Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability

Abstract

We consider integrals on unitary groups $U_d$ of the form $${\int }_{U_d}{U}_{i_1{j}_1}...U_{i_q{j}_q}{U}_{j_1^{\prime }{i}_1^{\prime }}^{\ast }...U_{j_{q^{\prime }}^{\prime }{i}_{q^{\prime }}^{\prime }}^{\ast }dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to \infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\underset{d\to \infty }{\lim }\frac{{\partial }^n}{\partial z^n}{d}^{-2}\log {\int }_{U_d}{e}^{zdTr(XUY{U}^{\ast })}dU{|}_{z=0}$$ assuming that the normalized traces $d^{-1}Tr(X^k)$ and $d^{-1}Tr(Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

arXiv comment34 pages, 1 figure

Cited by in corpus (43)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Chaos and complexity by design
• Efficient classical simulation of random shallow 2D quantum circuits
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Random matrix techniques in quantum information theory
• The Tensor Track, III
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Random quantum channels I: graphical calculus and the Bell state phenomenon
• Integration over compact quantum groups
• Integration over quantum permutation groups
• Localization with random time-periodic quantum circuits
• Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
• Asymptotics of unitary and othogonal matrix integrals
• Deformations of Polyhedra and Polygons by the Unitary Group
• Weingarten calculus for matrix ensembles associated with compact symmetric spaces
• Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy
• Integer moments of complex Wishart matrices and Hurwitz numbers
• Integration of invariant matrices and application to statistics
• Towards a state minimizing the output entropy of a tensor product of random quantum channels
• Random Unitary Representations of Surface Groups I: Asymptotic expansions
• Page curves and typical entanglement in linear optics
• Equilibration in low-dimensional quantum matrix models
• Revisiting SU(N) integrals
• The orthogonal Weingarten formula in compact form
• On the family of Wigner functions for N-level quantum system
• Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes
• Matrix addition and the Dunkl transform at high temperature
• On the convergence of output sets of quantum channels
• Additivity rates and PPT property for random quantum channels
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
• Novel charges in CFT's
• Quantum groups and generalized circular elements
• Zooming in on local level statistics by supersymmetric extension of free probability
• Matrix models for $\epsilon$-free independence
• The Holographic Map of an Evaporating Black Hole
• The tensor Harish-Chandra-Itzykson-Zuber integral I: Weingarten calculus and a generalization of monotone Hurwitz numbers
• Moments of a single entry of circular orthogonal ensembles and Weingarten calculus
• Random Circuits in the Black Hole Interior
• A Variational Quantum Algorithm For Approximating Convex Roofs
• Moment Methods on compact groups: Weingarten calculus and its applications
• Linear algebra and group theory
• Topological expansion of unitary integrals and maps
• Convergence of moments of twisted COE matrices