When does propagation of chaos in the critical Curie-Weiss model stop?
Abstract
We study increasing propagation of chaos for the critical Curie- Weiss model (i.e. the mean-field Ising model at critical inverse temperature 1, with no external field). We give a simple way to see that for windows of size k(N) of smaller order than sqrt{N} we still have propagation of chaos (reproving earlier results, see e.g.[1]), while for k(N) of order sqrt{N} the propagation of chaos breaks down. The law of a single spin converges to π, the Bernoulli law with parameter 1/2. If k(N) = alpha sqrt{N}, we give an explicit formula for the limiting distance in total variation of the law of the first k(N) spins with respect to the k-fold product of pi, as a function of alpha. For even larger window sizes, the distribution of the spins has, in the thermodynamical limit, maximal distance to the k-fold product of pi. One of the ingredients of the proof is a result about the unimodality/non- unimodality of the law of the number of positive spins among the first k = k(N) spins, which may be of independent interest.
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BibTeX
@misc{gantert2026when,
title = {{When does propagation of chaos in the critical Curie-Weiss model stop?}},
author = {Nina Gantert and Matthias Löwe and Kilian Schnappinger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02464},
eprint = {2609.02464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02464},
}
RIS
TY - UNPB
TI - When does propagation of chaos in the critical Curie-Weiss model stop?
AU - Nina Gantert
AU - Matthias Löwe
AU - Kilian Schnappinger
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02464
UR - https://arxiv.org/abs/2609.02464
AB - We study increasing propagation of chaos for the critical Curie- Weiss model (i.e. the mean-field Ising model at critical inverse temperature 1, with no external field). We give a simple way to see that for windows of size k(N) of smaller order than sqrt{N} we still have propagation of chaos (reproving earlier results, see e.g.[1]), while for k(N) of order sqrt{N} the propagation of chaos breaks down. The law of a single spin converges to π, the Bernoulli law with parameter 1/2. If k(N) = alpha sqrt{N}, we give an explicit formula for the limiting distance in total variation of the law of the first k(N) spins with respect to the k-fold product of pi, as a function of alpha. For even larger window sizes, the distribution of the spins has, in the thermodynamical limit, maximal distance to the k-fold product of pi. One of the ingredients of the proof is a result about the unimodality/non- unimodality of the law of the number of positive spins among the first k = k(N) spins, which may be of independent interest.
ER -