Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther

Abstract. In this paper two independent and unitarily invariant projection matrices P (N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size N converges to in nity. The result is formulated on the tracial state space T S(A) of the universal C-algebra A generated by two selfadjoint projections. The random pair (P (N); Q(N)) determines a random tracial state N 2 T S(A) and N satis es the large deviation. The rate function is in close connection with Voiculescu's free entropy de ned for pairs of projections.

Abstract. An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the large-N limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops. 1.

Abstract. In his seminal 1962 paper on the “threefold way”, Freeman Dyson classified the spaces of matrices that support the random matrix ensembles deemed relevant from the point of view of classical quantum mechanics. Recently, Heinzner, Huckleberry and Zirnbauer have obtained a similar classification based on less restrictive assumptions, thus taking care of the needs of modern mesoscopic physics. Their list is in one-to-one correspondence with the infinite families of Riemannian symmetric spaces as classified by Cartan. The present paper develops the corresponding random matrix theories, with a special emphasis on large deviation principles. Half a century ago, when physicists started to explore the usefulness of random matrix ensembles for the study of statistical properties of the spectra of heavy nuclei, their approach was firmly rooted in the classical framework of quantum mechanics. The Hamiltonian of a system was replaced by a random hermitian matrix each realization of which was supposed to commute with the appropriate unitary symmetries and with the correct “time reversals”, i.e. certain antiunitary operators. Concretely, the most general hermitian matrices that commute with time reversal in the literal sense are real symmetric matrices, whereas another

Tail inequalities for sums of random matrices that depend on the intrinsic dimension

Abstract not found

We give an upper bound in O(d (n+1)/2) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d (n+1)/2)Pn, where Pn is the (unknown) measure of the set of symmetric positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions. 1

Abstract. In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size N converges to infinity. The result is formulated on the tracial state space TS(A) of the universal C ∗-algebra A generated by two selfadjoint projections. The random pair (P(N), Q(N)) determines a random tracial state τN ∈ TS(A) and τN satisfies the large deviation. The rate function is in close connection with Voiculescu’s free entropy defined for pairs of projections.

Abstract. We study the microstate free entropy χproj(p1,..., pn) of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of χproj (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to χproj.

We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered. 1