Random walks on quasisymmetric functions

TL;DR

This paper explores how certain endomorphisms on quasisymmetric functions induce random walks on the descent algebra and permutations, providing spectral analysis and connecting to well-known shuffling methods.

Contribution

It introduces conditions under which endomorphisms generate random walks on the descent algebra, linking them to permutation shuffles and spectral properties, and proves a conjecture on spectra.

Findings

Realized several known random walks via endomorphisms on quasisymmetric functions

Proved a conjecture regarding spectra of random walks on ab-words

Recovered a theorem of Stembridge related to enriched P-partitions

Abstract

Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several well-studied random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the a-th convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case.