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Exponential Localization of Spatial Random Permutations in One Dimension

Reuben Drogin, Felipe Hernández

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 $W^2.$

Exponential Localization of Spatial Random Permutations in One Dimension

Abstract

We consider a class of random permutations of the interval , in which points are typically displaced a distance . We show the cycles are localized on the scale , 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
Paper Structure (5 sections, 3 theorems, 25 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Proof of Theorem \ref{['thm:main-boltzman']}
  4. Proof of Theorem 1 when $p=\infty$
  5. Proof of Theorem \ref{['thm:main-boltzman']} for $p\in [1,\infty)$.

Key Result

Theorem 1

For any $p\in [1,\infty]$, $\lambda, n,W\geq 1$ and $j\in [-n,n]$ we have where $c>0$ is uniform in all parameters, and $C_{\pi}(j)$ is the cycle of $j$ in $\pi$ given by

Figures (1)

  • Figure 1: The action of the map $\Phi_{\lambda}$

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof