# Fluctuation around the circular law for random matrices with real entries

Giorgio Cipolloni, László Erdős, Dominik Schröder

Electron. J. Probab.Vol. 26 (2021)

## Summary

This is the companion paper to the corresponding result for complex random matrices. The limiting distribution in this case is a symmetrized version of the Gaussian Free Field.## Abstract

We extend our recent result [Cipolloni, Erdős, Schröder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [O’Rourke, Renfrew 2016] or the first four moments of the matrix elements match the real Gaussian [Tao, Vu 2015; Kopel 2015]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [Cipolloni, Erdős, Schröder 2019] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.