This note discusses the spectral gap of the Fredrickson–Andersen one spin facilitated model in two different settings. The model describes an interacting particle system on a graph, where each site is either occupied or empty; and a site may change its occupation when at least one of its neighbors is empty. We will first consider the model on the infinite lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^{d}$$\end{document}Zd, with density close to 1. The second result is on finite graphs, with density that grows with the size of the graph in a way that guarantees O(1) empty sites. In both models lower and upper bounds on the spectral gap were known, but in general did not match. The purpose of this paper is to present new upper bounds that have the same asymptotics as the known lower bounds.

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关于 Fredrickson-Andersen One 自旋促进模型的光谱间隙的注释

本笔记讨论了 Fredrickson-Andersen 单自旋促进模型在两种不同设置下的光谱间隙。该模型在图形上描述了一个相互作用的粒子系统，其中每个站点要么被占用要么是空的；当至少一个邻居是空的时，一个站点可能会改变它的职业。我们将首先考虑无限格上的模型 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage {upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^{d}$$\end{document}Zd，密度接近 1。第二个结果是在有限图上，密度随着图的大小而增长，以保证 O(1) 空位点的方式。在两个模型中，光谱间隙的下限和上限都是已知的，但总的来说不匹配。本文的目的是提出与已知下界具有相同渐近线的新上界。