On a model of random cycles

TL;DR

This paper investigates a model of random permutations on a cubic lattice, providing numerical evidence for a phase transition to a state with infinite, macroscopic cycles, influenced by site adjacency preferences.

Contribution

It introduces a weighted permutation model on a cubic lattice and demonstrates a phase transition to macroscopic cycles through numerical analysis.

Findings

Evidence of a phase transition to infinite cycles

Macroscopic cycles emerge under certain weighting conditions

Numerical simulations support the phase transition hypothesis

Abstract

We consider a model of random permutations of the sites of the cubic lattice. Permutations are weighted so that sites are preferably sent onto neighbors. We present numerical evidence for the occurrence of a transition to a phase with infinite, macroscopic cycles.