Explicit density for singular values of Lévy's area matrix, determinantal point process characterization, and d to infinity asymptotics including absolute Cauchy limit.
On some properties of orthogonal Weingarten functions
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abstract
We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar measure over classical groups. Although a Fourier-type formula was known in the unitary case, the orthogonal counterpart was not known. It relies on the Jack polynomial generalization of both Schur and zonal polynomials. This formula substantially reduces the complexity involved in the computation of Weingarten formulas. We also describe a few more new properties of the Weingarten formula, state a conjecture and give a table of values.
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2026 1verdicts
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The Singular Values of L\'evy's Area Matrix
Explicit density for singular values of Lévy's area matrix, determinantal point process characterization, and d to infinity asymptotics including absolute Cauchy limit.