diff 999db1f..f6bf760 o3_model_massive_continuum_v2.txt
diff --git a/o3_model_massive_continuum_v2.txt b/o3_model_massive_continuum_v2.txt
index 09a558c..62ce8e4 100644
--- a/o3_model_massive_continuum_v2.txt
+++ b/o3_model_massive_continuum_v2.txt
@@ -53,7 +53,7 @@ is unique; denote it mu_beta. Two facts are used constantly:
To pass to the continuum, shrink the lattice spacing a > 0 to
zero and simultaneously choose a positive field normalisation
B > 0 (both to be specified). For a smooth compactly supported
-vector test f in C_c^infty(R^2; R^3) and a spin configuration on
+vector test f in C_c^infinity(R^2; R^3) and a spin configuration on
the lattice aZ^2, define the smeared field
phi_{a,B}(f) = B a^2 sum_{x in Z^2} f(a x) . q_x. (0.6)
@@ -143,7 +143,7 @@ normalisations B_N > 0 such that, with a_N = L^{-N}, the fields
phi_N(f) = B_N a_N^2 sum_{x in Z^2} f(a_N x) q_x (0.10)
converge jointly in law and in all joint moments as N -> infinity,
-for every finite list of tests f_r in C_c^infty(R^2; R^3) and every
+for every finite list of tests f_r in C_c^infinity(R^2; R^3) and every
list of component indices i_r in {1,2,3}.
The limiting hierarchy S = ( S_n^{i_1 ... i_n} )_{n>=1} on
@@ -199,7 +199,7 @@ THEOREM B. For every sequence beta -> infinity the joint law of
Psi_beta^{i_1}(f_1), ..., Psi_beta^{i_n}(f_n) (0.14)
converges, for every finite list of tests f_1,...,f_n in
-C_c^infty(R^2; R^3) and every list of component indices i_r in
+C_c^infinity(R^2; R^3) and every list of component indices i_r in
{1,2,3}. It does so jointly in law and in all joint moments. The
limit is the same for every such sequence, and equals a hierarchy G
which satisfies (A1)-(A6) of Theorem A.
@@ -256,8 +256,8 @@ coupling obeys the recursion
b' = b + alpha_h + kappa_h / b + Delta, |Delta| <= C_L H_j^{-1.05},
where alpha_h, kappa_h are precomputable functions of a finite
-canonical orbit. The limiting increments are alpha_infty -> -gamma,
-kappa_infty / b -> 0. Therefore the running coupling recovers the
+canonical orbit. The limiting increments are alpha_infinity -> -gamma,
+kappa_infinity / b -> 0. Therefore the running coupling recovers the
reference sequence beta_N = H_N + bounded offset, and the resulting
lattice length satisfies xi_{beta_N} ~ L^N = a_N^{-1}.
@@ -314,7 +314,7 @@ adaptation only):
Sec 2 (free kernels) ---> Sec 3 (exact RG step, q < L^{-2+u})
|
v
- Sec 5 (shooting: alpha_infty, beta_N)
+ Sec 5 (shooting: alpha_infinity, beta_N)
|
v
Sec 4 (integrated comparison)
@@ -366,7 +366,7 @@ every estimate uniform. The order is:
Whenever a constant is written C, c, C_L, or c_L, it is understood
fixed at the indicated stage and independent of later choices. A
-symbol such as o_H(1) means a quantity tending to zero as H -> infty
+symbol such as o_H(1) means a quantity tending to zero as H -> infinity
at the point where H was chosen large, with all constants kept in
the sense of stages 1-2. A symbol such as O(log H) denotes a
bounded multiple of log H at the point of use.
@@ -425,7 +425,7 @@ including all-zero choices.
C ( 1 + b + |A| + |A'| + d )^p
L(F) L(G) exp( - c d / X(b) ), (1.2)
- where L(F) = ||F||_infty + Lip(F) is a local Lipschitz norm
+ where L(F) = ||F||_infinity + Lip(F) is a local Lipschitz norm
(max over neighbouring lattice points). In particular the
homogeneous periodic infinite-plane limit exists and is
unique.
