On the function \(\text{Tr} e^{H+itK}\)
type: Item
instance of: scholarly article
title: On the function \(\text{Tr} e^{H+itK}\)
zbMATH Open document ID: 1001.15003
DOI: 10.1155/S0161171202011468
author: Mark Fannes
author: Dénes Petz
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-13T00:00:00Z
full work available at URL: https://eudml.org/doc/49585
full work available at URL: S0161171202011468.html
review text: Let \(H\) and \(K\) be selfadjoint \(n \times n\) matrices. The Bessis-Moussa-Villani conjecture is widely known [cf. \textit{D. Bessis, P. Moussa}, and \textit{M. Villani}, J. Math. Phys. 16, 2318-2325 (1975; Zbl 0976.82501)]; it means that the function \(t \mapsto \text{Tr}e^{H+itK}\) is positive definite on \(\mathbb{R}\). The aim of the paper is to show that this conjecture holds asymptotically for random choices of high-dimensional matrices. When \(H\) and \(K\) have independent Gaussian entries and their distribution is invariant under unitary conjugation, the authors constitute the so-called random matrix model for semicircular selfadjoint operators \(a\) and \(b\) in type \(II_1\) von Neumann algebra with faithful normal tracial state \(\tau\). Then \(a\) and \(b\) are in free relation and it is proved that the function \(t \mapsto \tau(e^{a+itb})\) is also positive definite.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1780357
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: random matrices
zbMATH Keywords: semicircular selfadjoint operator
zbMATH Keywords: trace
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1780357