Equality in Fill's spectral gap problem

Abstract: We study the adjacent-transposition chain on the symmetric group $\mathfrak{S}n$ with a regular parameter vector $\vec{p} = (p{i,j}){i\neq j}$. Fill's spectral gap conjecture, recently resolved in the affirmative by Greaves-Zhu, states that among all regular parameter vectors, the spectral gap of the transition matrix is minimized by the uniform vector $p{i,j}= 1/2$ for all $i\neq j$. We prove the stronger statement that among all regular parameter vectors, the spectral gap is minimized if and only if $\vec{p}$ has a neutral label, i.e., there exists $c \in [n]$ such that $p_{c,i} = 1/2$ for all $i\neq c$. Moreover, in this case, we show that the multiplicity of the second largest eigenvalue is equal to the number of neutral labels, unless the number of neutral labels is $n-2$ or $n$, in which case the multiplicity is $n-1$. This confirms a conjecture of Fill.

• The paper proves that only neutral label configurations yield the minimal spectral gap in adjacent-transposition Markov chains.
• It employs quadratic form comparisons and local subgroup orbit analysis to establish rigid equality cases.
• The study details how the number of neutral labels governs the eigenvalue multiplicity in the transition matrix.

Equality in Fill's Spectral Gap Problem: Summary and Analysis

Introduction and Context

This paper addresses the equality cases within Fill's spectral gap problem for the adjacent-transposition Markov chain on the symmetric group $\mathfrak{S}_n$ parametrized by a regular vector $\vec{p}$, where each $p_{i,j}$ describes the transition probabilities between adjacent elements. The original conjecture by Fill posited that the spectral gap of the transition matrix attains its minimum for the uniform parameter vector, i.e., $p_{i,j} = 1/2$ for $i \neq j$. This conjecture was recently verified by Greaves and Zhu (Greaves et al., 27 Mar 2026). The present work rigorously classifies all parameterizations that achieve this minimal spectral gap, fully characterizing both the minimizers and the multiplicity structure of the second largest eigenvalue.

Problem Formulation and Background

Given the Markov chain defined by adjacent transpositions and a regular parameter vector $\vec{p}$ (subject to monotonicity and lower bound constraints), the spectral gap $\lambda_K$ is defined as $1-\beta_K$, where $\beta_K$ is the second largest eigenvalue of the transition matrix $K$. The uniform case $\vec{p}$0 corresponds to the classic adjacent-transposition shuffle with known gap $\vec{p}$1.

Prior to this work, Fill formulated several conjectures on the spectral gap structure:

• The minimal spectral gap among all regular $\vec{p}$2 is achieved only when all transitions are equally likely ("the gap conjecture").
• A stronger conjecture specified that the spectral gap is minimized if and only if $\vec{p}$3 admits a "neutral label": that is, some $\vec{p}$4 with $\vec{p}$5 for all $\vec{p}$6.
• A further multiplicity conjecture described precisely how the number of neutral labels governs the multiplicity of the second largest eigenvalue.

Main Contributions

Characterization of Minimizing Parameter Vectors

The core result (Theorem) demonstrates that for regular parameter vectors, the spectral gap equals $\vec{p}$7 if and only if the parameter vector contains at least one neutral label. This completely settles the question of which regular Markov chains have the same mixing characteristics as the uniform chain. Formally,

$\vec{p}$8

where $\vec{p}$9 counts neutral labels.

Multiplicity Structure

The paper proves that the algebraic multiplicity $p_{i,j}$0 of the second largest eigenvalue is governed by $p_{i,j}$1 as follows:

• If $p_{i,j}$2, then $p_{i,j}$3,
• If $p_{i,j}$4, then $p_{i,j}$5.

Thus, in all but the exceptional parameter counts, each neutral label provides a distinct eigenfunction—these correspond to Wilson's "single-card eigenfunctions" in the shuffling literature [Wilson04].

Rigidity Analysis of the Minimizing Case

The proof employs the quadratic form comparison framework of Greaves--Zhu, but with enhancements to classify the strict equality cases. If $p_{i,j}$6 and $p_{i,j}$7, the equality structure forces significant rigidity in the eigenvector $p_{i,j}$8:

• The induced vector $p_{i,j}$9 (projection on transpositions at position 1) is uniquely parameterized by all unordered pairs $p_{i,j} = 1/2$0, but for nonzero entries, the regularity and the equality constraints force the existence of neutral labels.
• The analysis of orbits under the local subgroup $p_{i,j} = 1/2$1 restricts nonzero coefficients to those involving neutral labels and imposes linear dependencies among them.
• Further combinatorial structure deduced from regularity conditions propagates these neutrality constraints, ensuring the entire family of transition probabilities associated with a neutral label must be $p_{i,j} = 1/2$2.

Explicit Structure for Neutral Intervals

The paper determines that neutral labels must occur as a consecutive block (interval). Non-neutral labels off this interval must have transition probabilities strictly bounded away from $p_{i,j} = 1/2$3 in certain directions, enforcing the strong dichotomy between neutral and non-neutral regimes.

Moreover, the algebraic analysis reveals that aside from the uniform case, the only nonzero components of the eigenfunctions are associated with "crossing pairs" (indices straddling the neutral interval), further constraining the possible structure and supporting the multiplicity count.

Construction of All Minimizing Eigenfunctions

All extremal eigenvectors are explicitly constructed. For each neutral label, the classic coordinate function $p_{i,j} = 1/2$4 is an eigenfunction with the minimal eigenvalue. In the exceptional case $p_{i,j} = 1/2$5, an additional eigenfunction involving the relative positions of the two non-neutral labels is constructed (this explicit function was checked for algebraic independence against the per-label eigenfunctions).

Strong Claims and Contradictory Elements

A bold claim of the work is the if and only if characterization: for regular parameter vectors, no other configuration than the presence of a neutral label can minimize the spectral gap. This sharply contradicts any conjecture that even marginal deviation from uniformity or partial symmetry could generate the same relaxation time. The full multiplicity statement provides a closure to any speculation about accidental spectrum multiplicity arising from special combinations of transition probabilities.

Implications and Future Directions

The rigorous resolution of the equality cases in Fill's spectral gap problem brings closure to a line of work connecting Markov chain spectral analysis, card shuffling, and algebraic combinatorics.

The key implications are:

• Mixing time universality: Only parameter regimes that are locally uniform for at least one label achieve optimal mixing speeds.
• Spectral control: Any attempt to "accelerate" the adjacent-transposition walk by non-uniform biasing of some local transitions necessarily worsens the relaxation time, unless it preserves neutrality.
• Multiplicity management: Control over neutral labels allows precise tuning of degeneracies in the spectrum, relevant to understanding cutoff phenomena and convergence rates in combinatorial Markov chains.

The methodology (combining detailed quadratic form analysis, local group orbit techniques, and combinatorial propagation) is likely extensible to other parameter-modified random walks and their extremal spectral behavior. An immediate future direction is the extension to other families of shuffling chains (e.g., more general Coxeter groups), or to non-reversible deformations, where similar rigidity phenomena might occur under appropriate symmetry or monotonicity constraints.

Conclusion

This work gives a complete solution to the equality structure in Fill's spectral gap problem, providing a definitive classification of minimizers and their spectral multiplicity for regular adjacent-transposition chains. The results demonstrate that any biasing of transition probabilities away from a neutral configuration precludes optimal mixing, and pin down the precise algebraic mechanism by which the spectrum is degenerate. These conclusions sharpen understanding of Markov chain mixing on symmetric groups and provide foundational clarity for the role of symmetry and neutrality in spectral optimization.