Abstract for the talk at 05.11.2013 (15:15 h)Oberseminar ANALYSIS - PROBABILITY
Zakhar Kabluchko (Universität Ulm, Institut für Stochastik, Germany)
Zeros of random polynomials and random partition functions
We will be interested in the distribution of complex zeros of the random polynomials of the form <center class="math-display"> <img src="/fileadmin/lecture_img/tex_14516c0x.png" alt=" &#x2211;n k Pn(z) = fk,n&#x03BE;kz , k=0 " class="math-display"></center> where fk,n are deterministic complex coefficients and ξ1,ξ2,… are independent identically distributed random variables such that E log |ξk| < ∞. Under suitable conditions on the coefficients fk,n we will derive a formula for the limiting distribution of complex zeros of Pn as n →∞. For example, for the Weyl polynomials of the form <center class="math-display"> <img src="/fileadmin/lecture_img/tex_14516c1x.png" alt=" n P (z) = &#x2211; &#x03BE; z&#x221A;k-, n k=0 k k! " class="math-display"></center> the limiting distribution of zeros is the uniform distribution on the unit disc. This is an analogue of the celebrated circular law for random matrices.
In the last part of the talk we will briefly consider the distribution of the partition function zeros (Lee–Yang–Fisher zeros) for two models of spin glass: the Random Energy Model and the Generalized Random Energy Model.