Narrow Results

Let $G$ be an $N\times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij}\sim {𝒩}(0,1)$. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if ${\lambda }_1,\dots ,{\lambda }_{N_{ℝ}}$ are the real eigenvalues of $G$, then for any even polynomial function $P(x)$ and even $N=2n$, we have the convergence in distribution to a normal random variable

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the $p(n)$-dimensional volume of the simplex with $p(n)+1$ points in ${ℝ}^n$ distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-$\phi$ convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.

In joint work with Etienne Sandier, we studied the statistical mechanics of a classical two-dimensional Coulomb gas, particular cases of which also correspond to random matrix ensembles. We connect the problem to the “renormalized energy” W, a Coulombian interaction for an infinite set of points in the plane that we introduced in connection to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the “Abrikosov” triangular lattice. Results include a next order asymptotic expansion of the partition function, and various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero we show that the system tends to “crystallize” to a minimizer of W.