Words Maps and Spectra of Random Graph Lifts
Abstract
We begin with a new analysis of formal words. Let w be a formal word in letters g1,...,gk. The word map associated with w maps the permutations s1,...,sk in Sn to the permutation obtained by replacing for each i, every occurrence of gi in w by si. We investigate the random variable Xwn that counts the fixed points in this permutation when the si are selected uniformly at random. A major ingredient of our work is a new categorization of words which considerably extends the dichotomy of primitive vs. imprimitive words. We establish some results and make a few conjectures about the relation between the expectation E(Xwn) and this new categorization. This analysis contributes deeply to our study of the spectra of random lifts of graphs. Let G be a connected graph, and let the infinite tree T be its universal cover space. If L and R are the spectral radii of G and T respectively, then, as shown by J. Friedman, for almost every n-lift H of G, all "new" eigenvalues of H are < O(L(1/2)R(1/2)). We improve this upper bound to O(L(1/3)R(2/3)), and our aforementioned conjectures suggest a possible approach to proving an upper bound of O(R). This is a generalization of the problem of bounding the second eigenvalue in a random 2d-regular graph. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica, which determines, for every fixed w, the limit distribution (as n ∞) of Xwn. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w=ud for some word u.