Rigidity of one-dimensional point processes via optimal transport

2025/10/20 by Dereudre, David, Digneaux, Rafaël
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.17257

Abstract

We investigate rigidity phenomena in one-dimensional point processes. We show that the existence of an L¹ 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∈(-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).

Citations

• Phase transitions in one-dimensional Riesz gases with long-range interaction
• The two-dimensional one-component plasma is hyperuniform
• On strongly rigid hyperfluctuating random measures
• DLR equations and rigidity for the Sine-beta process
• Tranport estimates for random measures in dimension one

Cited by

• Box-Covariances of Hyperuniform Point Processes

Related

• DLR equations, number-rigidity and translation-invariance for infinite-volume limit points of the 2DOCP
• Number-Rigidity and β-Circular Riesz gas
• Non-rigidity Properties of the Coulomb Gas
• Rigidity of random stationary measures and applications to point processes
• Local laws and rigidity for Coulomb gases at any temperature