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Register a new userThe Two-Parameter Free Unitary Segal-Bargmann Transform and its Biane-Gross-Malliavin Identification
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Ching-Wei Ho
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Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the large-$N$ limit of the two-parameter Segal-Bargmann transform on the unitary group $mathbb{U}(N)$ and explore its limiting behavior. We also extend the notion of circular systems in order to define a two-parameter free Segal-Bargmann transform and prove a version of Biane-Gross-Malliavin Theorem of the two-parameter free unitary Segal-Bargmann transform.
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We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of Sniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.
The Bargmann-Fock space of slice hyperholomorphic functions is recently introduced by Alpay, Colombo, Sabadini and Salomon. In this paper, we reconsider this space and present a direct proof of its independence of the slice. We also introduce a quaternionic analogue of the classical Segal-Bargmann transform and discuss some of its basic properties. The explicit expression of its inverse is obtained and the connection to the left one-dimensional quaternionic Fourier transform is given.
We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.
A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality of Lust-Piquard and Pisier, yields a nonasymptotic bound on the spectral norm of general Gaussian random matrices $X=sum_i g_i A_i$ where $g_i$ are independent standard Gaussian variables and $A_i$ are matrix coefficients. This bound exhibits a logarithmic dependence on dimension that is sharp when the matrices $A_i$ commute, but often proves to be suboptimal in the presence of noncommutativity. In this paper, we develop nonasymptotic bounds on the spectrum of arbitrary Gaussian random matrices that can capture noncommutativity. These bounds quantify the degree to which the deterministic matrices $A_i$ behave as though they are freely independent. This intrinsic freeness phenomenon provides a powerful tool for the study of various questions that are outside the reach of classical methods of random matrix theory. Our nonasymptotic bounds are easily applicable in concrete situations, and yield sharp results in examples where the noncommutative Khintchine inequality is suboptimal. When combined with a linearization argument, our bounds imply strong asymptotic freeness (in the sense of Haagerup-Thorbj{o}rnsen) for a remarkably general class of Gaussian random matrix models, including matrices that may be very sparse and that lack any special symmetries. Beyond the Gaussian setting, we develop matrix concentration inequalities that capture noncommutativity for general sums of independent random matrices, which arise in many problems of pure and applied mathematics.
Let G/K be a Riemannian symmetric space of the complex type, meaning that G is complex semisimple and K is a compact real form. Now let {Gamma} be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the Segal--Bargmann transform, defined in terms of the heat equation, on the compact quotient {Gamma}G/K. We obtain isometry and inversion formulas precisely parallel to the results we obtained previously for globally symmetric spaces of the complex type. Our results are as parallel as possible to the results one has in the dual compact case. Since there is no known Gutzmer formula in this setting, our proofs make use of double coset integrals and a holomorphic change of variable.
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