@@ -653,7 +653,7 @@ Recovering the definition (1.7) of X_Q from F = (h e^{i theta_mu})_mu
(each direction contributes independently and the C^1 norm of F
dominates the sup of the C^1 norms of its two components), gives
(1.16). The bounded-width boundary layer is estimated by the
-trivial bound ||X_Q(F)||_{L^r} <= C b S ||F||_infty, which is
+trivial bound ||X_Q(F)||_{L^r} <= C b S ||F||_infinity, which is
dominated by the log factor since S <= exp(16b) and the extra
S-factor is absorbed by a small constant. That completes the
proof of the lemma. []
@@ -1119,7 +1119,7 @@ Let O, O' be bounded Haar-centered functions on S^2 (mean zero
against the normalised area measure). Let x, y be lattice sites at
distance d = |x - y|. The claim is
- | E_mu O(q_x) O'(q_y) | <= C ||O||_infty ||O'||_infty
+ | E_mu O(q_x) O'(q_y) | <= C ||O||_infinity ||O'||_infinity
exp( - c d / s ), (1.52)
where s is the coarse cell side (1.47), and C, c are absolute
@@ -1264,8 +1264,8 @@ term vanishes identically.
Therefore
| E_mu[ O(q_x) O'(q_y) ] |
- <= ||O||_infty ||O'||_infty Pr( disagree at y )
- <= ||O||_infty ||O'||_infty 2^{-ceil(d/s)}. (1.58)
+ <= ||O||_infinity ||O'||_infinity Pr( disagree at y )
+ <= ||O||_infinity ||O'||_infinity 2^{-ceil(d/s)}. (1.58)
Absorbing 2^{-ceil(d/s)} <= 2 * 2^{-d/s} into the constant,
(1.52) follows. []
@@ -1846,8 +1846,8 @@ Precise-coupling constants. At H_j large, alpha_h and kappa_h are
precomputable in the limit h -> infinity (they converge; see
the shooting construction). Their limits are the Kosterlitz-Thouless increments
- alpha_infty -> -gamma = -log L / (2 pi),
- kappa_infty / b -> --- (of order 1) * (1 / b).
+ alpha_infinity -> -gamma = -log L / (2 pi),
+ kappa_infinity / b -> --- (of order 1) * (1 / b).
The first is negative: each step DECREASES the coupling by
(log L)/(2 pi). This is the asymptotically-free running of the O(3)
@@ -1986,8 +1986,8 @@ bad-bond set) and pairwise disjoint supports.
Discrepancies. Define for each label lambda:
- d_lambda = || tk_lambda - k_lambda ||_infty, (4.2)
- p_lambda = || k_lambda ||_infty + || tk_lambda ||_infty. (4.3)
+ d_lambda = || tk_lambda - k_lambda ||_infinity, (4.2)
+ p_lambda = || k_lambda ||_infinity + || tk_lambda ||_infinity. (4.3)
Covering-cap assumption. There is an absolute constant
Cap0 > 0 (determined by L, P_0, and the fixed support constant A)
@@ -2213,7 +2213,7 @@ gain of the energy decomposition, one has
(1/Z) int e^X | tXi - Xi | dq
<= exp( sum_lambda d_lambda e^{2 s_lambda} ) - 1. (4.16)
-The sum is over all label, and d_lambda = || tk_lambda - k_lambda ||_infty
+The sum is over all label, and d_lambda = || tk_lambda - k_lambda ||_infinity
as in (4.2).
Sketch of proof.
@@ -2307,14 +2307,14 @@ the canonical extraction.
LEMMA 5.1 (Canonical orbit convergence). The orbit P^{(h)}
converges as h -> infinity, and the increments alpha_h + gamma,
kappa_h converge exponentially fast in h. Concretely there exist a
-fixed point P^{(infty)} and constants C, c > 0 (depending only on
+fixed point P^{(infinity)} and constants C, c > 0 (depending only on
L) such that for every h >= 0,
- | P^{(h)} - P^{(infty)} | <= C e^{-c h}, (5.4)
+ | P^{(h)} - P^{(infinity)} | <= C e^{-c h}, (5.4)
| alpha_h + gamma | <= C e^{-c h}, (5.5)
- | kappa_h - kappa_infty | <= C e^{-c h}, (5.6)
+ | kappa_h - kappa_infinity | <= C e^{-c h}, (5.6)
-where kappa_infty is a finite fixed number.
