Benedek Valkó

Continuum limits of random matrices and the Brownian carousel

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine β is continuous in the gap size and β, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at β=2.

Onsager Relations and Eulerian Hydrodynamic Limit for Systems with Several Conservation Laws

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yaus relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsagers reciprocity relations must hold. We check validity of these identities whenever a stationary measure with product structure exists. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena. The Onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel. The fact that equilibrium thermodynamic entropy is Lax entropy for the arising Euler equations was noticed earlier in the context of Hamiltonian systems with weak noise, see ref. 7.

t1/3 Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on {\mathbb{Z}}

We consider finite-range asymmetric exclusion processes on

Large gaps between random eigenvalues.

{\mathbb{Z}}

Random Schrödinger operators on long boxes, noise explosion and the GOE

with non-zero drift. The diffusivity D(t) is expected to be of

Diffusivity of Lattice Gases

{\mathcal{O}}(t^{1/3})