随机图提升的词图和谱
Words Maps and Spectra of Random Graph Lifts
摘要：
我们从正式词汇的新分析开始。设w是一个正式的单词，用字母g_1，...，g_k。与w相关联的词映射映射排列s_1，.，S_n中的s_k到通过对每个i用s_i替换w中的g_i的每次出现而获得的置换。当s_i是均匀随机选取的时，我们研究了在这个排列中计数不动点的随机变量X_w^n.我们工作的一个主要组成部分是一个新的词的分类，大大扩展了原始与非原始词的二分法。我们建立了一些结果，并对期望E（X_w^n）与这一新的范畴化之间的关系作了一些说明。这一分析对我们研究图的随机提升的谱有很大的贡献。设G是一个连通图，无限树T是它的泛覆盖空间.如果L和R分别是G和T的谱半径，则如J. Friedman所示，对于G的几乎每个n-提升H，H的所有“新”特征值都< O（L^（1/2）R^（1/2））。我们将这个上界改进为O（L^（1/3）R^（2/3）），并且我们前面提到的结构提供了一种可能的方法来证明O（R）的上界.这是随机二维正则图第二特征值定界问题的推广。作为题外话，我们得到了A. Nica，它确定，对于每个固定的w，极限分布（作为n \to \infty）的X_w^n。这个定理的一个令人惊讶的方面是，答案只取决于最大的整数d，因此对于某个单词u，w=u^d。
英文摘要：
We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by s_i. We investigate the random variable X_w^n that counts the fixed points in this permutation when the s_i 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(X_w^n) 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 \to \infty) of X_w^n. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w=u^d for some word u.
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