Spectral Gap for the Binary Fixed-Margin Swap Chain

2026-06-21

AI Summary

This work resolves a long-standing open problem concerning whether the lazy swap Markov chain on binary matrices with prescribed row and column sums exhibits a polynomial spectral gap—a question central to the rapid mixing of null-model samplers in ecology, statistics, and related fields. By reducing the general $m \times n$ case to the three-row setting and combining a comparison with a two-row heat-bath chain, analysis of column-count variables, and a harmonic decomposition over Johnson graphs, the authors establish—for the first time—a tight inverse-polynomial lower bound on the spectral gap of $\Omega\big((\binom{m}{2}\binom{n}{2})^{-1}\big)$ for all feasible margins. This result confirms the rapid mixing conjecture of Kannan–Tetali–Vempala (1997) and establishes that the lazy swap chain mixes rapidly under any feasible marginal constraints.

Abstract

We prove an inverse-polynomial spectral-gap bound for the lazy swap chain on binary matrices with prescribed row and column sums. This chain is a standard sampler for fixed-margin null models in ecology, statistics, and network analysis, and its rapid mixing for arbitrary feasible margins was conjectured by Kannan, Tetali, and Vempala in 1997. We show that for every feasible set of margins on an $m\times n$ binary matrix, the lazy swap chain has spectral gap at least $$ \binom{m}{2}^{-1}\binom{n}{2}^{-1}, $$ which is tight in the worst case. The proof compares the swap chain with a two-row heat-bath chain, reduces the analysis from arbitrary $m\times n$ matrices to the case of three rows, and proves the resulting three-row inequality by decomposing functions according to the column-count variable and the associated Johnson harmonic sectors. The proof itself was generated by ChatGPT 5.5 Pro. ChatGPT proposed the whole proof strategy, including the comparison with the two-row heat-bath chain, the reduction to the three-row case, and the decomposition of the three-row function space into the count sector and the Johnson harmonic sectors. It also generated all the technical lemmas and initial proofs. The author's role was to pose the problem, guide the search direction, evaluate the AI-generated arguments, rewrite the proof, and take responsibility for the final form and validity of the result.

Problem

• spectral gap
• swap chain
• binary matrices
• fixed margins
• rapid mixing

Innovation

• spectral gap
• lazy swap chain
• fixed-margin binary matrices
• rapid mixing
• Johnson harmonic decomposition