Record-dependent measures on the symmetric groups

TL;DR

This paper characterizes the extreme points of record-dependent probability measures on symmetric groups, linking them to permutation growth processes, partial orders, and measures on the Young-Fibonacci lattice.

Contribution

It provides a complete description of the extreme record-dependent measures on symmetric groups, connecting them to various combinatorial and probabilistic structures.

Findings

Characterization of extreme record-dependent measures

Connection to permutation growth processes

Relation to measures on the Young-Fibonacci lattice

Abstract

A probability measure $P_n$ on the symmetric group ${\mathfrak S}_n$ is said to be record-dependent if $P_n(\sigma)$ depends only on the set of records of a permutation $\sigma\in{\mathfrak S}_n$. A sequence $P=(P_n)_{n\in{\mathbb N}}$ of consistent record-dependent measures determines a random order on $\mathbb N$. In this paper we describe the extreme elements of the convex set of such $P$. This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice.