On the function \(\text{Tr} e^{H+itK}\)
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.
scientific article; zbMATH DE number 1780357
scientific article