diff f6bf760..6fbd3ac low_temp_sk_glauber_typical_v2.txt
diff --git a/low_temp_sk_glauber_typical_v2.txt b/low_temp_sk_glauber_typical_v2.txt
index dc4bb14..45430e7 100644
--- a/low_temp_sk_glauber_typical_v2.txt
+++ b/low_temp_sk_glauber_typical_v2.txt
@@ -114,7 +114,7 @@ let
T^J_{n,beta}(sigma)
=infBigl{tge0:
bigl|P_t^{J,beta}(sigma,·)-pi^J_{n,beta}bigr|_{TV}
- letfrac14Bigr}
+ le1/4Bigr}
be the mixing time from that start. The joint law of interest samples
the disorder and one Gibbs configuration,
@@ -191,13 +191,13 @@ to its section.
querying the matrix W=W(G) along orthogonalized directions. Each query
is conditionally Gaussian given the past and constructs a fresh
coordinate of a driving Brownian motion. The proposal Q_l has two
- controlled features: sublinear relative entropy {rm KL}(Q_lVert P),
+ controlled features: sublinear relative entropy {KL}(Q_lVert P),
and a likelihood bound outside an exponentially unlikely event. See
Section~sec:proposal.
Reversing the entropy: coverage. Forward entropy alone says
nothing about reaching a prescribed set. Conditioning the Gibbs joint law
on the spin makes the remaining disorder law strongly log-concave; a
- small Gaussian smoothing plus the quadratic transport inequality then
+ small Gaussian smoothing plus the quadratic transport ineuality then
reverses the entropy comparison. The outcome is a coverage statement:
any Gibbs set of mass at least n^{-d} receives at least
exp(-n^{1-2d}) proposal mass for the realized target disorder and
@@ -253,7 +253,7 @@ orthogonalize, reveal. The reverse-entropy step uses triangular transport
under strong convexity cite[Corollary~3.10]{BKM2005} together with the
convolution comparison recorded in cite[Lemma~11]{LugosiNeu2022}. The
mixing step combines the coarea proof of the reversible Cheeger
-inequality cite[Theorem~3.5]{LawlerSokal1988} with entropy dissipation
+ineuality cite[Theorem~3.5]{LawlerSokal1988} with entropy dissipation
for finite Markov chains cite[Lemmas~2.5,~2.7]{DiaconisSaloffCoste1996}.
%======================================================================
@@ -266,7 +266,7 @@ logarithms are natural.
Distances and inner products.: For probability measures
mullnu on a common space, set
-{rm KL}(muVertnu)=intlog(dmu/dnu),dmu. On the spin cube,
+{KL}(muVertnu)=intlog(dmu/dnu),dmu. On the spin cube,
total variation is one half of the ell^1 distance. For vectors
v,winR^n write
[
@@ -284,8 +284,8 @@ diagonal entries are independent N(0,2/n). Write G for the vector of
independent standard Gaussian drivers of all these entries and
gamma for its law. Since x_i^2=1 for every spin,
[
- tfrac12,x^{T},x
- =tfrac12sum_i W_{ii}
+ 1/2,x^{T},x
+ =1/2sum_i W_{ii}
]
is the same random constant for every configuration. Hence the diagonal
contributes the same additive constant to every Gibbs weight; it changes
@@ -295,15 +295,15 @@ we absorb into the pressure functional P below.
Gibbs weights on the completed matrix.: Define, with W=W(G),
[
- widehatwidehat H_G(x)=H_G(x)=tfrac12 x^{T},x,
+ Hhat_G(x)=H_G(x)=1/2 x^{T},x,
- F(G)=logsum_{xin{-1,1}^n}e^{betawidehatwidehat H_G(x)}H_G(x),
+ F(G)=logsum_{xin{-1,1}^n}e^{betaHhat_G(x)}H_G(x),
- pi_G(x)=e^{betawidehatwidehat H_G(x)-F(G)}H_G(x)-F(G).
+ pi_G(x)=e^{betaHhat_G(x)-F(G)}H_G(x)-F(G).
]
Because the diagonal is a configuration-independent constant,
pi_G=pi^J_{n,beta} as probability measures. We work with
-widehatwidehat H_GH_G everywhere. The Gibbs joint law of disorder and spin is
+Hhat_GH_G everywhere. The Gibbs joint law of disorder and spin is
[
P(dG,x)=gamma(dG),pi_G(dx).
