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2018 A random walk on the symmetric group generated by random involutions
Megan Bernstein
Electron. J. Probab. 23: 1-28 (2018). DOI: 10.1214/18-EJP140


The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this paper to mix for $\frac{1} {2} \leq p \leq 1$ fixed, $n$ sufficiently large in between $\log _{1/p}(n)$ steps and $\log _{2/(1+p)}(n)$ steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. Monotonicity relations used in the bound also give after sufficient time the likelihood order, the asymptotic order from most likely to least likely permutation. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of black holes.


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Megan Bernstein. "A random walk on the symmetric group generated by random involutions." Electron. J. Probab. 23 1 - 28, 2018.


Received: 17 December 2016; Accepted: 9 January 2018; Published: 2018
First available in Project Euclid: 15 March 2018

zbMATH: 1391.60168
MathSciNet: MR3779819
Digital Object Identifier: 10.1214/18-EJP140

Primary: 60J10
Secondary: 05E10


Vol.23 • 2018
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