Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models

• 成果类型: Article
• 署名作者: Caputo, P; Martinelli, F
• 署名单位: Roma Tre University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2003
• 页码: 691-721
• 关键词: asymmetric diffusion; ground-state; spectral gap; xxz model

摘要

Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with N particles in a rectangle of Z(2). Every particle at row h tries to jump to an arbitrary empty site at row h +/- 1 with rate q(+1), where q is an element of (0, 1) is a measure of the drift driving the particles toward the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in N and in the size of the rectangle. The proof is inspired by a recent interesting technique envisioned by E. Carlen, M. C. Carvalho and M. Loss to analyze the Kac model for the nonlinear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin-S, S is an element of (1)/N-2, XXZ chain and for the 111 interface of the spin-S XXZ Heisenberg model, thus generalizing previous results valid only for spin (1)/(2).