A Relativized Alon Second Eigenvalue Conjecture for Regular Base Graphs IV: An Improved Sidestepping Theorem

Abstract

This is the fourth 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 prove a {\em Sidestepping Theorem} that is more general and easier to use than earlier theorems of this kind. Such theorems concerns a family probability spaces $\{\mathcal{M}_n\}$ of $n\times n$ matrices, where $n$ varies over some infinite set, $N$, of natural numbers. Many trace methods use simple "Markov bounds" to bound the expected spectral radius of elements of ${\mathcal{M}}_n$: this consists of choosing one value, $k=k(n)$, for each $n\in N$, and proving expected spectral radius bounds based on the expected value of the trace of the $k=k(n)$-power of elements of ${\mathcal{M}}_n$. {\em Sidestepping} refers to bypassing such simple Markov bounds, obtaining improved results using a number of values of $k$ for each fixed $n\in N$. In more detail, if the $M\in {\mathcal{M}}_n$ expected value of ${\rm Trace}(M^k)$ has an asymptotic expansion in powers of $1/n$, whose coefficients are "well behaved" functions of $k$, then one can get improved bounds on the spectral radius of elements of ${\mathcal{M}}_n$ that hold with high probability. Such asymptotic expansions are shown to exist in the third article in this series for the families of matrices that interest us; in the fifth and sixth article in this series we will apply the Sidestepping Theorem in this article to prove the main results in this series of articles. This article is independent of all other articles in this series; it can be viewed as a theorem purely in probability theory, concerning random matrices or, equivalently, the $n$ random variables that are the eigenvalues of the elements of ${\mathcal{M}}_n$.