Spatial random permutations with small cycle weights

TL;DR

This paper studies two related models of random permutations with cycle weights, demonstrating the emergence of infinite macroscopic cycles beyond a critical density threshold.

Contribution

It introduces and analyzes two models of weighted random permutations, establishing conditions for the appearance of infinite cycles.

Findings

Infinite macroscopic cycles occur above a critical density

Cycle weights influence the cycle structure in permutations

Models relate to spatial permutations with jump lengths

Abstract

We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps. We prove the occurrence of infinite macroscopic cycles above a certain critical density.