Weingarten Calculus

By Georg Kostenberger. 2021

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We consider the problem of computing the integral ∫ U (d) ui1 j1 · · · uin jn ūi′ 1 j′ 1 · · · ūi′ n′ j′ n′ dU, (1) where the integration takes place with respect to the probability Haar measure on the unitary group, and the uij denotes the ij-th entry of a unitary matrix U. We present a unified approach connecting classical results ([11], [22], [18]), the explicit formula for the integral (1) given by B. Collins and P. Sniady [6] and subsequent works of various authors providing different points of view. Finally we are able to provide an explicit formula for the 2n-th moment of the trace of a unitary Haar random matrix, generalizing a result of P. Diaconis [8].

View Paper: https://www.semanticscholar.org/paper/23f93b70a21f96bff9469b63c70c23e4b31205c3