Random Matrices and Statistical Mechanics of Charged Particle Systems

We work on two outstanding open problems in the statistical mechanics of charged particle systems. The first one is equivalent to the universality conjecture of the local eigenvalue statistics of random matrices, an equilibrium problem. The second one is the construction of relativistic Vlasov kinetic theory from a microscopic model, a nonequilibrium problem. As to the first problem, we use a new, entirely analytic strategy to extend the known universality for the unitary matrices also to the other types of random matrices: real symmetric, complex normal, and quaternionic self-dual. The strategy applies to the bulk and to the edge of the spectrum. We also apply our method to the study of the Laughlin wave function of superconductivity, which is of a related structure. As for the second problem, we use the recently laid microscopic dynamical foundations of relativistic many-particle theory to establish the first derivation of relativistic Vlasov kinetic theory in form of a weak law of large numbers. We also study its fluctuations around the limit in form of a central limit theorem.

The universality conjecture is currently one of the top priority problems of random matrix theory, a subfield of probability and mathematical physics. Its applications range from nuclear physics, nanotechnology and superconductivity on the physics and technology side to deep number theoretical implications on the mathematical side - which in turn have applications in cryptography and related fields. The conjecture has been proven so far for the simplest type of matrices, but a proof for more general matrices has so far been elusive. The relativistic Vlasov theory of charged particle systems forms the dynamical basis for a large part of high temperature plasma physics, with applications ranging from thermonuclear fusion research to space plasma research, e.g. about the solar wind and magnetic storms. Its microscopic atomic underpinnings, which have so far eluded researchers, will make it possible for the first time to systematically study the accuracy of Vlasov theory and in particular to compute its leading corrections.