Random Matrices: Theory and Applications

Let $X$ be an $n\times n$ symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of $X$ with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of $X$ with sub-exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere.

Voiculescu’s freeness emerges when computing the asymptotic spectra of polynomials on $N\times N$ random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix $U_N$. In this paper, we elaborate on the previous result by proposing a random matrix model, which we name the Vortex model, where $U_N$ has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector $v_N$. In the limit $N\to +\infty$, we show that $N\times N$ matrices randomly rotated by the matrix $U_N$ are asymptotically conditionally free with respect to the normalized trace and the state vector $v_N$. We define a new concept called cyclic-conditional freeness “unifying” three independences: infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. Infinitesimal distributions in the Vortex model can be computed thanks to this new independence. Finally, we elaborate on the Vortex model in order to build random matrix models for $\alpha$-freeness and for $\beta \gamma$-freeness (formerly named indented independence and ordered freeness).

In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

We present a simple stochastic representation for the joint distribution of sample estimates of three scalar parameters and two vectors of portfolio weights that characterize the minimum-variance frontier. This stochastic representation is useful for sampling observations efficiently, deriving moments in closed-form, and studying the distribution and performance of many portfolio strategies that are functions of these five variables. We also present the asymptotic joint distributions of these five variables for both the standard regime and the high-dimensional regime. Both asymptotic distributions are simpler than the finite-sample one, and the one for the high-dimensional regime, i.e. when the number of assets and the sample size go together to infinity at a constant rate, reveals the high-dimensional properties of the considered estimators. Our results extend upon T. Bodnar, H. Dette, N. Parolya and E. Thorstén [Sampling distributions of optimal portfolio weights and characteristics in low and large dimensions, Random Matrices: Theory Appl. 11 (2022) 2250008].

In the paper, we consider the extended Gross–Witten–Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the $(i,j)$-entry being the modified Bessel functions of order $i-j-\nu$, $\nu \in ℂ$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $\tau$-function of the Painlevé III equation. As a double scaling limit, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings–McLeod solution of the inhomogeneous Painlevé II equation with parameter $\nu +\frac{1}{2}$. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift–Zhou nonlinear steepest descent method to the Riemann–Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point $z=-1$, where the $\psi$-function of the Jimbo–Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.

In [W. Ejsmont and F. Lehner, The free tangent law, Adv. Appl. Math. 121 (2020) 102093], we study the limit sums of free commutators and anticommutators and show that the generalized tangent function

In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.