Zur Seitenansicht

Titelaufnahme

Zugriffsbeschränkung
 Das Dokument ist frei verfügbar.
Links
Abstract

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.