Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for t...

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Autores principales:	Leonardo Santilli, Miguel Tierz
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2021
Materias:	Nuclear and particle physics. Atomic energy. Radioactivity QC770-798
Acceso en línea:	https://doaj.org/article/6a181fa0640c4ca6ab092525f9fbbe10
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id	oai:doaj.org-article:6a181fa0640c4ca6ab092525f9fbbe10
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spelling	oai:doaj.org-article:6a181fa0640c4ca6ab092525f9fbbe102021-12-04T04:32:52ZRiemannian Gaussian distributions, random matrix ensembles and diffusion kernels0550-321310.1016/j.nuclphysb.2021.115582https://doaj.org/article/6a181fa0640c4ca6ab092525f9fbbe102021-12-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S0550321321002790https://doaj.org/toc/0550-3213We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of m×m symmetric positive definite matrices, we show how to efficiently compute densities of eigenvalues by evaluating Pfaffians at specific values of m. Equivalently, we can obtain the same result by constructing specific skew orthogonal polynomials with regards to the log-normal weight function (skew Stieltjes-Wigert polynomials). Other symmetric spaces are studied and the same type of result is obtained for the quaternionic case. Moreover, we show how the probability density functions are a particular case of diffusion reproducing kernels of the Karlin-McGregor type, describing non-intersecting Brownian motions, which are also diffusion processes in the Weyl chamber of Lie groups.Leonardo SantilliMiguel TierzElsevierarticleNuclear and particle physics. Atomic energy. RadioactivityQC770-798ENNuclear Physics B, Vol 973, Iss , Pp 115582- (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Nuclear and particle physics. Atomic energy. Radioactivity QC770-798
spellingShingle	Nuclear and particle physics. Atomic energy. Radioactivity QC770-798 Leonardo Santilli Miguel Tierz Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
description	We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of m×m symmetric positive definite matrices, we show how to efficiently compute densities of eigenvalues by evaluating Pfaffians at specific values of m. Equivalently, we can obtain the same result by constructing specific skew orthogonal polynomials with regards to the log-normal weight function (skew Stieltjes-Wigert polynomials). Other symmetric spaces are studied and the same type of result is obtained for the quaternionic case. Moreover, we show how the probability density functions are a particular case of diffusion reproducing kernels of the Karlin-McGregor type, describing non-intersecting Brownian motions, which are also diffusion processes in the Weyl chamber of Lie groups.
format	article
author	Leonardo Santilli Miguel Tierz
author_facet	Leonardo Santilli Miguel Tierz
author_sort	Leonardo Santilli
title	Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
title_short	Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
title_full	Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
title_fullStr	Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
title_full_unstemmed	Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
title_sort	riemannian gaussian distributions, random matrix ensembles and diffusion kernels
publisher	Elsevier
publishDate	2021
url	https://doaj.org/article/6a181fa0640c4ca6ab092525f9fbbe10
work_keys_str_mv	AT leonardosantilli riemanniangaussiandistributionsrandommatrixensemblesanddiffusionkernels AT migueltierz riemanniangaussiandistributionsrandommatrixensemblesanddiffusionkernels
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Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

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