]
@@ -361,7 +361,7 @@ Use the scalar variance time v=beta^2 s with s the Parisi time
variable, and put alpha_v=mu_beta([0,v/beta^2]) for
0le vlebeta^2. Let Phi solve the backward Parisi equation
- Phi_v=-tfrac12(Phi_{yy}+alpha_vPhi_y^2),
+ Phi_v=-1/2(Phi_{yy}+alpha_vPhi_y^2),
Phi(beta^2,y)=log(2cosh y),
@@ -532,7 +532,7 @@ r(v_*)=1/beta^2 and monotonicity of r imply rhole1.
Let H(z) be the entropy of a sign with mean zin(-1,1):
[
- H(z)=-(1+z)/(2)calH(z)=-(1+z)/(2)log(1+z)/(2)
+ H(z)=-(1+z)/(2)H(z)=-(1+z)/(2)log(1+z)/(2)
-(1-z)/(2)log(1-z)/(2),
]
with continuous values at z=pm1. For V>v_* define the two scalars
@@ -542,7 +542,7 @@ matrix proposal:
e(V)=int_0^V√(r(v)),p(v),rho(v),dv,
j(V)=frac12int_0^Vrho(v)^2,
- E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv .
+ E(a_v^2)/(1-m_v^2)E(a_v^2)/(1-m_v^2),dv .
The factor rho(v)^2 inside j accounts for the independent noise we
introduce in the proposal. Below the support endpoint, It^o's formula
@@ -558,20 +558,20 @@ spatial derivatives. Substituting the endpoint value (eq:phi-at-vstar)
and using H(tanh y)=log(2cosh y)-ytanh y gives
int_0^{v_*}p(v),dv+E,H(m_{v_*})
- =P-(calP-(beta^2)/(4)(1-q_*)^2 .
+ =P-(P-(beta^2)/(4)(1-q_*)^2 .
[Entropy--energy identity]
For every V>v_*,
- log2-j(V)+beta e(V)=P-(calP-(beta^2)/(4),p(V)^2 .
+ log2-j(V)+beta e(V)=P-(P-(beta^2)/(4),p(V)^2 .
Proof.
The same martingale calculation applied to binary entropy yields
[
- frac12int_0^{v_*}E(a_v^2)/(1-m_v^2)bbE(a_v^2)/(1-m_v^2),dv
+ frac12int_0^{v_*}E(a_v^2)/(1-m_v^2)E(a_v^2)/(1-m_v^2),dv
=log2-E,H(m_{v_*}).
]
Localization justifies It^o's formula near z=pm1; the integrand is
@@ -706,7 +706,7 @@ cut. Both are stated for the joint law Q_l on (G,x)-space.
[Proposal comparison]
For every fixed 0<lle1/100 and all sufficiently large n,
- {rm KL}(Q_lVert P)le2n^{1-l/2} .
+ {KL}(Q_lVert P)le2n^{1-l/2} .
Moreover, there is an event in the proposal experiment of probability at
least 1-exp(-n^{1-10l}) such that the output subprobability
@@ -748,16 +748,16 @@ quadratic-form tail
2exp(-c_beta neta^2e^{-4V}).
]
A 1/4-net of the unit sphere in dimension O(K) has exp(O(K))
-points, by ball packing. The quadratic-form norm inequality and a union
+points, by ball packing. The quadratic-form norm ineuality and a union
bound therefore put the whole empirical matrix within eta of its
expectation, with exceptional probability at most exp(-n^{1-8l}).
The population identities are
- E[T_jT_k]_bbE[T_jT_k]_{rm row}=mathbf1_bf1_{j=k},
- E[T_jg_k]_bbE[T_jg_k]_{rm row}=0 (jle k),
- E[T_kx]_bbE[T_kx]_{rm row}=s_k,
- E[g_kx]_bbE[g_kx]_{rm row}=b_k,
+ E[T_jT_k]_bbE[T_jT_k]_{row}=mathbf1_bf1_{j=k},
+ E[T_jg_k]_bbE[T_jg_k]_{row}=0 (jle k),
+ E[T_kx]_bbE[T_kx]_{row}=s_k,
+ E[g_kx]_bbE[g_kx]_{row}=b_k,
where
b_k=h^{-1/2}int_{t_k}^{t_{k+1}}p(v)rho(v),dv. These follow from the
@@ -805,10 +805,10 @@ directions are orthonormal in |·|_n.
