Determinantal point processes and fermions on complex manifolds: Bulk universality

Abstract

We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Ampère operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials.

v2: Substantial revision. The CLT now holds for Lipschitz continuous test functions (thanks to the simple Remark 4.2). Moreover, a new section has been included, giving an outlook on relations to large deviatiations and phase transitions. The paper will appear in "Algebraic and Analytic Microlocal Analysis", M. Hitrik, D. Tamarkin, B. Tsygan, and S. Zelditch, eds. Springer

References in corpus (7)

• Fluctuations of eigenvalues of random normal matrices
• Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
• Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
• Universality for mathematical and physical systems
• Large deviations and stochastic calculus for large random matrices
• Complex determinantal processes and H1 noise
• Transfinite diameter notions in C^N and integrals of Vandermonde determinants

Cited by in corpus (22)

• Fluctuations of eigenvalues of random normal matrices
• Recent exact and asymptotic results for products of independent random matrices
• Random normal matrices and Ward identities
• The polyanalytic Ginibre ensembles
• Exponential Estimate for the asymptotics of Bergman kernels
• Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates
• Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble
• Products of Independent Gaussian Random Matrices
• Determinantal Probability: Basic Properties and Conjectures
• Asymptotic expansion of polyanalytic Bergman kernels
• The high temperature crossover for general 2D Coulomb gases
• The Interpolating Airy Kernels for the beta=1 and beta=4 Elliptic Ginibre Ensembles
• Analytic torsion, vortices and positive Ricci curvature
• Polynomial Ensembles and Recurrence Coefficients
• Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles
• Growth of balls of holomorphic sections and energy at equilibrium
• Sampling of real multivariate polynomials and pluripotential theory
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Large deviations of empirical measures of zeros on Riemann surfaces
• The Bergman kernel in constant curvature
• Voiculescu's entropy and potential theory
• Bulk asymptotics for polyanalytic correlation kernels