# The interchange process with reversals on the complete graph

## TL;DR

This paper studies an extended interchange process on the complete graph that includes reversals, demonstrating convergence of cycle sizes to a Poisson-Dirichlet distribution, which advances understanding of related stochastic models in physics.

## Contribution

It introduces a generalized interchange process with reversals and proves convergence to PD(1/2), extending Schramm's results for the standard process.

## Key findings

- Cycle sizes converge to PD(1/2) distribution
- Results apply above the critical point for macroscopic cycles
- Extends known convergence results to a new model

## Abstract

We consider an extension of the interchange process on the complete graph, in which a fraction of the transpositions are replaced by `reversals'. The model is motivated by statistical physics, where it plays a role in stochastic representations of $XXZ$-models. We prove convergence to PD($\tfrac12$) of the rescaled cycle sizes, above the critical point for the appearance of macroscopic cycles. This extends a result of Schramm on convergence to PD(1) for the usual interchange process.