Oliveira’s interchange-process mixing-time conjecture

Prove that there exists a universal constant \(A\) such that every connected weighted graph \(G\) on \(n\geq 2\) vertices has interchange-process mixing time \(t_{\mathrm{mix}}(G)\leq A(\log n)/\gamma\), where \(\gamma=\lambda_{\min}(G,[n-1,1])\) is the ordinary spectral gap.

Background

For the continuous-time interchange process, the standard spectral estimate gives the unconditional upper bound $t_{\mathrm{mix}}(G)=O(n\log n/\gamma)$. Oliveira’s conjecture seeks to remove the factor $n$, yielding a logarithmic dependence on the number of vertices.

The paper shows that the two-thirds spectral comparison conjecture would imply this mixing-time conjecture. Without assuming it, the paper proves the weaker unconditional bound $t_{\mathrm{mix}}(G)=O((\log n)^2/\gamma)$.