Wasserstein Stability, Couplings Across Volumes, and the 1+1/d Moment Thresholds in the Edwards--Anderson Model
Abstract
We study the quenched pressure of the nearest-neighbor Edwards--Anderson Ising model with free and periodic boundary conditions. First, we prove that the infinite-volume pressure is βd-Lipschitz in the coupling law for the $1$-Wasserstein distance, yielding quantitative thermodynamic limits for spatially inhomogeneous disorder. Second, for periodic volumes, we prove that almost-sure convergence under every joint coupling of the finite-volume disorder arrays is equivalent to complete convergence of the one-volume pressure laws. Third, we prove that 𝔼|J|1+1/d<∞ guarantees this universal-coupling conclusion. We further show that this exponent is optimal among uniform power-moment assumptions: for every 1≤ q<1+1/d, there is a centered symmetric law with finite q-th moment for which canonical nested volumes converge almost surely, whereas independently resampled volumes with the same fixed-volume marginals converge in probability but not almost surely. Finally, in dimension one, we prove that the first-moment condition is also necessary for a finite limiting pressure.
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BibTeX
@misc{chueng2026wasserstein,
title = {{Wasserstein Stability, Couplings Across Volumes, and the $1+1/d$ Moment Thresholds in the Edwards--Anderson Model}},
author = {Mauris Chueng and Hexiang Wang and Keheng Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09680},
eprint = {2609.09680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09680},
}
RIS
TY - UNPB
TI - Wasserstein Stability, Couplings Across Volumes, and the 1+1/d Moment Thresholds in the Edwards--Anderson Model
AU - Mauris Chueng
AU - Hexiang Wang
AU - Keheng Zhu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09680
UR - https://arxiv.org/abs/2609.09680
AB - We study the quenched pressure of the nearest-neighbor Edwards--Anderson Ising model with free and periodic boundary conditions. First, we prove that the infinite-volume pressure is βd-Lipschitz in the coupling law for the $1$-Wasserstein distance, yielding quantitative thermodynamic limits for spatially inhomogeneous disorder. Second, for periodic volumes, we prove that almost-sure convergence under every joint coupling of the finite-volume disorder arrays is equivalent to complete convergence of the one-volume pressure laws. Third, we prove that 𝔼|J|^1+1/d<∞ guarantees this universal-coupling conclusion. We further show that this exponent is optimal among uniform power-moment assumptions: for every 1≤ q<1+1/d, there is a centered symmetric law with finite q-th moment for which canonical nested volumes converge almost surely, whereas independently resampled volumes with the same fixed-volume marginals converge in probability but not almost surely. Finally, in dimension one, we prove that the first-moment condition is also necessary for a finite limiting pressure.
ER -