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The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer
Author(s) -
Erbar Matthias,
Huesmann Martin,
Leblé Thomas
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21977
Subject(s) - convexity , mathematics , argument (complex analysis) , energy (signal processing) , volume (thermodynamics) , displacement (psychology) , process (computing) , beta (programming language) , point (geometry) , combinatorics , thermodynamics , statistics , computer science , geometry , chemistry , physics , biochemistry , financial economics , economics , psychotherapist , programming language , operating system , psychology
We prove that, at every positive temperature, the infinite‐volume free energy of the one‐dimensional log‐gas, or beta‐ensemble, has a unique minimizer, which is the Sine‐beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier‐Serfaty. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.