+where kappa_infinity is a finite fixed number.
Proof. Define P^{(n)} := C_n o C_{n-1} o ... o C_1 applied to
P^{(0)} = 0. The bounds (5.4)-(5.6) are established.
@@ -2348,18 +2348,18 @@ Taking h' = h + 1 (histories share the first h steps, so r = h) yields
| alpha_{h+1} - alpha_h | <= C (A_0/L^2)^h. Summing the tail
gives
- | alpha_h - alpha_infty | <= C' (A_0/L^2)^h,
+ | alpha_h - alpha_infinity | <= C' (A_0/L^2)^h,
which is (5.5), with exponential rate c = log(L^2/A_0) > 0. The
Kosterlitz-Thouless increment identification of the canonical extraction gives
-alpha_infty = -gamma.
+alpha_infinity = -gamma.
Step 4 (kappa limit). The same history-stability argument applied
to the b^{-1} coefficient gives
- | kappa_h - kappa_infty | <= C_L (A_0/L^2)^h,
+ | kappa_h - kappa_infinity | <= C_L (A_0/L^2)^h,
-which is (5.6). The limiting value kappa_infty is determined by the
+which is (5.6). The limiting value kappa_infinity is determined by the
free principal symbol of the block kernel (the block observation) times a fixed
multiplicative factor. []
@@ -2379,13 +2379,13 @@ exponential convergence of alpha_m + gamma and kappa_m from
Lemma 5.1 gives
b_j - b_k = sum_{m=j+1}^k (alpha_m + gamma)
- + kappa_infty sum_{m=j+1}^k 1/b_m
+ + kappa_infinity sum_{m=j+1}^k 1/b_m
+ sum_{m=j+1}^k Delta_m + error.
The first sum is O(1) (bounded absolutely by the exponentially
convergent series). The third sum is <= C_L sum H_m^{-1.05} <=
-O(1). The middle sum is at least kappa_infty (k-j)/(2 H_k) and at
-most kappa_infty (k-j)/(H_j/2). Absorbing the alpha-difference and
+O(1). The middle sum is at least kappa_infinity (k-j)/(2 H_k) and at
+most kappa_infinity (k-j)/(H_j/2). Absorbing the alpha-difference and
Delta sums into the constants:
| (b_j - H_j) - (b_k - H_k) | <= C_L + C_L (k-j)/H. (5.8)
@@ -2458,9 +2458,9 @@ such beta_N. []
Define inclined increments alpha_h^+, kappa_h^+ using the inclined
first-step canonical map (the canonical extraction with L_* = 5L, the inclined first step).
Lemma 5.1 continues to hold; and the KEY equality is
-alpha_infty^+ = alpha_infty = -gamma (the Kosterlitz-Thouless
+alpha_infinity^+ = alpha_infinity = -gamma (the Kosterlitz-Thouless
increment is unchanged by the choice of first block; only
-kappa_infty changes, by a fixed bounded amount).
+kappa_infinity changes, by a fixed bounded amount).
PROPOSITION 5.2 (Inclined shooting). For every sufficiently large
N there is an admitted coupling beta_N^+ in [.6 H_N, 1.4 H_N]
@@ -2482,8 +2482,8 @@ both admit trajectories ending at layer 0, then b_0(beta) <
b_0(beta'). If b_0(beta) = H and b_0(beta') = H, then beta = beta'.
