From Random Matrices to Stochastic Operators

By Alan Edelman, Brian D. Sutton. 2006

137 citations recorded by the source corpus.

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/49b053b2e0893e5e9f5f906af67630c2893ddfbc

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Beta ensembles, stochastic Airy spectrum, and a diffusion (2006, background): “JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 24, Number 4, October 2011, Pages 919–944 S 0894-0347(2011)00703-0 Article electronically published on May 6, 2011 BETA ENSEMBLES, STOCHASTIC AIRY SPECTRUM, AND A DIFFUSION JOSÉ A. RAMÍREZ, BRIAN RIDER, AND BÁLINT VIRÁG”
Julia: A Fresh Approach to Numerical Computing (2014, methodology): “In[38]: # An example S=SymArrow([1,2,3,4,5],[6,7,8,9])”
Random matrix theory in statistics: A review (2014, background): “Ramírez et al. (2011) described such limit laws in terms of the eigenstates of a stochastic Airy operator introduced by Edelman and Sutton (2007) and Sutton (2005).”
Small deviations for beta ensembles (2009, methodology): “3) This result makes essential use of a (tridiagonal) matrix model valid at all beta due to Dumitriu-Edelman [5], and proves the conjecture of Edelman-Sutton [6].”
Limits of spiked random matrices I (2010, methodology): “We would also like to thank Jinho Baik, Alexei Borodin, Peter Forrester, Arno Kuijlaars, Eric Rains, Brian Rider, Brian Sutton, Dong Wang and Ofer Zeitouni for discussions, as well as AIM and MSRI for providing stimulating environments in December 2009 and September 2010 workshops.”
Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles (2009, background): “In this way, we will find a natural process k 7→ ψk, whose values are increasing functions ψk : (−π, π) → R, with the property that the eigenvalues of C(n) lie at the angles where ψn−1 matches the random boundary condition η.”
Universality of Wigner random matrices: a survey of recent results (2010, background): “For example, the universality at the edge eigenvalues is understood in a sense that they are shown to converge to the lowest eigenvalues for a one dimensional Schrödinger operator with a white noise drift and, in particular, the β-analogue of the Tracy-Widom distribution has been identified in [79, ”
Universality of the Stochastic Airy Operator (2013, methodology): “The heuristic behind the Edelman-Sutton conjecture can then be gleaned from the asymptotic assessment: in distribution, n k p n C k=2n C 1= p 2g for a standard Gaussian random variable g.”
Extreme value statistics of correlated random variables: A pedagogical review (2019, background): “For other values of β, it can be shown that Fβ describes the fluctuations of the ground state of a one-dimensional random Schrödinger operator, called the “stochastic Airy operator” [181, 182].”
Continuum limits of random matrices and the Brownian carousel (2007, background): “Sutton (2005) and Edelman and Sutton (2007) present heuristics that the operators given by the tridiagonal matrices have a limit whose eigenvalues give the Sine and Airy processes.”