Determinantal point processes and fermions on complex manifolds: Bulk universality

Determinantal point processes on a compact complex manifold X are considered 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. It is shown that the defining random measure on X of the process, describing the particle locations, converges in probability towards a pluripotential equilibrium measure, expressed as a Monge-Ampere measure. Its smooth fluctuations in the bulk are shown to be asymptotically normal and the limiting variance is explicitly computed. A scaling limit of the correlation functions is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This setting applies in particular to normal random matrix ensembles and multivariate orthogonal polynomials. Relation to phase transitions, direct image bundles and tunneling of ground state fermions in strong magnetic fields (i.e. exponentially small eigenvalues of the Dolbeault Laplacian) are also explored.