Two-thirds spectral comparison conjecture

Prove that for every pair of nontrivial partitions \(\tau,\sigma\vdash n\) satisfying \(\sigma\trianglerighteq\tau\), the optimal comparison constant \(C(\tau,\sigma)\), defined as the infimum over weighted graphs \(G\) of \(\lambda_{\min}(G,\tau)/\lambda_{\min}(G,\sigma)\) whenever the denominator is positive, satisfies \(C(\tau,\sigma)\geq 2/3\), equivalently \(\lambda_{\min}(G,\tau)\geq(2/3)\lambda_{\min}(G,\sigma)\) for every weighted graph \(G\).

Background

The paper characterizes precisely when the minimum eigenvalues of weighted-graph interchange-process Laplacians on two Specht modules admit a uniform positive multiplicative comparison: such a comparison exists exactly when the corresponding partitions are ordered by dominance. It proves the general lower bound $C(\tau,\sigma)\geq n^{-3}$.

The unresolved conjecture asserts the substantially stronger dimension-free bound $2/3$. The paper verifies this bound for complete and quasi-complete graphs and for cases covered by the Aldous order, and proves that $2/3$ would be sharp using an explicit family of star graphs. The conjecture is also used to derive a conjectural optimal-order mixing-time bound for the interchange process.