Rigidity of one-dimensional point processes via optimal transport

TL;DR

This paper explores how optimal transport maps can establish rigidity properties in one-dimensional point processes, including classical models like the Coulomb gas, by linking transport maps to number and cyclic rigidity phenomena.

Contribution

It demonstrates that the existence of an L^1 transport map from a lattice or Lebesgue measure ensures rigidity properties in one-dimensional point processes, extending to Riesz gases with specific parameters.

Findings

Existence of L^1 transport maps implies number and cyclic rigidity.

Application to Riesz gases with s in (-2,-1], including Coulomb gas.

Provides a new connection between optimal transport and point process rigidity.

Abstract

We investigate rigidity phenomena in one-dimensional point processes. We show that the existence of an $L^1$ transport map from a stationary lattice or the Lebesgue measure to a point process is sufficient to guarantee the properties of Number-Rigidity and Cyclic-Factor. We then apply this result to non-singular Riesz gases with parameter $s\in(-2,-1]$, defined in infinite volume as accumulation points of stationarized finite-volume Riesz gases. This includes, for $s=-1$, the well-known one-dimensional Coulomb gas (also called Jellium plasma, or the one-component 1D plasma).