01699nas a2200121   4500008004100000245011400041210006900155260001000224520117400234653002501408100001701433856012701450       2015                        en  d00aNormal matrix models and orthogonal polynomials for a class of potentials with discrete rotational symmetries0 aNormal matrix models and orthogonal polynomials for a class of p  bSISSA3 aIn this thesis we are going to study normal random matrix models which generalize naturally the polynomially perturbed Ginibre ensamble, focusing in particular on their eigenvalue distribution and on the asymptotics of the associated orthogonal polynomials. \\
The main result we are going to present are the following:
\begin{itemize}
\item we describe the explicit derivation of the equilibrium measure for a class of potentials with discrete rotational symmetries, namely of the form
\[V(z)=|z|^{2n}-t(z^{d}+\bar{z}^{d})\qquad n,d\in\mathbb{N},\ \ d\leq2n\ \ t>0 .\]
\item We obtain the strong asymptotics for the orthogonal polynomials associated to the weight
\[ e^{-NV(z)},\quad V(z)=|z|^{2s}-t(z^s+\bar{z}^{s}) \qquad z \in \mathbb{C},\;s\in \mathbb{N},\quad t>0,\]
and we will show how the density of their zeroes is related to the  eigenvalue distribution of the corresponding matrix model;
\item We show how the conformal maps used to describe the support of the equilibrium measure for polynomial perturbation of the potential $V(z)=|z|^{2n}$ lead to a natural generalization of the concept of polynomial curves introduced in by Elbau.
\end{itemize}10aMathematical Physics1 aMerzi, Dario  uhttps://www.math.sissa.it/publication/normal-matrix-models-and-orthogonal-polynomials-class-potentials-discrete-rotational