2020/02/06 by Giorgio Cipolloni, László Erdős, Cipolloni, Giorgio +3 · 2 citations
Chemistry · Mathematics · Physics and Astronomy · #15B52 #60B20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Probability (math.PR) #Quantum Mechanics and Applications #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2002.02438
openalex publication_date 2020/02/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We extend our recent result [Cipolloni, Erd Hos, Schr "oder 2019] on the\ncentral limit theorem for the linear eigenvalue statistics of non-Hermitian\nmatrices X with independent, identically distributed complex entries to the\nreal symmetry class. We find that the expectation and variance substantially\ndiffer from their complex counterparts, reflecting (i) the special spectral\nsymmetry of real matrices onto the real axis; and (ii) the fact that real\ni.i.d. matrices have many real eigenvalues. Our result generalizes the\npreviously known special cases where either the test function is analytic\n[O'Rourke, Renfrew 2016] or the first four moments of the matrix elements match\nthe real Gaussian [Tao, Vu 2015; Kopel 2015]. The key element of the proof is\nthe analysis of several weakly dependent Dyson Brownian motions (DBMs). The\nconceptual novelty of the real case compared with [Cipolloni, Erd Hos,\nSchr "oder 2019] is that the correlation structure of the stochastic\ndifferentials in each individual DBM is non-trivial, potentially even\njeopardising its well-posedness.\n