Energy of the final spin.: Let H_0=Pi_{>K-1}. Revealing the
columns in order decomposes the quadratic form exactly as
[
- frac{widehatwidehat H_G(x)}H_G(x)n
+ frac{Hhat_G(x)}H_G(x)n
=sum_kc_k< x,Pi_{>k}g_k>_n
- +tfrac12sum_kc_k^2d_k
- +tfrac12< x,H_0WH_0x>_n .
+ +1/2sum_kc_k^2d_k
+ +1/2< x,H_0WH_0x>_n .
]
The last two terms are O(eta) outside an event of probability at most
CKexp(-cneta^2): the diagonal estimate follows from the conditional
@@ -821,7 +821,7 @@ remaining value is s· b up to the same error. The bound on
interval contributes O(h), and e(V) is bounded. Hence s· b=e(V)+O_beta(h),
and altogether
- widehatwidehat H_G(x)/n=e(V)+O_H_G(x)/n=e(V)+O_beta(h+etalog(2K)).
+ Hhat_G(x)/n=e(V)+O_H_G(x)/n=e(V)+O_beta(h+etalog(2K)).
This is the energy contribution to the comparison. We next isolate the
entropy paid for the matrix information alone.
@@ -870,7 +870,7 @@ Interval by interval, every query direction is determined by past D and
by information in G_0. Each new endpoint and its entire bridge
are known before using the current D-increment. Deterministic-coefficient
bridge integrals are linear Gaussian integrals with the endpoint term
-fixed by that past. Thus F_v⊆calF_v⊆G_v. One can
+fixed by that past. Thus F_v⊆F_v⊆G_v. One can
also see this pathwise: rho=1 before v_*, and (eq:1) gives
-rholerho'le0 afterward, so rho has bounded variation and the
identity intrho,dB=rho B-int B,drho, applied on successive
@@ -891,7 +891,7 @@ experiment relative to the auxiliary reference is E_B, whose logarithm
is
[
L=log E_B=sum_i(int_0^Vxi_irho,dB_i
- -tfrac12int_0^Vxi_i^2rho^2,dv).
+ -1/2int_0^Vxi_i^2rho^2,dv).
]
Under the linked law, compensating the B_i-drift by rhoxi_i leaves
a centered integral with bracket at most 4nV, so it has
@@ -917,10 +917,10 @@ independent disorder and uniform spin. The new reference has
(G,x)-marginal P, and the log density of the linked experiment
relative to it is
[
- L-nlog2+F(G)-betawidehatwidehat H_G(x).H_G(x).
+ L-nlog2+F(G)-betaHhat_G(x).H_G(x).
]
Differentiating the log partition function gives a Gaussian Lipschitz
-constant at most C_betasqrt n. The Gaussian concentration inequality
+constant at most C_betasqrt n. The Gaussian concentration ineuality
cite[Section~5.4, Theorem~5.6]{BLM2013} together with the pressure bound
(eq:pressure) therefore implies
[
@@ -956,7 +956,7 @@ established.
# Reversing the entropy: coverage
%======================================================================
-Proposition~prop:proposal controls {rm KL}(Q_lVert P), the
+Proposition~prop:proposal controls {KL}(Q_lVert P), the
entropy cost of producing a proposal spin. But entropy in the
forward direction is not enough: a set of large Gibbs mass might still
receive negligible proposal mass, and we will need in
@@ -966,7 +966,7 @@ appreciable probability. We therefore need a reverse comparison.
The idea is to condition the Gibbs joint law on the spin. With the spin
fixed, the remaining law of the disorder is strongly log-concave. A small
Gaussian smoothing then turns the forward comparison into the reverse one
-via the quadratic transport inequality.
+via the quadratic transport ineuality.
## Horizons and the smoothing map
@@ -982,7 +982,7 @@ connecting sequence in Section~sec:connections: its exceptional-event
estimate exp(-n^{1-10b}) is far stronger, even though its entropy bound
(eq:5) is weaker.
-[Smoothing map S]calS]
+[Smoothing map S]S]
For any joint law of disorder and spin, let S retain the spin
and replace its disorder by
@@ -1004,12 +1004,12 @@ smoothed disorder, by the Bayes rule for Gaussian vectors.
## Why smoothing reverses the entropy comparison
-The central reverse inequality is the following.
