Equality in Fill's spectral gap problem
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915916519833600 |
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| author | Jain, Vishesh Mizgerd, Clayton |
| author_facet | Jain, Vishesh Mizgerd, Clayton |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03937 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equality in Fill's spectral gap problem Jain, Vishesh Mizgerd, Clayton Combinatorics Probability 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. |
| title | Equality in Fill's spectral gap problem |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2604.03937 |