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We study determinantal random point processes on a compact complex manifold Xassociated to an Hermitian metric on a line bundle over X and a probabilitymeasure on X. Physically, this setup describes a free fermion gas on X subjectto a U(1)- gauge field and when X is the Riemann sphere it specializes tovarious random matrix ensembles. It is shown that, in the many particle limit,the empirical random measures on X converge exponentially towards thedeterministic pluripotential equilibrium measure, defined in terms of theMonge-Ampere operator of complex pluripotential theory. More precisely, a largedeviation principle (LDP) is established with a good rate functional. We alsoexpress the LDP in terms of the Ray-Singer analytic torsion and theexponentially small eigenvalues of dbar-Laplacians. This can be seen as aneffective bosonization formula, generalizing the previously known formula inthe Riemann surface case to higher dimensions and the paper is concluded with aheuristic quantum field theory intepretation of the resulting effectiveboson-fermion correspondence.

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