Record-dependent measures on the symmetric groups

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.

23 pages. v2: minor corrections, to appear in Random Structures and Algorithms

References in corpus (1)

• The problem of harmonic analysis on the infinite-dimensional unitary group