On the Relativized Alon Second Eigenvalue Conjecture III: Asymptotic Expansions for Tangle-Free Hashimoto Traces

TL;DR

This paper establishes asymptotic expansions for expected counts of non-backtracking walks in large random covering graphs, advancing understanding of eigenvalue bounds related to Alon's conjecture.

Contribution

It introduces simplified 'certified traces' for analyzing tangle-free random covering graphs, extending previous methods to asymptotic expansions in $1/n$.

Findings

Proves asymptotic expansion for expected non-backtracking walks in large graphs.

Shows coefficients are sums of polynomials times exponentials in walk length.

Simplifies previous trace methods for analyzing eigenvalues in random graphs.

Abstract

This is the third in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this paper we consider random graphs that are random covering graphs of large degree $n$ of a fixed base graph. We prove the existence of asympototic expansion in $1/n$ for the expected value of the number of strictly non-backtracking closed walks of length $k$ times the indicator function that the graph is free of certain {\em tangles}; moreover, we prove that the coefficients of these expansions are "nice functions" of $k$, namely approximately equal to a sum of polynomials in $k$ times exponential functions of $k$. Our results use the methods of Friedman used to resolve Alon's original conjecture, combined with the results of Article~II in this series of articles. One simplification in this article over the previous methods of Friedman is that the "regularlized traces" used in this article, which we call {\em certified traces}, are far easier to define and work with than the previously utilized {\em selective traces}.