The oriented Kesten--McKay law for random regular digraphs

Yukun He; Jiaoyang Huang

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arXiv · 2609.05297

The oriented Kesten--McKay law for random regular digraphs

Yukun He·Jiaoyang Huang

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Abstract

We consider the adjacency matrix of a random directed $d$-regular graph on $N$ vertices. For fixed $d\geq 2$, we prove that the empirical eigenvalue density converges in probability to the oriented Kesten--McKay law as $N\to \infty$. The key technical input is the small-ball probability estimate for the smallest singular value. The proof combines a fixed-rank transposition argument with finite-field anticoncentration for shifted inverse compressions. We also prove a polynomial hard-edge estimate, which allows us to deduce the global law from the vanishing small-ball probability.

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Yukun He, Jiaoyang Huang. 2026-09-04. The oriented Kesten--McKay law for random regular digraphs. https://arxiv.org/abs/2609.05297

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