Bulk universality for sparse complex non-Hermitian random matrices

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 N - 1 - ( r - 2 ) ϵ for some ε >0 ϵ > 0 are universal. This includes sparse matrices whose entries are the product of a Bernouilli random variable with mean N-1+ε 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 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 2 N × 2 N matrices whose N× N N × N blocks are multiples of the identity.