LAUSR

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.