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The oriented Kesten--McKay law for random regular digraphs
Abstract
We consider the adjacency matrix of a random directed d-regular graph on N vertices. For fixed d≥ 2, we prove that the empirical eigenvalue density converges in probability to the oriented Kesten--McKay law as N→ ∞. 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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BibTeX
@misc{he2026oriented,
title = {{The oriented Kesten--McKay law for random regular digraphs}},
author = {Yukun He and Jiaoyang Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05297},
eprint = {2609.05297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05297},
}
RIS
TY - UNPB
TI - The oriented Kesten--McKay law for random regular digraphs
AU - Yukun He
AU - Jiaoyang Huang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05297
UR - https://arxiv.org/abs/2609.05297
AB - We consider the adjacency matrix of a random directed d-regular graph on N vertices. For fixed d≥ 2, we prove that the empirical eigenvalue density converges in probability to the oriented Kesten--McKay law as N→ ∞. 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.
ER -