Spectral Gap for Random-to-Random Shuffling on Linear Extensions

Arvind Ayyer, Anne Schilling, Nicolas M. Thiéry
2016 Experimental Mathematics  
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 n).
more » ... We conjecture that the mixing time is in fact O(n n) as for the usual random-to-random shuffling.
doi:10.1080/10586458.2015.1107868 fatcat:imprvcasnnbshghdg3nkngp6ie