Spectral gap for the interchange process in a box

We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree.Comment: 8 pages. I learned after completing a draft of this paper that its main result had recently been obtained by Starr and Conomo

Spectral gap for the zero range process with constant rate

We solve an open problem concerning the relaxation time (inverse spectral gap) of the zero range process in $\mathbf {Z}^d/L\mathbf {Z}^d$ with constant rate, proving a tight upper bound of $O((\rho +1)^2L^2)$, where $\rho$ is the density of particles.Comment: Published at http://dx.doi.org/10.1214/009117906000000304 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The mixing time for simple exclusion

We obtain a tight bound of $O(L^2\log k)$ for the mixing time of the exclusion process in $\mathbf{Z}^d/L\mathbf{Z}^d$ with $k\leq{1/2}L^d$ particles. Previously the best bound, based on the log Sobolev constant determined by Yau, was not tight for small $k$. When dependence on the dimension $d$ is considered, our bounds are an improvement for all $k$. We also get bounds for the relaxation time that are lower order in $d$ than previous estimates: our bound of $O(L^2\log d)$ improves on the earlier bound $O(L^2d)$ obtained by Quastel. Our proof is based on an auxiliary Markov chain we call the chameleon process, which may be of independent interest.Comment: Published at http://dx.doi.org/10.1214/105051605000000728 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The mixing time of the Thorp shuffle

The Thorp shuffle is defined as follows. Cut the deck into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip; then drop from the other pile. Continue this way until both piles are empty. We show that the mixing time for the Thorp shuffle with $2^d$ cards is polynomial in $d$.Comment: 21 page

Improved bounds for the mixing time of the random-to-random insertion shuffle

We prove an upper bound of $1.5324 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling.Comment: 7 pages, 1 figur

The mixing time of the fifteen puzzle

We show that there are universal positive constants c and C such that the mixing time T_{mix} for the fifteen puzzle in an n by n torus satisfies cn^4 log n < T_{mix} < Cn^4 log^2 n.Comment: 30 pages, 9 figure