Universal singularity at the closure of a gap in a random matrix theory

Abstract

We consider a Hamiltonian $H=H_0+V$, in which $H_0$ is a given non-random Hermitian matrix,and $V$ is an $N\times N$ Hermitian random matrix with a Gaussian probability distribution.We had shown before that Dyson's universality of the short-range correlations between energy levels holds at generic points of the spectrum independently of $H_0$. We consider here the case in which the spectrum of $H_0$ is such that there is a gap in the average density of eigenvalues of $H$ which is thus split into two pieces. When the spectrum of $H_0$ is tuned so that the gap closes, a new class of universality appears for the energy correlations in the vicinity of this singular point.

20pages, Revtex, to be published in Phys. Rev. E

References in corpus (1)

• An Extension of Level-spacing Universality

Cited by in corpus (88)

• Level Spacing of Random Matrices in an External Source
• The Pearcey Process
• Large $n$ limit of Gaussian random matrices with external source, part II
• Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
• Large n limit of Gaussian random matrices with external source, part I
• Algebraic methods in random matrices and enumerative geometry
• Quadratic vector equations on complex upper half-plane
• Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach
• Noncolliding Brownian Motion and Determinantal Processes
• The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach
• Intersection theory from duality and replica
• Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• Vertices from replica in a random matrix theory
• Universality of the Pearcey process
• Non-colliding Brownian Motions and the extended tacnode process
• New correlation functions for random matrices and integrals over supergroups
• Universal shocks in random matrix theory
• Characteristic polynomials of random matrices at edge singularities
• Lectures on random matrix models. The Riemann-Hilbert approach
• Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
• Polynuclear growth model, GOE${}^2$ and random matrix with deterministic source
• Asymptotic analysis of random matrices with external source and a family of algebraic curves
• Multiplying unitary random matrices - universality and spectral properties
• The Higher-Order Heat-Type Equations via signed Lévy stable and generalized Airy functions
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• Cusp Universality for Random Matrices II: The Real Symmetric Case
• Random matrices with external source and multiple orthogonal polynomials
• Asymptotics of Fredholm determinant associated with the Pearcey kernel
• From the Pearcey to the Airy process
• Multiplicative functionals on ensembles of non-intersecting paths
• Nonlinear PDEs for gap probabilities in random matrices and KP theory
• New Multicritical Random Matrix Ensembles
• A Matrix model for plane partitions
• The rank 1 real Wishart spiked model
• On an Airy matrix model with a logarithmic potential
• A critical phenomenon in the two-matrix model in the quartic/quadratic case
• The intersection numbers of the p-spin curves from random matrix theory
• Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD
• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
• Computing topological invariants with one and two-matrix models
• Gravity without averaging
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• New Critical Matrix Models and Generalized Universality
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Unitary matrix models and random partitions: Universality and multi-criticality
• Exact and explicit evaluation of Brezin-Hikami kernels
• Density of State in a Complex Random Matrix Theory with External Source
• Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
• Exact spectral densities of complex noise-plus-structure random matrices
• Asymptotics of Plancherel measures for the infinite-dimensional unitary group
• Low energy excitations of mean-field glasses
• The Gaussian free field in an interlacing particle system with two jump rates
• A phase transition for non-intersecting Brownian motions, and the Painleve II equation
• Gap probability for the hard edge Pearcey process
• On Certain Wronskians of Multiple Orthogonal Polynomials
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• On the deformed Pearcey determinant
• Critical Behavior of Non-Intersecting Brownian Motions
• Spectral Form Factor for Time-dependent Matrix model
• Pearcey universality at cusps of polygonal lozenge tiling
• Dyson's non-intersecting Brownian motions with a few outliers
• Gaussian perturbations of hard edge random matrix ensembles
• Hard-edge asymptotics of the Jacobi growth process
• Punctures and p-spin curves from matrix models II
• Asymptotic analysis of the two matrix model with a quartic potential
• Joint probability for the Pearcey process
• Edge Universality for Nonintersecting Brownian Bridges
• Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles
• Random Matrix, Singularities and Open/Close Intersection Numbers
• Punctures and p-spin curves from matrix models
• Random surface growth with a wall and Plancherel measures for O(infinity)
• Multicritical Matrix Models and the Chiral Phase Transition
• Narain transform for spectral deformations of random matrix models
• Universal Cusp Scaling in Random Partitions
• Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions
• Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions
• The Hermitian two matrix model with an even quartic potential
• Non-intersecting squared Bessel paths with one positive starting and ending point
• Critical behavior in Angelesco ensembles
• Phase transitions for products of characteristic polynomials under Dyson Brownian motion
• PDE's for the Gaussian ensemble with external source and the Pearcey distribution
• A PDE for Nonintersecting Brownian Motions and Applications
• Global Fluctuations in General Beta Dyson Brownian Motion
• Limits of determinantal processes near a tacnode
• Nonintersecting Brownian bridges between reflecting or absorbing walls
• Average characteristic polynomials for multiple orthogonal polynomial ensembles
• Painlevé kernels in Hermitian matrix models