=ε^{-2}$; so the exceptional
+product is at most $ε^{6/5}|\logε|^{C_*}$ (using
+$\delta g-2s_4\ge6/5$ in both the $ε^{13/10}$ and the $ε^{6/5}$
branches). All the extra variables inside the containing set are then
removed by the coarea and time-order bounds: the causal-order bound
-gives $\prod_i\tau^{q_i}(2q_iP_\varepsilon)^{q_i/2}/q_i!$, and the ordinary
+gives $\prod_i\tau^{q_i}(2q_iP_ε)^{q_i/2}/q_i!$, and the ordinary
velocity box by the birth-energy bound. Combining
the exceptional-piece summation, the extraction lemma, the low-complexity sum is
bounded by
$$
-\varepsilon^{6/5}|\log\varepsilon|^{C_*}(C\tau\sqrt L)^{b-O_K(1)} .
+ε^{6/5}|\logε|^{C_*}(C\tau\sqrt L)^{b-O_K(1)} .
$$
Choose $L$ so large that $C\tau\sqrt L=C T/\sqrt L<1/2$, absorbing
constants depending on $K,L$ into the fixed prefactor. The exponential atom
@@ -2462,31 +2462,31 @@ counting ($C_1^b$ with $C_1$ independent of $K,L$, per
the marked component expansion, part (iii) and the terminal-witness expansion) together
with the bounded crossing labels, selected pairs, and initial coefficients
costs only a fixed-degree polynomial; the resulting sum over $b$ is a
-convergent geometric series, with the $\Lambda_i$ cutoff removing the large-$b$
+convergent geometric series, with the $Λ_i$ cutoff removing the large-$b$
part. Thus the total sum is at most
-$$C\varepsilon^{6/5}|\log\varepsilon|^{C_*}$$
+$$Cε^{6/5}|\logε|^{C_*}$$
-Since $\tfrac{6}{5}=1.2>1+\tfrac{1}{20}=1.05$, this is
-$\le C\varepsilon^{1+1/20}$, proving the discrepancy bound. (The witnesses
-with terminal component larger than $\Lambda_i$ are smaller than any power
+Since $(6)/(5)=1.2>1+(1)/(20)=1.05$, this is
+$\le Cε^{1+1/20}$, proving the discrepancy bound. (The witnesses
+with terminal component larger than $Λ_i$ are smaller than any power
by the energy tail, the analytic package, via the activity cost
-$\mu=\varepsilon^{-2}$ versus the Gaussian gain over $\Lambda_i\ge|\log\varepsilon|^{30}$
+$\mu=ε^{-2}$ versus the Gaussian gain over $Λ_i\ge|\logε|^{30}$
births, summing geometrically as in the coarse step.)
Remark (Order of choices). The local exponent $\delta$ and the finite hierarchy of
the amplified cut fix $\Gamma_0$ from dimensional constants. Then
choose $\Gamma$, then $L$ for the imported estimates and the common
exponential base; only afterward choose $K$, the localization powers, and the
-small upper bound on $\varepsilon$. No observation number or moment order enters the
+small upper bound on $ε$. No observation number or moment order enters the
first two choices. This is what allows a single pasted dynamics to be fixed in
the Gaussian limit of the model dynamics, below.
The pasted process of the pasted Gaussian limit is centered by its own
microscopic mean. The real gas is centered by a slightly different mean,
since the pasted and true laws charge the discrepancy event (probability
-$\le C\varepsilon^{1+\eta}$, the pasted accuracy bound) with different mass. At the
-fluctuation scale $\sqrt\mu=\varepsilon^{-1}$ this difference of means must go to
+$\le Cε^{1+\eta}$, the pasted accuracy bound) with different mass. At the
+fluctuation scale $\sqrt\mu=ε^{-1}$ this difference of means must go to
zero, and that is exactly the sharp estimate of the pasted accuracy bound. This is made
quantitative, then the fluctuation limit theorem is assembled.
@@ -2494,30 +2494,30 @@ Lemma (Discrepancy and centering). Couple the true and pasted dynamics from the
For every fixed time $t$ and bounded continuous test $\phi$, with
$\|N/\mu\|_{L^2}\le C$,
-$$\sqrt\mu |E_{true}\pi_t^\varepsilon(\phi)
--E_{pasted}\pi_t^\varepsilon(\phi)|
-\le C_\phi \varepsilon^{(\eta-1)/2}\longrightarrow0$$
+$$\sqrt\mu |E_{true}\pi_t^ε(\phi)
+-E_{pasted}\pi_t^ε(\phi)|
+\le C_\phi ε^{(\eta-1)/2}→0$$
and moreover
-$$\zeta_t^{\varepsilon,true}(\phi)
-=\zeta_t^{\varepsilon,pasted}(\phi)+o_{L^1}(1)$$
+$$\zeta_t^{ε,true}(\phi)
+=\zeta_t^{ε,pasted}(\phi)+o_{L^1}(1)$$
Proof.
