We give expansions of reproducing kernels the Christoffel-Darboux type in terms Schur polynomials. For this, we use evaluations averages characteristic polynomials and random matrix ensembles. explicitly compute new averages, such as average a $q$-Laguerre ensemble, ensuing kernels. In addition to classical $q$-deformed cases on real line, extensions Dotsenko-Fateev integrals obtain expressions...

Random-matrix theory gained attention during the 1950s due to work by Eugene Wigner in mathematical physics. Specifically, Wigner wished to describe the general properties of the energy levels (or of their spacings) of highly excited states of heavy nuclei as measured in nuclear reactions (Wigner, 1957). Such a complex nuclear system is represented by an Hermitian operator H (called the Hamilto...

3 Universality 4 3.1 Macroscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Wigner’s semicircle law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Microscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Bulk universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

This summary will briefly describe some recent results in random matrix theory and their applications. 1 Motivation 1.1 Multiple Antenna Gaussian Channels 1.1.1 The deterministic case Consider a gaussian channel with t transmitting and r receiving antennas. The received vector y ∈Cr will depend on the transmitted vector x ∈Ct by y = Hx+n where H is an rxt complex matrix of gains and n is a vect...

which is the Central Limit Theorem. In principle, all the random variables X1, X2, · · · , XN can be of order 1, hence SN ∼ 1 as well, but the probability of having such a rare event is incredibly small. We can even estimate the bound on the probability for the rare event from the large deviation principle. A similar phenomenon happens when we form a large matrix from i.i.d. random variables an...

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These...

This handout provides the essential elements needed to understand finite random matrix theory. A cursory observation should reveal that the tools for infinite random matrix theory are quite different from the tools for finite random matrix theory. Nonetheless, there are significantly more published applications that use finite random matrix theory as opposed to infinite random matrix theory. Ou...

There is a wedge product notation that can facilitate the computation of matrix Jacobians. In point of fact, it is never needed at all. The reader who can follow the derivation of the Jacobian in Handout #2 is well equipped to never use wedge products. The notation also expresses the concept of volume on curved surfaces. For advanced readers who truly wish to understand exterior products from a...