• phase diagram
• random matrix
• QCD
• phase transitions
• RMT

Random Matrix theories have proven to be an excellent tool to study the phase diagram of Quantum Chromodynamics (QCD). The thermodynamics of QCD predicts a phase transition from the confined hadron matter to a state of deconfined weakly interacting quark and gluons, the so-called Quark Gluon Plasma (QGP). Such transition can be studied in the plane parametrized by the temperature T and the baryon chemical potential. The formulation of QCD on the Lattice (LQCD), which relies on Monte Carlo techniques of numerical integration, fails to investigate directly the phase diagram at nonzero chemical potential because of the notorious sign problem.
On the other hand, effective theories are able to provide useful information about the existence and the nature of such transition.
Chiral Random Matrix Theory(CRMT) is one of them. In particular one of the main challenges is to determine theoretically and experimentally the position of the critical endpoint of QCD, a second order point which represents the endpoint of a supposed first order line that continues at lower temperature and higher chemical potential.
Born to investigate the spectral properties of the Euclidean Dirac operator, CRMT reproduces some of the main features of the QCD phase structure at nonzero temperature and chemical potential with two flavors. Using a Landau-Ginzburg-like effective potential, in this work we are going to study the properties of a RM model introduced in the final form in a paper of Halasz, Jackson, Shrock, Stephanov and Verbaarschot (arXiv:hep-ph/9804290v2). The virtue of such model is the opportunity to study in an analytical way the chiral transition, especially analyzing how the zeros of the partition function in the complex chemical potential plane (and the singularities of the potential) behave in thermodynamic limit of infinite volume according to the Yang-Mills theory. In RM theories the volume is replaced by the dimension of the matrices, N. The advantage to have an explicit expression of the partition function both in the finite and infinite size case allows to study finite size effects.
The central part of the thesis is focused on the extrapolation of the radius of convergence of a Taylor series of the effective potential (i.e. the free energy of the system) at fixed temperature in the baryon chemical potential centered around zero. This is one of the techniques even used in LQCD, because of the chance to perform simulations at nonzero chemical potential and reconstruct partially the power series with the available coefficients. The knowledge of the radius of convergence would allow to estimate the position of the critical endpoint of the QCD phase diagram and the spinodal lines of the first order transition.
In principle such Taylor expansion is not essential for the RM model because we are exactly able to determine through other ways the location of the critical point and the phase boundary. However this toy model can be interesting to test the advantages and the limits of some mathematical methods of power series analysis and decide what is the best to be used even in LQCD.
The second part of the work consist of a modification of the model including all (or at least a great number of) the Matsubara frequencies, indeed the initial model describes successfully a mean-field transition using only the lowest one.
The third part analyzes the potentialities and the limits of the model to reproduce the phase structure of QCD at imaginary chemical potential, already known from LQCD simulations.
The final part of the work represents the effort to use the previous techniques of power series analysis with real LQCD data, in particular to detect the Roberge-Weiss singularity at imaginary chemical potential.