+The central reverse ineuality is the following.
[Smoothing reverses KL]
- {rm KL}(SP SQ_f)calSQ_f)
- ledelta^{-1},{rm KL}(Q_fVert P) .
+ {KL}(SP SQ_f)SQ_f)
+ ledelta^{-1},{KL}(Q_fVert P) .
@@ -1018,7 +1018,7 @@ The two unsmoothed laws have the same uniform spin marginal (see
Section~sec:proposal). Given x, the conditional disorder density
of P has negative logarithm, up to a normalizing constant,
[
- U_x(G)=tfrac12|G|^2-betawidehatwidehat H_G(x)+F(G).H_G(x)+F(G).
+ U_x(G)=1/2|G|^2-betaHhat_G(x)+F(G).H_G(x)+F(G).
]
Its Hessian is at least the identity: the Hamiltonian is linear in G
(hence contributes nothing to the Hessian), and F is a convex
@@ -1027,8 +1027,8 @@ log-sum-exp of linear functions.
We use the following consequence of strong convexity. Suppose mu has
density proportional to e^{-U} with nabla^2Uge I, and lambdallmu.
Then there are couplings of mu,lambda whose mean squared displacement
-is at most 2{rm KL}(lambdaVertmu) (or tends to that bound). This is
-the quadratic transport inequality; the triangular-transport form is
+is at most 2{KL}(lambdaVertmu) (or tends to that bound). This is
+the quadratic transport ineuality; the triangular-transport form is
cite[Corollary~3.10]{BKM2005}. We recall the proof for the
finite-dimensional densities used here.
@@ -1038,11 +1038,11 @@ to lambda by successive increasing conditional quantile maps. The
resulting triangular map T has positive diagonal derivatives, whose
product is its Jacobian determinant. The entropy formula is
[
- {rm KL}(lambdaVertmu)
+ {KL}(lambdaVertmu)
=int(U(T(z))-U(z)-logdetnabla T(z)),dmu(z).
]
Strong convexity bounds the first difference below by
-nabla U(z)·(T(z)-z)+tfrac12|T(z)-z|^2. Integration by parts
+nabla U(z)·(T(z)-z)+1/2|T(z)-z|^2. Integration by parts
replaces the linear term by div(T-id). There is
no boundary term: on a face where the i-th coordinate is an endpoint of
the box, the conditional quantile map fixes that endpoint, so the normal
@@ -1066,7 +1066,7 @@ mu=P(·mid x) and lambda=Q_f(·mid x). For any coupling of
two disorder values g,widetilde g, the Gaussian smoothing kernels
satisfy
[
- {rm KL}(N(√(1-delta),g,delta I)
+ {KL}(N(√(1-delta),g,delta I)
Vert N(√(1-delta),widetilde g,delta I))
=(1-delta)/(2delta)|g-widetilde g|^2 .
]
@@ -1091,12 +1091,12 @@ G^circ=√(1-delta),G+sqrtdelta,barbar G,G, and let
nu(·mid G^circ) be the conditional spin kernel of
SQ_f (eq:smoothing). Then
- E,{rm KL}(pi_GVertnu(·mid G^circ))
+ E,{KL}(pi_GVertnu(·mid G^circ))
le O_beta(nsqrtdelta)+2n^{1-f/2}/delta .
With probability tending to one over the target/base pair,
- {rm KL}(pi_GVertnu(·mid G^circ))le n^{1-4d} .
+ {KL}(pi_GVertnu(·mid G^circ))le n^{1-4d} .
On the same event, simultaneously for every set
A⊆{-1,1}^n with pi_G(A)ge n^{-d},
@@ -1113,16 +1113,16 @@ produced the base G^circ. Inserting this law into the logarithmic ratio
gives the exact decomposition
[
- E,{rm KL}(pi_GVertnu(·mid G^circ))
- &=E,{rm KL}(pi_GVert
+ E,{KL}(pi_GVertnu(·mid G^circ))
+ &=E,{KL}(pi_GVert
overlinepi(·mid G^circ))
- &+{rm KL}(SP SQ_f).calSQ_f).
+ &+{KL}(SP SQ_f).SQ_f).
]
Both expectations are over the target/base pair (G,G^circ) of the
statement. The conditional mixture minimizes averaged divergence among
spin laws depending only on the base, so the first term on the right is
-at most E,{rm KL}(pi_GVertpi_{G^circ}).
