2008/12/22 by Robert J. Berman, Berman, Robert J. · 2 citations
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.0812.4224
openalex publication_date 2008/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study determinantal random point processes on a compact complex manifold X\nassociated to an Hermitian metric on a line bundle over X and a probability\nmeasure on X. Physically, this setup describes a free fermion gas on X subject\nto a U(1)- gauge field and when X is the Riemann sphere it specializes to\nvarious random matrix ensembles. It is shown that, in the many particle limit,\nthe empirical random measures on X converge exponentially towards the\ndeterministic pluripotential equilibrium measure, defined in terms of the\nMonge-Ampere operator of complex pluripotential theory. More precisely, a large\ndeviation principle (LDP) is established with a good rate functional. We also\nexpress the LDP in terms of the Ray-Singer analytic torsion and the\nexponentially small eigenvalues of dbar-Laplacians. This can be seen as an\neffective bosonization formula, generalizing the previously known formula in\nthe Riemann surface case to higher dimensions and the paper is concluded with a\nheuristic quantum field theory intepretation of the resulting effective\nboson-fermion correspondence.\n