An estimate of the spectral gap for zero-range-exclusion dynamics (Q1769152)

The authors consider a Markovian system of particles carrying energy in \(\mathbb{Z}^d\). The dynamics is a combination of an exclusion process for the particles and a zero-range process for their energy. The process has two conserved quantities, the number of particles and the total energy. In this paper an estimate of the spectral gap is derived. It is proved that for the process in a cube of width \(n\) the spectral gap is bounded below by \(Cn^{-2}\), where \(C\) is independent of \(n\), but depends on the two local conserved quantities, the number of particles per site and the energy per particle. The estimation of the spectral gap of similar models is commonly reduced to two steps: an estimate of the spectral gap of the corresponding mean field dynamics and a moving-particle lemma. For the present model the former is obtained by an adaptation of the methods of \textit{C. Landim, S. Sethuraman} and \textit{S. Varadhan} [Ann. Probab. 24, No. 4, 1871--1902 (1996; Zbl 0870.60095)], but the moving-particle lemma is not so simple as for exclusion or zero-range processes and causes the dependence of the constant \(C\) on the conserved quantities.