论文标题
常规图上的随机群集模型
Random cluster model on regular graphs
论文作者
论文摘要
对于图形$ g =（v，e）$带有$ v（g）$ pertices $ z_g（q，q，w）= \ sum_ {a \ subseteq e（g）} q^{k（a）} w^{| a | a | a | a | a | a | a $ k（a $ k（a）$ norge a contection $ norge a concontect此外，令$ g（g）$表示图$ g $的围绕最短周期的长度。 在本文中，我们表明，如果$（g_n）_n $是$ d $的序列序列，使girth $ g（g_n）\ to \ infty $，则限制$ \ lim_ \ lim_ {n \ to \ infty} \ frac} \ frac {1}} {v（g_n）} {v（g_n）} {如果$ q \ geq 2 $和$ w \ geq 0 $，则存在φ_{d，q，w} $$。数量$φ_{d，q，w} $可以如下计算。让$φ_{d，q，w}（t）：= \ left（\ sqrt {1+ \ frac {w} {q}}}} \ cos（t）+\ sqrt {\ sqrt {\ frac {（q-1） ^{d}+（q-1）\ left（\ sqrt {1+ \ frac {w} {q}}}} \ cos（t） - \ sqrt {\ frac {w} {q（q-1）{q（q-1）}}}}}} \ sin（t）\ sin（t）\ right）然后$$φ_{d，q，w}：= \ max_ {t \ in [-π，π]}φ_{d，q，w}（t），$$，对于随机$ d $ d $ rightigarch的顺序，相同的结论也是如此，概率为一。 我们的结果将Dembo，Montanari，Sly和Sun的工作扩展了Potts模型（Integer $ Q $），我们证明了Helmuth，Jenssen和Perkins的猜想，内容涉及使用固定$ Q $的随机群集模型的相变。
For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum_{A\subseteq E(G)}q^{k(A)}w^{|A|},$$ where $k(A)$ denotes the number of connected components of the graph $(V,A)$. Furthermore, let $g(G)$ denote the girth of the graph $G$, that is, the length of the shortest cycle. In this paper we show that if $(G_n)_n$ is a sequence of $d$-regular graphs such that the girth $g(G_n)\to \infty$, then the limit $$\lim_{n\to \infty} \frac{1}{v(G_n)}\ln Z_{G_n}(q,w)=\ln Φ_{d,q,w}$$ exists if $q\geq 2$ and $w\geq 0$. The quantity $Φ_{d,q,w}$ can be computed as follows. Let $$Φ_{d,q,w}(t):=\left(\sqrt{1+\frac{w}{q}}\cos(t)+\sqrt{\frac{(q-1)w}{q}}\sin(t)\right)^{d}+(q-1)\left(\sqrt{1+\frac{w}{q}}\cos(t)-\sqrt{\frac{w}{q(q-1)}}\sin(t)\right)^{d},$$ then $$Φ_{d,q,w}:=\max_{t\in [-π,π]}Φ_{d,q,w}(t),$$ The same conclusion holds true for a sequence of random $d$-regular graphs with probability one. Our result extends the work of Dembo, Montanari, Sly and Sun for the Potts model (integer $q$), and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed $q$.