Proof. The signed increments at each step m are
-alpha_m + gamma + kappa_infty / b_m + Delta_m. The mapping
-b -> b + kappa_infty / b is increasing in b (when kappa_infty > 0),
+alpha_m + gamma + kappa_infinity / b_m + Delta_m. The mapping
+b -> b + kappa_infinity / b is increasing in b (when kappa_infinity > 0),
and the Gronwall propagation is monotone. So better starting beta
gives better final value. The case b_0(beta) = b_0(beta') = H
with beta != beta' contradicts the strict monotone increase. []
@@ -2602,7 +2602,7 @@ K_0 and M_0 by Prop 6.1 for U. Then:
rectangle,
| E_{torus_N(reference)} F - omega_{beta_N}(F) |
- <= C ||F||_infty exp( - c 2^k ), (6.7)
+ <= C ||F||_infinity exp( - c 2^k ), (6.7)
where torus_N(reference) is the physical torus of side
M_0 L^N * 2^k, and omega_{beta_N} is the periodic
@@ -2613,13 +2613,13 @@ K_0 and M_0 by Prop 6.1 for U. Then:
R >> 1 on the plane,
| Cov_{omega_beta}(F, G) |
- <= C (1+b+|supp F|+|supp G|+R)^p ||F||_infty ||G||_infty
+ <= C (1+b+|supp F|+|supp G|+R)^p ||F||_infinity ||G||_infinity
* exp( - m R ), (6.8)
with m = c / X(b) > 0, uniformly in b.
Proof. Part (a) (torus-to-plane convergence). Fix F with compact
-support K and per-bound ||F||_infty <= 1. For k >= k_0(K), the
+support K and per-bound ||F||_infinity <= 1. For k >= k_0(K), the
support K fits inside the reference box of side 2^k M_0 a_N^{-1}.
Set w := 2^k M_0 a_N^{-1} (the width of the current torus in
physical units) and write T_w := K_w / lambda_1(w) for the
@@ -2629,12 +2629,12 @@ Decompose F in the Perron eigenbasis of T_w: F = c P_Omega + F_exc,
with F_exc orthogonal to the vacuum projection P_Omega. Then
E_{torus_N(F)} F = c + Tr(F_exc T_w^N) / Tr(T_w^N)
- = c + O( ||F_exc||_infty s_N(w) / (1+s_N(w)) )
+ = c + O( ||F_exc||_infinity s_N(w) / (1+s_N(w)) )
where s_N(w) = Tr(T_w^N - P_w). Also omega_{beta_N}(F) = c (the
Perron-eigenvalue component of F on the infinite plane). Hence
- | E_{torus_N(F)} F - omega_{beta_N}(F) | <= ||F||_infty s_N(w).
+ | E_{torus_N(F)} F - omega_{beta_N}(F) | <= ||F||_infinity s_N(w).
From Prop 6.1 (6.6) at scale (N, w) the excited-state trace obeys
@@ -2667,7 +2667,7 @@ finite-torus expectation differs from the plane expectation by
Passing to the plane limit and taking the minimum 2^k ~ R, this gives
| Cov_{omega_beta}(F, G) |
- <= C (1+b+|supp F|+|supp G|+R)^p ||F||_infty ||G||_infty exp( -m R )
+ <= C (1+b+|supp F|+|supp G|+R)^p ||F||_infinity ||G||_infinity exp( -m R )
with m = c / X(b).
@@ -2860,10 +2860,10 @@ Define
B_N = prod_{j=1}^N c_j^{-1}. (7.5)
-LEMMA 7.2 (Absolute convergence). The limit B_infty = lim B_N
+LEMMA 7.2 (Absolute convergence). The limit B_infinity = lim B_N
exists and is strictly positive, with
- | log B_N - log B_infty | <= C H^{-1} p_H^2, (7.6)
+ | log B_N - log B_infinity | <= C H^{-1} p_H^2, (7.6)
uniformly in N.
@@ -2926,7 +2926,7 @@ Let phi_N = phi_{a_N, B_N} be the depth-N smeared field
phi_N(f) = B_N a_N^2 sum_x f(a_N x) q_x with a_N = L^{-N} and B_N
the field normalization from the renormalization of the sources. Write X_r = phi_N^{k_r}(f_r)
for a finite list of smeared fields, component indices k_r in
-{1, 2, 3}, and test functions f_r in C_c^infty(R^2; R).
+{1, 2, 3}, and test functions f_r in C_c^infinity(R^2; R).
LEMMA 8.1 (Uniform moment bounds). For every m >= 1 there exists
a constant C_m > 0 (depending on L, H, m, the f_r, and the
@@ -2947,7 +2947,7 @@ Proof. Step 1 (representation). Each X_r = phi_N^{k_r}(f_r) equals
Since a_N^{-2} = L^{2N} counts the fine sites per retained-block
variable, the coefficient of each retained spin V_Y is
B_N * (sum_{x in block Y} f_r(a_N x)) / (block volume), which is
-bounded by B_N ||f_r||_infty.
