We reveal a phase transition with decreasing viscosity in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities reflecting a spontaneous one step replica symmetry breaking in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory. (Based on the joint work with Pierre Le Doussal and Alberto Rosso.)

Choose at random a matrix from the unitary group and cut a square block from it. The obtained matrix is a random contraction. Such matrices appear in a variety of mathematical contexts and are also used for modelling physical processes. After surveying some of these applications, I will talk about universal statistical patterns in the distribution of eigenvalues of random contractions in the limit of infinite matrix dimension.

Statistical properties of quantum transport are considered for a chaotic cavity with an arbitrary number of open channels. In the framework of the random matrix approach, we establish the relevance of the Selberg integral theory to the problematic and apply it to calculate exact explicit expressions of low-order cumulants of the conductance and shot-noise. By further exploiting the marriage of the Selberg integral with the theory of symmetric functions (Schur functions), we develop a powerful method for computing the moments of the conductance and shot-noise (including their joint moments) of arbitrary order and at any number of open channels. The approach is applicable equally well for systems with and without time-reversal symmetry. We also give a detailed discussion of the corresponding cumulants, the distribution functions, etc.

Linear statistics on ensembles of random matrices occur frequently in many applications. We present a general method to compute probability distributions of linear statistics for large matrix size N. This is applied to the calculation of conductance and shot noise for ballistic scattering in chaotic cavities, in the limit of large number of open channels. The method is based on a mapping to a Coulomb gas problem in Laplace space, displaying phase transitions as the Laplace parameter is varied. As a consequence, the sought distributions generally display a central Gaussian region flanked on both sides by non-Gaussian tails, and weak non-analytical points at the junction of the two regimes.

In this talk I will show how the characteristic polynomial of a random unitary matrix has been successfully used in number theory to model the Riemann zeta function, and then I will present some new work re-interpreting and generalizing the previous random matrix results in a probabilistic setting. This is joint work with Paul Bourgade, Ashkan Nikeghbali and Marc Yor.

It is well established numerically that the spectral statistics of pseudo-integrable models differ considerably from the reference statistics of integrable and chaotic systems. In a previous paper by Bogomolny and Schmit the statistical properties of a certain quantized pseudo-integrable map had been calculated analytically but only for a special sequence of matrix dimensions. This talk aims at describing the method in order to obtain the spectral statistics of the same quantum map for all matrix dimensions.

The speaker will show how Ulam's increasing subsequence problem is connected to a variety of classical integrable systems.

Phase transitions can often be linked to the spectral properties of certain physical operators. In the theory of the strong interactions the fundamental breaking of chiral symmetry is intimately linked to the spectrum of the Dirac operator. When, however, quarks are favored over anti quarks by means of a chemical potential the Dirac operator becomes non hermitian. In this case the link between the Dirac spectrum and breaking of chiral symmetry has remained a puzzle (The Silver Blaze Problem) for more than two decades. In this talk we discuss how Random Matrix Theory solves this problem.

The eigenvalue statistics at the edge of the spectrum of large random matrices from the Gaussian unitary ensemble (GUE) are described by the Airy point process and the maximal eigenvalue is asymptotically Tracy-Widom distributed. In contrast, for the complex Ginibre ensemble (consisting of matrices with iid complex Gaussian entries), extreme eigenvalues behave like a Poisson process and the maximal modulus (or maximal real part) of the eigenvalues converges to a Gumbel-distributed random variable. In this talk, a family of ensembles interpolating between these models is considered, and we show how a non-trivial transition between Airy and Poisson statistics occurs for the eigenvalues near the edge of the spectrum.

Tuck devised a function Q(x), associated with a function D(x), whose positivity guarantees the absence of complex zeros of D(x) close to the real x axis, and observed that large values of Q are very rare if D is associated with the Riemann zeros. In an unusual and challenging application of random-matrix theory with P Shukla, this is explained by studying the probability distribution P(Q) for functions D with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE).

The characteristic polynomials of random matrices from various random matrix ensembles have been investigated over the last few years. In this talk I will present some results on the correlations of the characteristic polynomials of general Wigner matrices. It turns out that the well-known results for Gaussian Wigner matrices essentially continue to hold in this setting. This talk is partly based on joint work with Friedrich Goetze.