Exponential Localization of Spatial Random Permutations in One Dimension

TL;DR

This paper studies the localization properties of spatial random permutations in one dimension, demonstrating that cycles are localized on a scale proportional to W^3 with exponential decay, and conjecturing a scale of W^2.

Contribution

It provides a rigorous analysis of cycle localization in 1D random permutations and proposes a conjecture relating to eigenfunction localization in random band matrices.

Findings

Cycles are localized on the scale W^3 with exponential tail bounds.

Conjecture: cycles are localized on the scale W^2, similar to eigenfunctions of 1D random band matrices.

Abstract

We consider a class of random permutations of the interval $[-n,n]$, in which points are typically displaced a distance $O(W)$. We show the cycles are localized on the scale $W^3$, with an exponentially decaying tail bound. Analogous to eigenfunctions of one dimensional random band matrices, the cycles are conjectured to be localized to the scale $W^2.$