+at most E,{KL}(pi_GVertpi_{G^circ}).
For any two Hamiltonians, the Gibbs divergence is bounded by twice beta
times the supremum of their absolute energy difference. Here those
@@ -1134,13 +1134,13 @@ supremum by O(nsqrtdelta). Combining this estimate with (eq:8)
and (eq:5) proves (eq:9).
The power-of-n event. Since 4d<f/8 and 4d<f/4 (recall
-d=f/100 and f=1/100), Markov's inequality applied to (eq:9) gives
+d=f/100 and f=1/100), Markov's ineuality applied to (eq:9) gives
(eq:10) with probability tending to one.
Coverage from the entropy. On the single event (eq:10),
let A be any set with pi_G(A)ge n^{-d}. For s=pi_G(A) and
t=nu(Amid G^circ), entropy contraction to the two-point partition
-gives slog(1/t)le{rm KL}(pi_GVertnu)+log2. Hence
+gives slog(1/t)le{KL}(pi_GVertnu)+log2. Hence
[
log(1/t)le n^d(n^{1-4d}+log2)le n^{1-2d}
]
@@ -1170,7 +1170,7 @@ pass near the boundary.
For distinct spins x,y, write c_G(x,y) for the heat-bath jump rate and
[
- Q_G^{rm edge}(S,S^{rm c})
+ Q_G^{edge}(S,S^{c})
=sum_{xin S,,ynotin S}pi_G(x),c_G(x,y)
]
for the stationary flow of the chain across the set S. Detailed balance
@@ -1181,9 +1181,9 @@ of the next section uses.
[Balanced-cut flow]
With probability tending to one in the target disorder,
- Q_G^{rm edge}(S,S^{rm c})>B_{min}:=exp(-n^{1-b/20})
+ Q_G^{edge}(S,S^{c})>B_{min}:=exp(-n^{1-b/20})
quadfor every S with
- min(pi_G(S),pi_G(S^{rm c}))ge n^{-d}.
+ min(pi_G(S),pi_G(S^{c}))ge n^{-d}.
@@ -1204,7 +1204,7 @@ exp(-O_beta(sqrt n)).
If a cut has stationary flow at most B_{min}, then its two-sided vertex
boundary
[
- ∂ S={x:,d_H(x,S)=1 and d_H(x,S^{rm c})=1}
+ ∂ S={x:,d_H(x,S)=1 and d_H(x,S^{c})=1}
]
has Gibbs mass at most 2B_{min}exp(O_beta(sqrt n)), since each
boundary vertex controls at most 2n unit-rate crossings each carrying a
@@ -1213,20 +1213,20 @@ heat-bath acceptance probability at least exp(-O_beta(sqrt n)).
Set R_n=lceil n^{1-b/4}rceil and enlarge ∂ S by Hamming
distance R_n. For two spins at Hamming distance at most R_n and with
|W(G)|le C, the logarithmic Gibbs weight ratio is bounded by
-O_beta(n^{1-b/8}): the change in widehatwidehat H_GH_G along any path of
+O_beta(n^{1-b/8}): the change in Hhat_GH_G along any path of
R_n flips is at most Csqrt n· R_n/sqrt n· n^{1/2}=O_beta(n^{1-b/8}),
and a Hamming ball of radius R_n has logarithmic cardinality
O(n^{1-b/4}log n). The enlarged boundary
D_R(G) therefore has target Gibbs mass at most
- exp(-tfrac12 n^{1-b/20})
+ exp(-1/2 n^{1-b/20})
for n large, since 1-b/20<1-b/8 and b/20<b/4.
Among cuts violating (eq:11), select the first one in a fixed ordering
of the subsets of the cube whenever such a cut exists and |W(G)|le C.
Let D(G) be its enlarged boundary, and set
-D(G)=∅calD(G)=∅ if no such cut exists or the norm bound fails.
+D(G)=∅D(G)=∅ if no such cut exists or the norm bound fails.
This is measurable because the cube has finitely many subsets. Let
E be the event that a witness cut exists, the target-norm bound
holds, and the coverage statement (eq:10) holds for the sampled
@@ -1316,16 +1316,16 @@ exp(O_beta(nsqrtdelta)).
