Source: Ann. Appl. Probab. Volume 13, Number 2
(2003), 691-721.
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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