Spherically Symmetric Random Permutations

TL;DR

This paper studies spherically symmetric random permutations under various metrics, identifying the extreme cases for infinite processes using a unified stochastic monotonicity approach.

Contribution

It characterizes the extreme infinitely spherically symmetric permutation processes for Hamming, Kendall-tau, and Caley metrics, providing a unified proof method.

Findings

Identified extreme permutation processes for three metrics.

Unified approach via stochastic monotonicity.

Applicable to infinite permutation processes.

Abstract

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group $S_n$ and are consistent as $n$ varies. The extreme infinitely spherically symmetric permutation-valued processes are identified for the Hamming, Kendall-tau and Caley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.