On the Relativized Alon Eigenvalue Conjecture II: Asymptotic Expansion Theorems for Walks

Abstract: This is the second 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. The first main result in this article concerns the function $f(k,n)$ defined as the number of SNBC (strictly non-backtracking closed) walks of length $k$ of a given homotopy type in a random covering graph of degree $n$ of a fixed graph. We prove the existence of asymptotic expansions in powers of $1/n$ for $f(k,n)$, where the coefficients---functions of $k$---are proven to have some desirable properties; namely, these coefficients are approximately a sum of polynomials times exponential functions. The second main result is a generalization of the first, where the number of SNBC walks of length $k$ is multiplied by an indicator function that the covering graph contains a certain type of {\em tangle}; the second result requires more terminology, although its proof uses the same basic tools used to prove the first result. % The motivation for the second main result will be clear in % the third article in this series of articles. The results in this article are mostly straightforward generalizations of methods used in previous works. However, this article (1) "factors" these methods into a number of short, conceptually simple, and independent parts, (2) writes each independent part in more general terms, and (3) significantly simplifies of one of the previous computations. As such we expect that this article will make it easier to apply trace methods to related models of random graphs.