Fluctuations in mean-field Ising models
Fluctuations in mean-field Ising models
复制标题
平均场伊辛模型的涨落
DOI：
10.1214/22-aap1857
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
S. Mukherjee
中科院分区：
文献类型：
--
作者：
Nabarun Deb;S. Mukherjee
在本文中，我们研究了 $N$ 顶点上大约 $d_N$ 正则图 $G_N$ 上 Ising 模型中平均磁化强度的波动。特别是，如果 $G_N$ 是 \enquote{well Connect}，我们表明每当 $d_N\gg \sqrt{N}$ 时，波动是普遍的，并且与整个铁磁参数范围内的 Curie-Weiss 模型的波动相同。我们给出一个反例来证明条件 $d_N\gg \sqrt{N}$ 是紧的，从某种意义上说，如果 $d_N\sim \sqrt{N}$ 除了在高温状态下之外，极限分布会发生变化。通过完善我们的论点，我们将高温状态下的普遍性扩展到 $d_N\gg N^{1/3}$。我们的结果得出了正则图、Erdős-Renyi 图（有向和无向）、随机块模型和稀疏正则图子上 Ising 模型中平均磁化强度的普遍涨落。事实上，我们的结果适用于具有非负项的一般矩阵，包括维格纳矩阵上的伊辛模型和块自旋伊辛模型。作为证明技术的副产品，我们获得了这些波动的 Berry-Esseen 界限、自旋平均值的指数浓度以及配分函数的平均场近似的严格误差界限。
In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately $d_N$ regular graph $G_N$ on $N$ vertices. In particular, if $G_N$ is \enquote{well connected}, we show that whenever $d_N\gg \sqrt{N}$, the fluctuations are universal and same as that of the Curie-Weiss model in the entire Ferro-magnetic parameter regime. We give a counterexample to demonstrate that the condition $d_N\gg \sqrt{N}$ is tight, in the sense that the limiting distribution changes if $d_N\sim \sqrt{N}$ except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to $d_N\gg N^{1/3}$. Our results conclude universal fluctuations of the average magnetization in Ising models on regular graphs, Erdős-Renyi graphs (directed and undirected), stochastic block models, and sparse regular graphons. In fact, our results apply to general matrices with non-negative entries, including Ising models on a Wigner matrix, and the block spin Ising model. As a by-product of our proof technique, we obtain Berry-Esseen bounds for these fluctuations, exponential concentration for the average of spins, and tight error bounds for the Mean-Field approximation of the partition function.
DOI：
10.1214/17-aos1612
发表时间：
2016-11
期刊：
The Annals of Statistics
影响因子：
--
作者：
R. Mukherjee;S. Mukherjee;Ming Yuan
通讯作者：
R. Mukherjee;S. Mukherjee;Ming Yuan
影响因子：
4.5
作者：
Berthet, Quentin;Rigollet, Philippe;Srivastava, Piyush
通讯作者：
Srivastava, Piyush