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May 2003 Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models
Pietro Caputo, Fabio Martinelli
Ann. Appl. Probab. 13(2): 691-721 (May 2003). DOI: 10.1214/aoap/1050689600

Abstract

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 $\bbZ^2$. Every particle at row $h$ tries to jump to an arbitrary empty site at row $h\pm 1$ with rate $q^{+ 1}$, where $q\in (0,1)$ 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-$S$, $S\in \ov2\bbN$, $\mbox{\textit{{XXZ}}}$ chain and for the 111 interface of the spin-$S$ $\mbox{\textit{{XXZ}}}$\vspace{-1pt} Heisenberg model, thus generalizing previous results valid only for spin $\ov2$.

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Pietro Caputo. Fabio Martinelli. "Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models." Ann. Appl. Probab. 13 (2) 691 - 721, May 2003. https://doi.org/10.1214/aoap/1050689600

Information

Published: May 2003
First available in Project Euclid: 18 April 2003

zbMATH: 1069.82002
MathSciNet: MR1970283
Digital Object Identifier: 10.1214/aoap/1050689600

Subjects:
Primary: 60J27 , 60K35 , 60K40 , 82B10 , 82B20

Keywords: $XXZ$ model , Asymmetric simple exclusion , diffusion limited chemical reactions , equivalence of ensembles , spectral gap

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.13 • No. 2 • May 2003
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