Spectral gap for random-to-random shuffling on linear extensions

TL;DR

This paper introduces a new Markov chain for sampling linear extensions of posets, extending random-to-random shuffling, with conjectured bounds on eigenvalues and mixing times, potentially improving sampling efficiency.

Contribution

It generalizes random-to-random shuffling to linear extensions of posets and provides conjectured bounds on spectral gap and mixing time.

Findings

Eigenvalue bound conjecture: (1+1/n)(1-2/n)

Relaxation time bounded by n^2/(n+2)

Mixing time conjectured to be O(n log n)

Abstract

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.