%0 Thesis
%D 2015
%T Normal matrix models and orthogonal polynomials for a class of potentials with discrete rotational symmetries
%A Dario Merzi
%K Mathematical Physics
%X In 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}
%I SISSA
%G en
%1 34938
%2 Mathematics
%4 1
%# MAT/07
%$ Submitted by Dario Merzi (dmerzi@sissa.it) on 2015-10-24T00:25:34Z
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%0 Report
%D 2013
%T Equilibrium measures for a class of potentials with discrete rotational symmetries
%A Ferenc Balogh
%A Dario Merzi
%X In 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 $|t|t_{cr}$.
%I SISSA
%G en
%U http://hdl.handle.net/1963/7230
%1 7270
%2 Mathematics
%4 1
%$ Submitted by Dario Merzi (dmerzi@sissa.it) on 2013-12-09T12:34:49Z
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