Apply the likelihood estimate (eq:6) with coarse horizon l=b. Its
good event depends only on this proposal run. If
-q_{rm good}(dxmid g) is the corresponding output subprobability kernel,
+q_{good}(dxmid g) is the corresponding output subprobability kernel,
disintegration against the Gaussian disorder marginal gives
- q_{rm good}(dxmid g)leexp(n^{1-b/2}),pi_g(dx)
+ q_{good}(dxmid g)leexp(n^{1-b/2}),pi_g(dx)
quadfor almost every g .
The primitive auxiliary inputs and the final spin draw for this run are
independent of all the matrix drivers. Thus, after fixing
G(theta)=g and adjoining (G,G^circ), this remains the conditional
-good-output kernel given the past. In particular, the inequality applies
+good-output kernel given the past. In particular, the ineuality applies
to the random set D(G) once those matrices have been fixed.
The separate energy-change event is used only to compare Gibbs weights;
it is not built into the proposal good event.
@@ -1333,7 +1333,7 @@ it is not built into the proposal good event.
By (eq:12), the resulting boundary probability on the complement of
the energy-change event is at most
[
- exp(n^{1-b/2}+O_beta(nsqrtdelta)-tfrac12 n^{1-b/20}).
+ exp(n^{1-b/2}+O_beta(nsqrtdelta)-1/2 n^{1-b/20}).
]
Add the proposal exceptional probability exp(-n^{1-10b}) and the
energy-change exception. Since sqrtdelta=n^{-f/8} and f=1/100, the
@@ -1388,7 +1388,7 @@ signals from (eq:4) differ by the primitive integral differences plus
the endpoint difference multiplied by a deterministic factor at most
sqrt h. Finally, the scalar integral equation (eq:Y) and its
spatially Lipschitz drift control the differences in Y and M over this
-interval at constant cost, by Gronwall's inequality. Starting with the
+interval at constant cost, by Gronwall's ineuality. Starting with the
entirely auxiliary first interval and taking successive maxima of these
differences proves (eq:13) after increasing C_1 if necessary.
@@ -1409,7 +1409,7 @@ follows from the martingale conditional mean through the continuation.
Under the unconditioned single-run law, each squared distance is an average
of bounded i.i.d. coordinate variables; consequently it is at most
n^{-b/3} except with probability O(exp(-n^{1-b})). The triangle
-inequality in |·|_n, along with the negligible magnetization
+ineuality in |·|_n, along with the negligible magnetization
increments above, then gives Hamming step sizes at most R_n for large n,
since n^{-b/3}ll R_n^2/n=n^{1-b/2}/n=n^{-b/2}.
@@ -1458,7 +1458,7 @@ centering
fill[blue!3] (7.3,-.75) rectangle (13.5,.9);
draw[densely dotted,black!55] (7.3,-.75) -- (7.3,.9);
node at (3.6,.63) {S};
- node at (10.4,.63) {S^{mathrm c}rmc};
+ node at (10.4,.63) {S^{c}rmc};
node[fine] (leftfine) at (.65,0) {};
node[coarse] (coarsezero) at (2.5,0) {};
node[coarse] (coarseone) at (4.4,0) {};
@@ -1469,8 +1469,8 @@ centering
draw[black!65] (leftfine) -- (coarsezero) -- (coarseone) -- (crossleft);
draw[black!65] (crossright) -- (coarselast) -- (rightfine);
draw[red!65!black,line width=1.1pt] (crossleft) -- (crossright);
- node[below=1mm] at (leftfine.south) {x_{rm f}^{(0)}};
- node[below=1mm] at (rightfine.south) {x_{rm f}^{(1)}};
+ node[below=1mm] at (leftfine.south) {x_{f}^{(0)}};
+ node[below=1mm] at (rightfine.south) {x_{f}^{(1)}};
node[below=1mm] at (crossleft.south) {u};
node[below=1mm] at (crossright.south) {v};
node[fill=white,inner sep=1.5pt] at (7.3,.48) {d_H(u,v)le R_n};
@@ -1492,7 +1492,7 @@ centering
node[below=1mm] at (boundaryright.south) {y_{t+1}};
node[below=1mm] at (pathright.south) {v=y_ell};
node at (2.6,-2.85) {S};
- node at (12,-2.85) {S^{mathrm c}rmc};
+ node at (12,-2.85) {S^{c}rmc};
node[red!65!black,align=center] at (7.3,-3.8)
{d_H(u,∂ S),,d_H(v,∂ S)leellle R_n
;⇒; u,vinD(G).};
@@ -1514,11 +1514,11 @@ The finite-state argument below has three steps:
[label=arabic*.,leftmargin=*]
Find a set C of large stationary mass with good internal expansion,
by repeatedly pruning away subsets that do not expand.
