On the Relativized Alon Second Eigenvalue Conjecture I: Main Theorems, Examples, and Outline of Proof

This is the first in a series of six 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. Many of the techniques we develop hold whether or not the base graph is regular. Our first main theorem in this series of articles is that if the base graph is $d$-regular, then for any $\epsilon>0$, as the degree, $n$, of the covering map tends to infinity, some new adjacency eigenvalue of the map is larger in absolute value that $2(d-1)^{1/2}+\epsilon$ with probability at most order $1/n$. Our second main theorem is that if, in addition, the base graph is Ramanujan, then this probability is bounded above and below by $1/n$ to the power of a positive integer that we call the {\em tangle power} of the model, i.e., of the probability spaces of random covering maps of degree $n$. The tangle power is fairly easy to bound from below, and at times to compute exactly; it measures the probability that certain {\em tangles} appear in the random covering graph, where a {\em tangle} is a local event that forces the covering graph to have a new eigenvalue strictly larger than $2(d-1)^{1/2}$. Our main theorems are relativizations of Alon's conjecture on the second eigenvalue of random regular graphs of large degree. In this first article of the series, we introduce all the terminology needed in this series, motivate this terminology, precisely state all the results in the remaining articles, and make some remarks about their proofs. As such, this article provides an overview of the entire series of articles; furthermore, the rest of the articles in this series may be read independently of one another.