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Random Unitary Representations of Surface Groups I: Asymptotic expansions

Magee, Michael

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In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let Σg denote a topological surface of genus g≥2. We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of π1(Σg) under a random representation of π1(Σg) into SU(n). Each such expected value involves a contribution from all irreducible representations of SU(n). The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.


Magee, M. (2022). Random Unitary Representations of Surface Groups I: Asymptotic expansions. Communications in Mathematical Physics, 391(1), 119-171.

Journal Article Type Article
Acceptance Date Nov 30, 2021
Online Publication Date Dec 31, 2021
Publication Date 2022-04
Deposit Date Jul 9, 2021
Publicly Available Date Feb 7, 2022
Journal Communications in Mathematical Physics
Print ISSN 0010-3616
Electronic ISSN 1432-0916
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 391
Issue 1
Pages 119-171


Published Journal Article (766 Kb)

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