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arXiv · 2601.21195

$q$-deformations of the Tsetlin library

Arvind Ayyer·Sarah Brauner·Jan de Gier·Anne Schilling
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Abstract

The Tsetlin library is a random shuffling process on permutations of $n$ letters, where each letter $i$ can be interpreted as a book; book $i$ is brought to the front of the bookshelf with an assigned probability $x_i$. We define a $q$-deformation of the Tsetlin library by replacing the symmetric group action on permutations by the action of the type $A$ Iwahori-Hecke algebra. We compute the stationary distribution and spectrum of this Markov chain by relating it to a Markov chain on complete flags over the finite field vector space $\mathbb{F}_q^n$ and applying techniques from semigroup theory. We prove that for a natural choice of $x_i$ the total variation distance mixing time of the $q$-Tsetlin library on permutations of $n$ is $O(n)$ compared to $\Theta(n \log n)$ for the Tsetlin library at $q=1$, which demonstrates a phase transition. We also generalize the $q$-Tsetlin library to words (with repeated letters), and compute its stationary distribution and spectrum.

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Arvind Ayyer, Sarah Brauner, Jan de Gier, Anne Schilling. 2026-01-29. $q$-deformations of the Tsetlin library. https://arxiv.org/abs/2601.21195

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