Random walk on the symplectic forms over a finite field

TL;DR

This paper studies a Markov chain generated by random transvections on symplectic forms over finite fields, establishing a cutoff phenomenon and analyzing eigenvalues to understand its mixing time.

Contribution

It introduces a new analysis of the cutoff phenomenon for a walk on symplectic forms, extending results on matchings to a $q$-deformed setting.

Findings

Established cutoff at n+c steps for the walk.

Bounded eigenvalues to prove the upper bound.

Showed support is exponentially small before n-c steps.

Abstract

Random transvections generate a walk on the space of symplectic forms on $\mathbf{F}_q^{2n}$. The main result is establishing cutoff for this Markov chain. After $n+c$ steps, the walk is close to uniform while before $n-c$, it is far from uniform. The upper bound is proved by explicitly finding and bounding the eigenvalues of the random walk. The lower bound is found by showing that the support of the walk is exponentially small if only $n-c$ steps are taken. The result can be viewed as a $q$-deformation of a result of Diaconis and Holmes on a random walk on matchings.