Sharp Asymptotics for Toeplitz Determinants and Convergence Towards the Gaussian Free Field on Riemann Surfaces
Journal article, 2012

We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal central limit theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limit, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szego-type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.

random matrices

bundles

positivity

kernel

Author

Robert Berman

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

International Mathematics Research Notices

1073-7928 (ISSN) 1687-0247 (eISSN)

22 5031-5062

Subject Categories

Mathematics

Physical Sciences

DOI

10.1093/imrn/rnr229

More information

Created

10/8/2017