The interchange process with reversals on the complete graph

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($\frac{1}{2}$) 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.

arXiv comment49 pages, journal version

References in corpus (8)

• Geometric Aspects of Quantum Spin States
• Random loop representations for quantum spin systems
• Infinite cycles in the random stirring model on trees
• The random interchange process on the hypercube
• Phase transition for the interchange and quantum Heisenberg models on the Hamming graph
• Phase transition for loop representations of Quantum spin systems on trees
• Critical parameter of random loop model on trees
• Critical temperature of Heisenberg models on regular trees, via random loops

Cited by in corpus (8)

• Phase transition for the interchange and quantum Heisenberg models on the Hamming graph
• Loop correlations in random wire models
• Split-and-Merge in Stationary Random Stirring on Lattice Torus
• Critical Parameters for Loop and Bernoulli Percolation
• Characterising random partitions by random colouring
• Cutoff for Rewiring Dynamics on Perfect Matchings
• On a class of orthogonal-invariant quantum spin systems on the complete graph
• Mixing of fast random walks on dynamic random permutations