On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$
On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$
复制标题
关于 ${mathbb Z}^d$ 上硬核模型中的相变
DOI：
10.1017/s0963548303006035
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
J. Kahn
中科院分区：
文献类型：
--
作者：
David J. Galvin;J. Kahn
结果表明，${{\mathbb Z}}^d$上的硬核模型在某些函数$\lambda(d)$以上的活动处呈现相变，该函数在$d\rightarrow \infty$趋向于零。更准确地说，考虑${{\mathbb Z}}^d$上通常的最近邻图，并将偶数和奇数顶点的集合（以显而易见的方式定义）写成${\cal E}$和${\cal O}$。为${\cal G}L_M$中的独立集合（不跨越边的顶点集合）设置$${\cal G}L_M={\cal G}L_M^d =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}\leq M\},\quad \partial^{\star} {\cal G}L_M =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}= M\},$$并写入${\cal I}({\cal G}L_M)$。对于$\lambda>0$，请从${\cal I}({\cal G}L_M)$和$\Pr({\bf I}=I) \propto \lambda^{|I|}$中选择${\bf I}$。定理存在一个常数$C$，如果$\lambda > Cd^{-1/4}\log^{3/4}d$，那么$$\lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}|{\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal E})~> \lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}| {\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal O}).$$因此，粗略地说，边界对原点行为的影响随着边界的消退而持续存在。
It is shown that the hard-core model on ${{\mathbb Z}}^d$ exhibits a phase transition at activities above some function $\lambda(d)$ which tends to zero as $d\rightarrow \infty$. More precisely, consider the usual nearest neighbour graph on ${{\mathbb Z}}^d$, and write ${\cal E}$ and ${\cal O}$ for the sets of even and odd vertices (defined in the obvious way). Set $${\cal G}L_M={\cal G}L_M^d =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}\leq M\},\quad \partial^{\star} {\cal G}L_M =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}= M\},$$ and write ${\cal I}({\cal G}L_M)$ for the collection of independent sets (sets of vertices spanning no edges) in ${\cal G}L_M$. For $\lambda>0$ let ${\bf I}$ be chosen from ${\cal I}({\cal G}L_M)$ with $\Pr({\bf I}=I) \propto \lambda^{|I|}$. TheoremThere is a constant$C$such that if$\lambda > Cd^{-1/4}\log^{3/4}d$, then$$\lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}|{\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal E})~> \lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}| {\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal O}).$$ Thus, roughly speaking, the influence of the boundary on behaviour at the origin persists as the boundary recedes.