On the logarithm of the characteristic polynomial of the Ginibre ensemble

TL;DR

This paper refines the understanding of the logarithm of the characteristic polynomial of the Ginibre ensemble, showing it converges to the Gaussian Free Field with a precise characterization of the zero mode.

Contribution

It provides a sharper version of previous convergence results, clarifying the zero mode of the Gaussian Free Field for the Ginibre ensemble.

Findings

Convergence of the logarithm of the characteristic polynomial to the GFF.

Removal of ambiguity regarding the zero mode of the field.

Enhanced understanding of linear statistics in the Ginibre ensemble.

Abstract

We prove a slightly sharper version of a result of Rider and Vir\'ag 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.