Fill’s eigenvalue-multiplicity conjecture for adjacent-transposition chains
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For a regular adjacent-transposition Markov chain on S_n, n≥3, what is the algebraic multiplicity of its second-largest eigenvalue? Fill’s conjecture relates it to the number N of neutral labels, with exceptional values when N=n−2 or N=n.
Why it matters and what counts as progress
Why it matters: Multiplicity describes the dimension of a slow eigenspace, information not determined by the second eigenvalue alone.
What counts as progress: Prove multiplicity N outside the exceptions and n−1 at the exceptions, or extend the spectral description to broader transposition chains.
How the problem developed
- 2003 · Conjecture [1]
James Allen Fill In unpublished notes, conjectured that at a minimizing regular chain the second-largest eigenvalue multiplicity is the number of neutral labels, except when that number is n or n−2, when it is n−1.
- 27 Mar 2026 · Theorem [2]
Gary Greaves and Haoran Zhu Resolved Fill’s spectral-gap conjecture and characterized regular chains attaining the minimum gap.
- 5 Apr 2026 · Theorem [3]
Vishesh Jain and Clayton Mizgerd Proved Fill’s exact multiplicity formula for the second-largest eigenvalue of every minimizing regular chain.
AI contribution
Equality in Fill's spectral gap problem
For regular adjacent-transposition chains on S_n with n≥3, the second-largest eigenvalue has algebraic multiplicity M equal to the number N of neutral labels, except when N=n−2 or n, when M=n−1.
Sources
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Publication dates and editorial changes.
ChatGPT proof settles Fill’s eigenvalue multiplicity conjecture