Random Matrices: Theory and Applications

In this paper, we first obtain an algebraic formula for the moments of a centered Wishart matrix, and apply it to obtain new convergence results in the large dimension limit when both parameters of the distribution tend to infinity at different speeds.

We use this result to investigate APPT (absolute positive partial transpose) quantum states. We show that the threshold for a bipartite random induced state on C^d = C^d₁ ⊗ C^d₂, obtained by partial tracing a random pure state on C^d ⊗ C^s, being APPT occurs if the environmental dimension s is of order s₀ = min(d₁, d₂)³max(d₁, d₂). That is, when s ≥ Cs₀, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when s ≤ cs₀. Besides, we compute effectively C and c and show that it is possible to replace them by the same sharp transition constant when min(d₁, d₂)² ≪ d.

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös–Rényi random graph G(n, p_n) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, p_n ∈ (0, 1) and np_n → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semi-circle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semi-circle law.

The aim of this paper is to present a systematic method for computing moments of matrix elements taken from circular orthogonal ensembles (COE). The formula is given as a sum of Weingarten functions for orthogonal groups but the technique for its proof involves Weingarten calculus for unitary groups. As an application, explicit expressions for the moments of a single matrix element of a COE matrix are given.

Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian β matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail of Tracy–Widom β laws, for all β > 0, by studying the double scaling limit.

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.

Consider an n × n non-Hermitian random matrix M_n whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of f(M_n) as n tends to infinity, where f is analytic on an appropriate domain. This extends the results in [19, 20, 23] from symmetric random matrices to the non-Hermitian case.