Diffusion at the random matrix hard edge

Abstract

We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian sample-covariance structures (the "beta ensembles") are described by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. This picture pertains to the so-called hard edge of random matrix theory and sits in complement to the recent work of the authors and B. Virag on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used here to prove there exists a transition between the soft and hard edge laws at all values of beta.

(More) typos corrected, added references

References in corpus (12)

• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• Matrix Models for Beta Ensembles
• The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
• Level-Spacing Distributions and the Bessel Kernel
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Continuum limits of random matrices and the Brownian carousel
• From Random Matrices to Stochastic Operators
• Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
• Universality for orthogonal and symplectic Laguerre-type ensembles
• Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Circular Beta Ensembles
• Spectral analysis of Sinai's walk for small eigenvalues

Cited by in corpus (17)

• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Universality of Wigner random matrices: a Survey of Recent Results
• Full counting statistics for interacting trapped fermions
• Localization of the continuous Anderson Hamiltonian in $1$-d
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• The Sine${}_{\beta }$ operator
• Edge scaling of the β-Jacobi ensemble
• Classical integrability for beta-ensembles and general Fokker-Planck equations
• Asymptotics for products of characteristic polynomials in classical $\beta$-Ensembles
• Operator limits of random matrices
• Operator limit of the circular beta ensemble
• Distribution of the Smallest Eigenvalue in Complex and Real Correlated Wishart Ensembles
• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial
• The gradient complexity of linear regression
• Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime
• The Tracy-Widom distribution at large Dyson index
• Inverting weak random operators