Mathematical Physics
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Research Akemann


Fields of Research: Random Matrix Theory (RMT); orthogonal polynomials in the complex plane; universality; lattice gauge theory, effects of finite volume, spacing and chemical potential; econo-physics

RMT has a long history with many applications in modern physics, mathematics and other sciences. The underlying strategy is often to replace a complicated, interacting physical theory by a simple, zero-dimensional RMT. The only ingredients used are global, anti-unitary symmetries such as time reversal or complex and charge conjugation. The problem to be described (Hamiltonian, Dirac operator, covariance matrix, etc.) is then replaced by a random matrix with the same symmetries, and a Gaussian distribution of its matrix elements in the simplest case.

The solution of the RMT under consideration is then a well defined mathematical problem that can be solved exactly for infinite or even finite matrix size, using e.g. classical orthogonal polynomials among others. Sometimes the RMT approach is not merely heuristic, but represents a well defined approximation, as an effective theory of the relevant degrees of freedom. An example for such a map is in Quantum Chromodynamics (QCD), the theory of strongly interacting quarks and gluons: at low energies in the confined phase with broken chiral symmetry QCD is well approximated by an effective theory of its pseudo-Goldstone bosons: chiral perturbation theory. This reduces further to a RMT in a certain limit, in which the effects of the chiral limit, chemical potential, finite volume, or very recently finite lattice spacing can be efficiently studied.

A prominent tool to solve QCD nonperturbatively is Lattice QCD, doing numerical simulations on a finite space-time lattice. This is one of the major activities in the High Energy Group here in Bielefeld. However, at nonzero chemical potential this approach suffers from the infamous sign problem, as the Euclidean Dirac operator becomes non-Hermitian. For that reason I have been particularly interested in non-Hermitian RMT which remains analytically solvable. By now we have completed the program to solve all corresponding 3 symmetry classes of orthogonal, unitary and symplectic RMT, in so-called Wishart two-matrix models. In the cases where Lattice QCD has no sign problem we have seen so far an impressive agreement with the data.

Some of my mathematical tools are (skew) orthogonal polynomials in the complex plane and their asymptotic, saddle-point methods, or supersymmetry. An important question is that of universality in RMT, which is to large extent open for correlation functions of complex eigenvalues .
Does RMT provide a unique answer when deforming a Gaussian distribution of matrix elements? This is crucial for the predictability of RMT, and I am very interested in heuristic and exact universality proofs. This and related questions are studied in parallel in the Probability Theory and Mathematical Statistics Group in Bielefeld, in particular for non-invariant ensembles.

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  • @ 2011 Universität Bielefeld
  • | letzte Aktualisierung: 16.02.2011
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