Abstract
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.
Highlights
1.1 Aldous’s conjectureThis subsection is taken from David Aldous’s web page
The interchange process is reversible, and its stationary distribution is uniform on all n! configurations
The red particles in the interchange process behave as the usual exclusion process
Summary
This subsection is taken (with minor alterations) from David Aldous’s web page. Consider an n-vertex graph G which is connected and undirected. There is one particle at each vertex. The interchange process is the following continuous-time Markov chain on configurations. For each edge (i, j), at rate 1 the particles at vertex i and vertex j are interchanged. The interchange process is reversible, and its stationary distribution is uniform on all n! There is a spectral gap λIP(G) > 0, which is the absolute value of the largest non-zero eigenvalue of the transition rate matrix. If instead we just watch a single particle, it performs a continuous-time random walk on G (hereafter referred to as “the continuoustime random walk on G”), which is reversible and has a spectral gap λRW(G) > 0. Simple arguments (the contraction principle) show λIP(G) ≤ λRW(G).
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