<p>We prove that the local eigenvalue statistics in the bulk for complex random matrices with independent entries whose <i>r</i>-th absolute moment decays as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N^{-1-(r-2)\epsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>ϵ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> are universal. This includes sparse matrices whose entries are the product of a Bernouilli random variable with mean <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N^{-1+\epsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mi>ϵ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and an independent complex-valued random variable. By a standard truncation argument, we can also conclude universality for complex random matrices with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$4+\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>+</mo> <mi>ϵ</mi> </mrow> </math></EquationSource> </InlineEquation> moments. The main ingredient is a sparse multi-resolvent local law for products involving any finite number of resolvents of the Hermitisation and deterministic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2N\times 2N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>N</mi> <mo>×</mo> <mn>2</mn> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> matrices whose <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$N\times N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> blocks are multiples of the identity.</p>

Bulk universality for sparse complex non-Hermitian random matrices

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2026年7月8日

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Mohammed Osman

摘要

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