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Title: Low-temperature random matrix theory at the soft edge

“Low temperature” random matrix theory is the study of random eigenvalues as energy is removed. In standard notation, β is identified with inverse temperature, and low temperatures are achieved through the limit β → ∞. In this paper, we derive statistics for low-temperature random matrices at the “soft edge,” which describes the extreme eigenvalues for many random matrix distributions. Specifically, new asymptotics are found for the expected value and standard deviation of the general-β Tracy-Widom distribution. The new techniques utilize beta ensembles, stochastic differential operators, and Riccati diffusions. The asymptotics fit known high-temperature statistics curiously well and contribute to the larger program of general-β random matrix theory.
Authors:
 [1] ;  [2] ;  [3]
  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)
  2. Department of Mathematics, University of California, Berkeley, California 94720 (United States)
  3. Department of Mathematics, Randolph-Macon College, Ashland, Virginia 23005 (United States)
Publication Date:
OSTI Identifier:
22306204
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 6; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFUSION; EIGENVALUES; MATHEMATICAL OPERATORS; MATRICES; RANDOMNESS; RICCATI EQUATION; STATISTICS; STOCHASTIC PROCESSES