Asymptotics of the Spectral Gap for the Interchange Process on Large Hypercubes

We consider the interchange process (IP) on the $d$-dimensional, discrete hypercube of side-length $n$. Specifically, we compare the spectral gap of the IP to the spectral gap of the random walk (RW) on the same graph. We prove that the two spectral gaps are asymptotically equivalent, in the limit $n \to \infty$. This result gives further supporting evidence for a conjecture of Aldous, that the spectral gap of the IP equals the spectral gap of the RW on all finite graphs. Our proof is based on an argument invented by Handjani and Jungreis, who proved Aldous's conjecture for all trees. This also has implications for the spectral gap of the quantum Heisenberg ferromagnet.