- On C, obtain a Poincar'e inequality from the coarea proof of the
- reversible Cheeger inequality.
+ On C, obtain a Poincar'e ineuality from the coarea proof of the
+ reversible Cheeger ineuality.
Use entropy dissipation of the full chain plus stationarity to
bound the pi-average of the fixed-start total variation at time T,
- and finish with Markov's inequality over the starting state.
+ and finish with Markov's ineuality over the starting state.
Crucially, the argument never asks a trajectory to stay inside C; that
@@ -1536,13 +1536,13 @@ respect to pi with jump rates c(x,y). Assume that for some R>0
]
Let 0<ε<1/3 and B>0. Suppose that
[
- Q(S,S^{rm c}):=sum_{xin S,,ynotin S}pi(x)c(x,y)>B
+ Q(S,S^{c}):=sum_{xin S,,ynotin S}pi(x)c(x,y)>B
quadwhenever εlepi(S)le1-ε.
]
Then for every T>0,
- sum_{xinΩ}pi(x)bigl|P_T(x,·)-pibigr|_{rm TV}
- lefrac{√(2R,H(pi))}{Bsqrt T}+2ε,
+ sum_{xinΩ}pi(x)bigl|P_T(x,·)-pibigr|_{TV}
+ le√(2R,H(pi))/Bsqrt T+2ε,
H(pi)=-sum_xpi(x)logpi(x).
@@ -1550,7 +1550,7 @@ Then for every T>0,
Proof.
Put q(x,y)=pi(x)c(x,y)=q(y,x), and use the unordered-edge Dirichlet form
[
- mathfrak D(u)=frakD(u)=sum_{{x,y}}q(x,y)(u(x)-u(y))^2.
+ D(u)=D(u)=sum_{{x,y}}q(x,y)(u(x)-u(y))^2.
]
Only distinct vertices occur, and every unordered pair is counted once.
@@ -1577,13 +1577,13 @@ set satisfies pi(C)>1-ε and
quadfor every Usubset C with pi(U)lepi(C)/2 .
-Step 2: a Poincar'e inequality on the retained set.: Use the
-coarea proof of the reversible Cheeger inequality
+Step 2: a Poincar'e ineuality on the retained set.: Use the
+coarea proof of the reversible Cheeger ineuality
cite[Theorem~3.5]{LawlerSokal1988}, keeping the original unnormalized
weights on C. The resulting estimate is
sum_{yin C}pi(y)(u(y)-barbar u_C)^2u_C)^2
- le(2R)/(B^2),mathfrak D(u),frakD(u),
+ le(2R)/(B^2),D(u),D(u),
barbar u_C=u_C=frac1{pi(C)}sum_{yin C}pi(y)u(y).
To prove it, choose a weighted median m of u on C. Each of
@@ -1602,7 +1602,7 @@ By Cauchy--Schwarz this is at most
since (w(x)+w(y))^2le2(w(x)^2+w(y)^2) and the total outgoing rate is at
most R. If sum_Cpi w^2 vanishes the estimate is immediate; otherwise
divide and square. The sum of the energies of the positive and negative
-parts is at most mathfrak D(u).frakD(u). The weighted mean minimizes squared
+parts is at most D(u).D(u). The weighted mean minimizes squared
error, proving (eq:14).
Step 3: entropy dissipation for the full chain.: For a fixed
@@ -1611,8 +1611,8 @@ with (a-b)(log a-log b)ge4(sqrt a-sqrt b)^2 for a,b>0 gives the
entropy-dissipation estimate
cite[Lemmas~2.5,~2.7]{DiaconisSaloffCoste1996}
- int_0^Tsum_xpi(x),mathfrak D(√(f_t^x))frakD(√(f_t^x)),dt
- letfrac14 H(pi).
+ int_0^Tsum_xpi(x),D(√(f_t^x))D(√(f_t^x)),dt
+ le1/4 H(pi).
Indeed, the derivative of sum_ypi(y)f_t^x(y)log f_t^x(y) is
[
@@ -1639,14 +1639,14 @@ Cauchy--Schwarz and a_x^2pi(C)lesum_Cpi f_t^xle1 imply
sum_Cpi|f_t^x-a_x^2|le2r_x .
