Title: The Uniform Even Subgraph and Its Connection to Phase Transitions of Graphical Representations of the Ising Model Show Chinese title

Title: 一致偶子图及其与伊辛模型图表示相变的关系

Abstract: The uniform even subgraph is intimately related to the Ising model, the random-cluster model, the random current model, and the loop $\mathrm{O}$(1) model. In this paper, we first prove that the uniform even subgraph of $Z^d$ percolates for $d \geq 2$ using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop $\mathrm{O}$(1) model on $Z^d$ percolates for $d \geq 2$ for edge-weights $x$ lying in some interval $(1-\varepsilon,1]$. Finally, our main theorem is that the loop $\mathrm{O}$(1) model and random current models corresponding to a supercritical Ising model are always at least critical, in the sense that their two-point correlation functions decay at most polynomially and the expected cluster sizes are infinite. Show Chinese abstract

Abstract: 均匀偶子图与伊辛模型、随机簇模型、随机电流模型以及环路模型$\mathrm{O}$(1) 密切相关。 本文首先利用均匀偶子图作为偶图群上的哈尔测度的特征，证明了当$Z^d$发生渗流时，均匀偶子图在$d \geq 2$下也发生渗流。 我们通过证明当边权 $x$ 属于某个区间 $(1-\varepsilon,1]$ 时，$Z^d$ 上的 \($\mathrm{O}$(1) \) 模型在 $d \geq 2$ 下渗透，从而进一步加强了这一结果。最后，我们的主要定理是：临界 Ising 模型对应的 \($\mathrm{O}$(1) \) 模型和随机电流模型总是至少处于临界状态，这意味着它们的两点关联函数最多以多项式速率衰减，并且预期聚类大小为无穷大。