The probability of long cycles in interchange processes

TL;DR

This paper analyzes the formation of long cycles in interchange processes by expressing cycle counts as characters of irreducible representations, providing formulas for probabilities of long cycles and estimates for shorter ones.

Contribution

It introduces a novel character-based approach to study cycle formation in interchange processes, including explicit probability formulas for long cycles.

Findings

Derived a formula for the probability of a permutation being one long cycle at time t

Provided estimates for the occurrence of shorter cycles

Linked cycle structure to irreducible representation characters

Abstract

We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.