]
Also, total normalization gives
-|sum_Cpi(f_t^x-1)|lesum_{C^{rm c}}pi(f_t^x+1).
+|sum_Cpi(f_t^x-1)|lesum_{C^{c}}pi(f_t^x+1).
Comparing f_t^x first with a_x^2 and then with 1 on C, and adding
the complement, yields
[
sum_Ωpi|f_t^x-1|
- le4r_x+2sum_{C^{rm c}}pi(f_t^x+1).
+ le4r_x+2sum_{C^{c}}pi(f_t^x+1).
]
-The last sum before its factor two has average 2pi(C^{rm c}) over
+The last sum before its factor two has average 2pi(C^{c}) over
xsimpi, because stationarity gives sum_xpi(x)f_t^x(y)=1. Dividing
by two for total variation and applying Cauchy--Schwarz in x proves the
upper bound at time t. Contraction under P_{T-t} gives the same bound
@@ -1665,8 +1665,8 @@ The last choice is valid because there are exactly n rate-one update
clocks, so the total jump rate at any vertex is n. Also
H(pi_G)le nlog2, since the state space has 2^n elements. Hence
[
- sum_xpi_G(x)bigl|P_T^G(x,·)-pi_Gbigr|_{rm TV}
- lefrac{√(2log2);n}{Bsqrt T}+2n^{-d}.
+ sum_xpi_G(x)bigl|P_T^G(x,·)-pi_Gbigr|_{TV}
+ le√(2log2);n/Bsqrt T+2n^{-d}.
]
Take
[
@@ -1675,11 +1675,11 @@ Take
Then the first term tends to zero, because
[
log(n)/(Bsqrt T)
- =log n+n^{1-b/20}-tfrac12 n^{1-b/40}→-∞ .
+ =log n+n^{1-b/20}-1/2 n^{1-b/40}→-∞ .
]
The flow event (eq:11) itself has disorder probability tending to
-one. Markov's inequality therefore shows that the Gibbs mass of fixed
+one. Markov's ineuality therefore shows that the Gibbs mass of fixed
states whose distance at time T exceeds 1/4 tends to zero in disorder
probability. The added diagonal changes neither this Gibbs law nor the
transition rates, so the same statement holds for the model H^J_n in
@@ -1692,7 +1692,7 @@ All threshold dimensions may depend on the fixed beta>1.
## Reading the final exponents
-For orientation, here is the chain of inequalities that governs the
+For orientation, here is the chain of ineualities that governs the
constants, all in the ``1-exponent'' scale:
[
10b ll b/4 ll b/2 ll b/20 ll d
@@ -1724,7 +1724,7 @@ arXiv:1303.3573v1.
M.~Boban, A.~Li and S.~Oveis Gharan,
newblock Rank-1-perturbed trickledown theorems: mixing time of Glauber
dynamics for the Sherrington--Kirkpatrick model up to
-betaletfrac12+ε,
+betale1/2+ε,
newblock preprint (2026),
arXiv:2609.13138v1.
@@ -1744,7 +1744,7 @@ arXiv:1201.2891.
S.~Boucheron, G.~Lugosi and P.~Massart,
-newblock Concentration Inequalities: A Nonasymptotic Theory of
+newblock Concentration Ineualities: A Nonasymptotic Theory of
Independence, Oxford University Press, 2013.
@@ -1757,7 +1757,7 @@ arXiv:1501.06635v2.
P.~Diaconis and L.~Saloff-Coste,
-newblock Logarithmic Sobolev inequalities for finite Markov chains,
+newblock Logarithmic Sobolev ineualities for finite Markov chains,
newblock Ann. Appl. Probab. 6 (1996), no.~3, 695--750.
newblock doi:10.1214/aoap/1034968224.
@@ -1789,7 +1789,7 @@ arXiv:1504.02731v2.
G.~F. Lawler and A.~D. Sokal,
newblock Bounds on the L^2 spectrum for Markov chains and Markov
-processes: a generalization of Cheeger's inequality,
+processes: a generalization of Cheeger's ineuality,
newblock Trans. Amer. Math. Soc. 309 (1988), no.~2,
557--580.
newblock doi:10.1090/S0002-9947-1988-0930082-9.