PUB - Publikationen an der Universität Bielefeld

The aim of this thesis is to investigate the impact of characteristic polynomials on the spectral eigenvalue statistics of random matrix models, with applications in effective field theory models of Quantum chromodynamics (QCD). The symmetries of the field theory lead to random matrix ensembles named chiral Gaussian Unitary Ensemble (chGUE(N)) and extensions thereof. The random matrix ensembles are comparable to the effective theory of QCD in a low-energy regime, where chiral symmetry breaking is predominant and it suffices to consider only the smallest eigenvalues of the QCD Dirac operator. We consider four members of the chGUE(N) symmetry class: the classical chGUE(N) consisting of Hermitian, chiral block matrices with complex entries and its extensions by N_f massive flavors describing dynamical quarks. Furthermore, we consider the chGUE(N) extended by external parameters describing effects of external sources like temperature and its combination with Nf massiv flavors. The correlations of the chGUE(N), and its extensions with external parameters, als well as its deformations with massive flavors, belong the class of determinantal point processes. This implies that correlation functions can be expressed as determinants of a correlation kernel. The random matrix ensembles we consider feature special biorthogonal structures leading to a sub-class of determinantal point processes called invertible polynomial ensembles. Such ensembles are characterised by a joint probability density function (JPDF) containing two determinants, which can be linked to orthogonal polynomials, if the considered model is independent of temperature. If temperature is present as an external source, the JPDF has biorthogonal structure and the usage of orthogonal polynomials becomes more involved. in this case, the correlation kernel can be expressed in terms of expectation values of ratios of characteristic polynomials. We will derive a multicontour- integral representation of the expectation value of an arbitrary ratio of characteristic polynomials for invertible polynomial ensembles at finite matrix size N. Additionally, we perform a saddle point analysis and derive the large N asymptotic form of the correlation kernel for the chGUE(N) matrix models including temperature as an external source. The limiting kernels show determinantal structures comparable to existing results partially derived with supersymmetry and orthogonal polynomial methods. We show that the limiting kernel for non-zero temperature models is indeed equivalent to existing results for temperature independent models. Furthermore, we show that the resulting correlation functions for both zero and non-zero temperature models agree with existing formulae of the correlation functions derived via supersymmetry. This answers the question wether the correlations of the underlying physical field model are indeed universal in the low-energy regime, where random matrices can be used to model QCD effective field theories.