Asymptotics of unitary and othogonal matrix integrals

Abstract

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the othogonal group.

41 pages, important modifications, new section about orthogonal integrals

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• Monotone Hurwitz numbers and the HCIZ integral II
• Large Deviation Principles via Spherical Integrals
• On the Complex Asymptotics of the HCIZ and BGW Integrals
• Free energies and fluctuations for the Unitary Brownian Motion
• Gaussian multiplicative Chaos for symmetric isotropic matrices
• The $1/N$ expansion for SO(N) lattice gauge theory at strong coupling
• Wilson loop expectations in $SU(N)$ lattice gauge theory
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
• A tracial Nullstellensatz
• Large Deviations Asymptotics of Rectangular Spherical Integral
• Universality in several-matrix models via approximate transport maps
• Moment Methods on compact groups: Weingarten calculus and its applications
• Orbital free pressure and its Legendre transform
• Commutators of random matrices from the unitary and orthogonal groups
• $SO(N)$ Lattice Gauge Theory, planar and beyond
• Joint convergence of several copies of different patterned random matrices
• Asymptotics of the partition function of a Laguerre-type random matrix model