+bounded by B_N ||f_r||_infinity.
Step 2 (effective measure structure). At depth N the effective
measure on the retained spins V_Y is, by Theorem 3.1, a Gaussian-
@@ -2980,10 +2980,10 @@ effective measure, using its O(3)-invariance from Theorem 3.1) gives
E X_1 ... X_m = -E X_1 ... X_m, hence the expectation vanishes.
Step 5 (final bound). Combining Step 3 with the two-point bound
-|E X_a X_b| <= B_N^2 ||f_a||_infty ||f_b||_infty (from unit modulus
+|E X_a X_b| <= B_N^2 ||f_a||_infinity ||f_b||_infinity (from unit modulus
of V_Y), one obtains
- | E X_1 ... X_m | <= (2m-1)!! * B_N^{2m} * prod ||f_r||_infty.
+ | E X_1 ... X_m | <= (2m-1)!! * B_N^{2m} * prod ||f_r||_infinity.
Since B_N is a fixed positive number (Lemma 7.2), this is the
stated bound (8.1) with C := max(B_N, 1) absorbed. []
@@ -3037,10 +3037,10 @@ LEMMA 8.3 (Analyticity and convergence). There is rho > 0
(depending on m, the f_r's supports and the physical constants,
but not on N) such that G_N(xi) is analytic in |xi| < rho for
every N, and G_N converges uniformly on compact subsets of this
-polydisc to a limit G_infty.
+polydisc to a limit G_infinity.
Proof. By (8.2a), |E prod_r (phi_N(f_r))^{k_r}| <=
-C^{|k|} (2|k|-1)!! prod_r ||f_r||_{infty}^{k_r}, so the Taylor
+C^{|k|} (2|k|-1)!! prod_r ||f_r||_{infinity}^{k_r}, so the Taylor
series has radius of convergence at least rho = c/C for a fixed c.
Term-by-term convergence of the moments (the local moment bounds together with the
martingale-like limit argument) gives uniform
@@ -3048,8 +3048,8 @@ convergence on compact subsets. Analytic continuation by the
identity theorem completes the proof. []
Corollary 8.4. The hierarchy S_m is the m-th Taylor coefficient
-of G_infty at xi = 0. In particular the moments of S at every
-finite order exist and are determined by G_infty.
+of G_infinity at xi = 0. In particular the moments of S at every
+finite order exist and are determined by G_infinity.
LEMMA 8.5 (Euclidean invariance of S). The limiting hierarchy S
@@ -3200,7 +3200,7 @@ S^(canonical) from the fixed-offset field construction, and let H_phys := P_+^0
Hamiltonian produced by the OS reconstruction of the reconstruction.
Step 1 (Feynman-Kac representation). By OS reconstruction (Section
-9.3), for two test functions f, g in C_c^infty(R^2) and any tau > 0,
+9.3), for two test functions f, g in C_c^infinity(R^2) and any tau > 0,
S_2(f, tau g) = < f_hat | exp(-tau H_phys) | g_hat >_{H},
@@ -3451,7 +3451,7 @@ LEMMA 11.2 (Tilted torus comparison). Let F be a bounded measurable
local observable supported in a fixed reference rectangle. Then
| E_{T_tilted(2^k L*)} F - E_{T_axis(25 * 2^k L*, 2^k L*)} F |
- <= C ||F||_infty exp( - c 2^k ), (11.3)
+ <= C ||F||_infinity exp( - c 2^k ), (11.3)
uniformly in k >= 0, with C, c fixed.
@@ -3469,7 +3469,7 @@ supported in a fixed slab of T_tilted and G in the same tilted
torus translated by distance R in the tilted coordinates. Then
| Cov_{T_tilted}(F, G) |
- <= C (1 + R)^p ||F||_infty ||G||_infty exp( - m R ), (11.4)
+ <= C (1 + R)^p ||F||_infinity ||G||_infinity exp( - m R ), (11.4)
with m = c / X(H) as in Theorem 6.3.