TY  - THES
AB  - 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.
DA  - 2021
LA  - eng
PY  - 2021
TI  - Characteristic polynomials of random matrices and their role in an effective theory of strong interactions
UR  - https://nbn-resolving.org/urn:nbn:de:0070-pub-29565034
Y2  - 2024-11-22T04:43:18
ER  -