Discrepancy properties for random regular digraphs

TL;DR

This paper establishes concentration inequalities and discrepancy properties for random regular directed graphs, using exchangeable pairs and novel involutions, extending results known for Erdős-Rényi digraphs.

Contribution

It introduces new involutions, including a reflection operation, to prove concentration inequalities for regular digraphs, advancing understanding of their edge distribution.

Findings

Concentration inequalities for codegrees in regular digraphs

Discrepancy properties similar to Erdős-Rényi graphs

Use of exchangeable pairs with novel involutions

Abstract

For the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edges essentially matching results for Erd\H{o}s-R\'enyi digraphs obtained from Chernoff-type bounds. The proofs make use of the method of exchangeable pairs, developed for concentration of measure by Chatterjee. Exchangeable pairs are constructed using two involutions on the set of regular digraphs: a well-known "simple switching" operation, as well as a novel "reflection" operation.