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-rotational01024nas a2200121 4500008004100000245008800041210006900129260001000198520062200208100001900830700001700849856003600866 2013 en d00aEquilibrium measures for a class of potentials with discrete rotational symmetries0 aEquilibrium measures for a class of potentials with discrete rot bSISSA3 aIn this note the logarithmic energy problem with external potential
$|z|^{2n}+tz^d+\bar{t}\bar{z}^d$ is considered in the complex plane, where $n$
and $d$ are positive integers satisfying $d\leq 2n$. Exploiting the discrete
rotational invariance of the potential, a simple symmetry reduction procedure
is used to calculate the equilibrium measure for all admissible values of $n,d$
and $t$.
It is shown that, for fixed $n$ and $d$, there is a critical value
$|t|=t_{cr}$ such that the support of the equilibrium measure is simply
connected for $|t|t_{cr}$.1 aBalogh, Ferenc1 aMerzi, Dario uhttp://hdl.handle.net/1963/7230