Cycle structure of the interchange process and representation theory

TL;DR

This paper uses representation theory to analyze the cycle structure of the random transposition process, revealing precise timing for the emergence of large cycles and connecting it to the phase transition of the associated random graph.

Contribution

It provides an exact formula for cycle counts using representation theory, linking cycle emergence to the giant component transition in random graphs, and offers new proofs of known theorems.

Findings

Expected cycle counts jump sharply at the phase transition

Giant cycles emerge simultaneously with the giant component

Transition window for cycle emergence is very narrow

Abstract

Consider the process of random transpositions on the complete graph. We use representation theory to give an exact, simple formula for the expected number of cycles of size k at time t, in terms of an incomplete Beta function. Using this we show that the expected number of cycles of size k jumps from 0 to its equilibrium value, 1/k, at the time where the giant component of the associated random graph first exceeds k. Consequently we deduce a new and simple proof of Schramm's theorem on random transpositions, that giant cycles emerge at the same time as the giant component in the random graph. We also calculate the "window" for this transition and find that it is quite thin. Finally, we give a new proof of a result by the first author and Durrett that the random transposition process exhibits a certain slowdown transition. The proof makes use of a recent formula for the character decomposition of the number of cycles of a given size in a permutation, and the Frobenius formula for the character ratios.