发表机构
Technische Universität München; University of Münster（慕尼黑工业大学; 明斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析临界Curie-Weiss模型的混沌传播，证明窗口大小k(N)小于√N时混沌传播存在、等于√N时失效，还给出对应全变差极限距离公式及更大窗口的相关结论。
AI 中文摘要
我们研究临界Curie-Weiss模型（即临界逆温度为1、无外场的平均场Ising模型）中不断增强的混沌传播。我们给出一种简单方法，用于证明当窗口大小k(N)的阶小于√N时，仍存在混沌传播（再次证明了早期结果，例如见文献[1]）；而当k(N)的阶为√N时，混沌传播会失效。单个自旋的分布收敛于π，即参数为1/2的伯努利分布。若k(N)=α√N，我们给出前k(N)个自旋的分布与π的k重乘积分布之间的全变差极限距离的显式公式，该公式是α的函数。对于更大的窗口大小，自旋的分布在热力学极限下与π的k重乘积分布的距离达到最大。证明的要素之一是关于前k=k(N)个自旋中正自旋数量的分布是否为单峰的结果，该结果可能具有独立研究价值。
英文摘要
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.
Comments22 pages
