Bulk Universality for Sparse Complex non-Hermitian Random Matrices

2025/08/05 by Osman, Mohammed
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2508.03631

Abstract

We prove that the local eigenvalue statistics in the bulk for complex random matrices with independent entries whose r-th absolute moment decays as N^-1-(r-2)ε for some ε>0 are universal. This includes sparse matrices whose entries are the product of a Bernouilli random variable with mean N^-1+ε and an independent complex-valued random variable. By a standard truncation argument, we can also conclude universality for complex random matrices with 4+ε moments. The main ingredient is a sparse multi-resolvent local law for products involving any finite number of resolvents of the Hermitisation and deterministic 2N×2N matrices whose N× N blocks are multiples of the identity.

Citations

• The Zigzag Strategy for Random Band Matrices
• Law of fractional logarithm for random matrices
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• The sparse circular law under minimal assumptions
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• Universality of the least singular value for sparse random matrices
• Fixed energy universality for Dyson Brownian motion
• Local law and Tracy-Widom limit for sparse random matrices
• The Local Semicircle Law for a General Class of Random Matrices
• Random Matrices: The circular Law

Related

• Bulk universality for complex eigenvalues of real non-symmetric random matrices with i.i.d. entries
• Towards the bulk universality of non-Hermitian random matrices
• Universality in the bulk of the spectrum for complex sample covariance matrices
• Bulk Universality for Real Matrices with Independent and Identically Distributed Entries
• Universality in the random matrix spectra in the regime of weak non-Hermiticity