## Spectral gap for the interchange process in a box.(English)Zbl 1189.60180

Summary: 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 S. Handjani and D. Jungreis [J. Theor. Probab. 9, No. 4, 983–993 (1996; Zbl 0878.60043)], who proved that Aldous’s conjecture holds when the graph is a tree.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J27 Continuous-time Markov processes on discrete state spaces

### Keywords:

spectral gap; interchange process

Zbl 0878.60043
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