paper

Determinantal point processes and fermions on complex manifolds: large deviations and bosonization

arXiv:0812.4224

Abstract

We study determinantal random point processes on a compact complex manifold X associated to an Hermitian metric on a line bundle over X and a probability measure on X. Physically, this setup describes a free fermion gas on X subject to a U(1)- gauge field and when X is the Riemann sphere it specializes to various random matrix ensembles. It is shown that, in the many particle limit, the empirical random measures on X converge exponentially towards the deterministic pluripotential equilibrium measure, defined in terms of the Monge-Ampere operator of complex pluripotential theory. More precisely, a large deviation principle (LDP) is established with a good rate functional. We also express the LDP in terms of the Ray-Singer analytic torsion and the exponentially small eigenvalues of dbar-Laplacians. This can be seen as an effective bosonization formula, generalizing the previously known formula in the Riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory intepretation of the resulting effective boson-fermion correspondence.

v2: 36 pages. Major revision, the main new features are: (1) a second proof of the LDP is added which uses the abstract Gärtner-Ellis theorem (2) the LDP has been extended to a non-compact setting and to beta-ensembles and (3) the relation of the LDP to bosonization is explained and explored. Accordingly, the title has been changed

Cited by in corpus (5)