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

Magee, Michael; Puder, Doron

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


Doron Puder


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 Fr be the free group on r generators. To sample a random element from U (n) according to the measure induced by w ∈ Fr, 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(, f ), where is an orientable surface with boundary, and f is a map from to the bouquet ofr circles, which sends the boundary components of to powers of w. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of . 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 [u1, v1]··· ug, vg = w in the free group, and deduce that one can “hear” the stable commutator length of a word through its unitary word measures.


Magee, M., & Puder, D. (2019). Matrix group integrals, surfaces, and mapping class groups I: U(n). Inventiones Mathematicae, 218(2), 341-411.

Journal Article Type Article
Acceptance Date Apr 11, 2019
Online Publication Date May 14, 2019
Publication Date Nov 30, 2019
Deposit Date May 8, 2019
Publicly Available Date May 14, 2020
Journal Inventiones Mathematicae
Print ISSN 0020-9910
Electronic ISSN 1432-1297
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 218
Issue 2
Pages 341-411


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