Matrix group integrals, surfaces, and mapping class groups I: U(n)

• 成果类型: Article
• 署名作者: Magee, Michael; Puder, Doron
• 署名单位: Durham University; Tel Aviv University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00891-4
• 发表日期: 2019
• 页码: 341-411
• 关键词: moduli space; EQUATIONS; dimension; unitary; limit

摘要

Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group U (n) of unitary matrices. More concretely, we study measures induced by free words on U (n). Let F-r be the free group on r generators. To sample a random element from U (n) according to the measure induced by w is an element of F-r, one substitutes the r letters in w by r independent, Haar-random elements from U (n). The main theme of this paper is that every moment of this measure is determined by families of pairs (Sigma, f), where Sigma is an orientable surface with boundary, and f is a map from Sigma to the bouquet of r circles, which sends the boundary components of Sigma to powers of w. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of Sigma. As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on U (n) bears information about the number of solutions to the equation [u(1), v(1)] ... [u(g), v(g)] = w in the free group, and deduce that one can hear the stable commutator length of a word through its unitary word measures.