论文标题
平均场模型的波动
Fluctuations in Mean-Field Ising models
论文作者
论文摘要
在本文中，我们研究了Ising模型中平均磁化的波动，大约$ d_n $常规图$ g_n $在$ n $ vertices上。特别是，如果$ g_n $是\ enquote {良好连接}，我们表明，每当$ d_n \ gg \ sqrt {n} $中，波动是通用的，并且与整个Ferro-Magnetic参数romeTime中的Curie-Weiss模型相同。我们给出一个反例，以证明条件$ d_n \ gg \ sqrt {n} $很紧，因为如果$ d_n \ sim \ sim \ sqrt {n} $在高温制度中，限制分布会改变。通过完善我们的论点，我们将高温制度中的通用性扩展到$ d_n \ gg n^{1/3} $。我们的结果得出了常规图，Erdős-rényi图（定向和无向导），随机块模型和稀疏的常规图形的常规图模型中平均磁化模型中平均磁化的通用波动。实际上，我们的结果适用于具有非阴性条目的一般矩阵，包括Wigner矩阵上的ISING模型和块旋转ISING模型。作为我们证明技术的一种副产品，我们获得了这些波动的浆果 - 埃斯尼界限，平均旋转的指数浓度以及分区函数的平均场近似值的紧密误差界限。
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-Rényi 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.
