Odd-degree spectral-gap ordering conjecture

Prove or disprove the odd-degree spectral-gap ordering \(\lambda_1(\alpha,w)\le \lambda_{3,*}(\alpha,w)\le \lambda_{5,*}(\alpha,w)\le\cdots\) for homogeneous site weights \(\alpha\equiv1\) and arbitrary block weights \(w\).

Background

The paper proves an ordering of the pure-degree gaps along the even degrees, thereby resolving the even-degree part of a conjecture of Alon and Puder in the homogeneous case. The corresponding odd-degree ordering is stated for the one-, three-, five-, and higher odd-particle sectors.

The authors neither prove nor disprove the odd-degree claim in general and then show that it fails for certain nonhomogeneous site weights. Thus, the explicitly unresolved version is the conjecture for α≡1\alpha\equiv1 and arbitrary block weights.

References

In the same conjecture, Alon and Puder also propose the odd-index analogue _1(\alpha,w)\le _{3,\ast}(\alpha,w)\le _{5,\ast}(\alpha,w)\le \ldots, for $\alpha \equiv 1$ and arbitrary block weights $w$. While our results neither prove or disprove their claim, it is not difficult to see that eq:odd-chain cannot hold for general weights.

eq:odd-chain:

1(α,w)≤3,∗(α,w)≤5,∗(α,w)≤…,_1(\alpha,w)\le _{3,\ast}(\alpha,w)\le _{5,\ast}(\alpha,w)\le \ldots,

— Aldous' spectral gap phenomena in stochastic exchange models  (2609.10450 - Caputo et al., 9 Sep 2026) in Remark 2.?? (Remark \ref{rem:new-ordering})