  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.
