On the logarithm of the characteristic polynomial of the Ginibre ensemble

Abstract

We prove a slightly sharper version of a result of Rider and Virág who proved that after centering, the logarithm of the absolute value of the characteristic polynomial of the Ginibre ensemble converges in law to the Gaussian Free Field on the unit disk with free boundary conditions (and continued harmonically outside of it). Using their results on the linear statistics of the Ginibre ensemble, we prove a result that removes the ambiguity concerning the "constant part" or zero mode of the field.

References in corpus (2)

• On the asymptotics of a Toeplitz determinant with singularities
• Liouville Quantum Gravity on the Riemann sphere

Cited by in corpus (1)

• Central limit theorems for the real eigenvalues of large Gaussian random matrices