Let $A=\{the two trajectories agree on [0,T]\}$, so
-$\mathbf{P}(A^c)\le C\varepsilon^{1+\eta}$, and both $N$ and $N/\mu$ have bounded second
+$P(A^c)\le Cε^{1+\eta}$, and both $N$ and $N/\mu$ have bounded second
moments (the initial ensemble). On $A$ the empirical measures agree for every
-$t$, hence so do $\pi_t^\varepsilon(\phi)$ and the running records. Write
-$\pi=(1/\mu)\sum_i\phi$. Off $A$, $|\pi|\le\|\phi\|_\infty N/\mu$ on either
+$t$, hence so do $\pi_t^ε(\phi)$ and the running records. Write
+$\pi=(1/\mu)\sum_i\phi$. Off $A$, $|\pi|\le\|\phi\|_∞ N/\mu$ on either
side, so $|\pi^{true}-\pi^{pasted}|
-\le2\|\phi\|_\infty (N/\mu) 1 _{A^c}$. By Cauchy--Schwarz,
+\le2\|\phi\|_∞ (N/\mu) 1 _{A^c}$. By Cauchy--Schwarz,
$$
|E_{true}\pi-E_{pasted}\pi|
-\le2\|\phi\|_\infty\|N/\mu\|_{L^2} \mathbf{P}(A^c)^{1/2}
-\le C_\phi \varepsilon^{(1+\eta)/2},
+\le2\|\phi\|_∞\|N/\mu\|_{L^2} P(A^c)^{1/2}
+\le C_\phi ε^{(1+\eta)/2},
$$
-and multiplying by $\sqrt\mu=\varepsilon^{-1}$ gives
-$\varepsilon^{(\eta-1)/2}\to0$, proving the mean comparison.
+and multiplying by $\sqrt\mu=ε^{-1}$ gives
+$ε^{(\eta-1)/2}\to0$, proving the mean comparison.
For the centered-difference estimate, insert the centered definitions:
$\zeta^{true}-\zeta^{pasted}
@@ -2527,11 +2527,11 @@ The second bracket is bounded by the mean comparison. For the first,
using the $A^c$ indicator,
$$
E\sqrt\mu |\pi^{true}-\pi^{pasted}|
-\le2\|\phi\|_\infty\sqrt\mu E[(N/\mu) 1 _{A^c}]
-\le2\|\phi\|_\infty\|N/\mu\|_{L^2}\sqrt\mu \mathbf{P}(A^c)^{1/2},
+\le2\|\phi\|_∞\sqrt\mu E[(N/\mu) 1 _{A^c}]
+\le2\|\phi\|_∞\|N/\mu\|_{L^2}\sqrt\mu P(A^c)^{1/2},
$$
-and $\sqrt\mu \varepsilon^{(1+\eta)/2}=\varepsilon^{-1}\varepsilon^{(1+\eta)/2}
-=\varepsilon^{(\eta-1)/2}\to0$. So the $L^1$ norm of the difference of the
+and $\sqrt\mu ε^{(1+\eta)/2}=ε^{-1}ε^{(1+\eta)/2}
+=ε^{(\eta-1)/2}\to0$. So the $L^1$ norm of the difference of the
centered fields tends to zero.
Proof of the fluctuation limit theorem.
@@ -2544,29 +2544,29 @@ $L^2$-continuous fields there.
Initial fluctuation limit. the initial ensemble, part (iii) gives, for any
fixed list of tests, joint convergence of
-$(\zeta_0^\varepsilon(\phi_1),...,\zeta_0^\varepsilon(\phi_m))$ to the centered Gaussian
+$(\zeta_0^ε(\phi_1),...,\zeta_0^ε(\phi_m))$ to the centered Gaussian
vector with covariance $\int f_0\phi_i\phi_j$.
Dynamical limit on the pasted dynamics. Fix a single pasted
dynamics as in the Gaussian limit of the model dynamics, below ($\Gamma,L$ independent of the
observation list and order). the pasted Gaussian limit gives, for every fixed
list of times and tests, convergence of the centered combination
-$\varepsilon\sum_i y_i(T)$ to $\sum_j\zeta_{t_j}(\phi_j)$, with convergence of all
+$ε\sum_i y_i(T)$ to $\sum_j\zeta_{t_j}(\phi_j)$, with convergence of all
centered moments. By the factorial-to-ordinary identity and
the sharp connected bounds, this is the finite-dimensional Gaussian limit
with the multi-time covariance the target covariance.
Transfer to the true flow. By the discrepancy and centering bound,
-$\zeta_t^{\varepsilon,true}(\phi)-\zeta_t^{\varepsilon,pasted}(\phi)
+$\zeta_t^{ε,true}(\phi)-\zeta_t^{ε,pasted}(\phi)
=o_{L^1}(1)$, while
-$|\zeta_t^\varepsilon(\phi)|\le\|\phi\|_\infty(\varepsilon N+\varepsilon E N)$ has uniformly
-bounded moments because $E(N/\mu)^p<\infty$ (the initial ensemble). Hence the
+$|\zeta_t^ε(\phi)|\le\|\phi\|_∞(ε N+ε E N)$ has uniformly
+bounded moments because $E(N/\mu)^p<∞$ (the initial ensemble). Hence the
$p$-moments of the difference vanish for every fixed $p$ by interpolation of
the $L^1$ bound with the uniform $L^{p+1}$ bound, and, for any bounded
continuous functional of a fixed finite collection
-$(\zeta_{t_1}^\varepsilon(\phi_1),...,\zeta_{t_m}^\varepsilon(\phi_m))$,
+$(\zeta_{t_1}^ε(\phi_1),...,\zeta_{t_m}^ε(\phi_m))$,
$$
-E_{true}F-E_{pasted}F\longrightarrow0
+E_{true}F-E_{pasted}F→0
$$
(local uniform approximation of a continuous $F$ on a compact by a smooth
one, then the moment control of the difference off the compact). Therefore