Commutators of random matrices from the unitary and orthogonal groups

Abstract

We investigate the statistical properties of $C=uv{u}^{-1}{v}^{-1}$, when $u$ and $v$ are independent random matrices, uniformly distributed with respect to the Haar measure of the groups $U(N)$ and $O(N)$. An exact formula is derived for the average value of power sum symmetric functions of $C$, and also for products of the matrix elements of $C$, similar to Weingarten functions. The density of eigenvalues of $C$ is shown to become constant in the large-$N$ limit, and the first $N^{-1}$ correction is found.

arXiv comment26 pages, 4 figures

References in corpus (33)

• A bound on chaos
• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator
• Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems
• How to generate random matrices from the classical compact groups
• Chaos signatures in the short and long time behavior of the out-of-time ordered correlator
• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Freezing Transitions and Extreme Values: Random Matrix Theory, $\zeta (1/2+it)$, and Disordered Landscapes
• On some properties of orthogonal Weingarten functions
• Autocorrelation of Random Matrix Polynomials
• On some integrals over the U(N) unitary group and their large N limit
• Asymptotics of unitary and othogonal matrix integrals
• Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices
• Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries
• Weingarten calculus for matrix ensembles associated with compact symmetric spaces
• Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs
• Jucys-Murphy elements and Weingarten matrices
• Jucys-Murphy elements, orthogonal matrix integrals, and Jack measures
• Invariant Integration over the Unitary Group
• Combinatorial theory of the semiclassical evaluation of transport moments I: Equivalence with the random matrix approach
• Asymptotics of characters of symmetric groups related to Stanley character formula
• Real Second Order Freeness and Haar Orthogonal Matrices
• On polynomial integrals over the orthogonal group
• On duality and negative dimensions in the theory of Lie groups and symmetric spaces
• Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$
• Monotone Hurwitz numbers and the HCIZ integral II
• On the large N limit of matrix integrals over the orthogonal group
• $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral
• The orthogonal Weingarten formula in compact form
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
• Relevant OTOC operators: footprints of the classical dynamics
• Word Measures on Unitary Groups
• On the immanants of blocks from random matrices in some unitary ensembles