RandPol · Zeros of random polynomials

Zeros of random polynomials

Summary of the project: The aim of this project is to study the zeros of random polynomials. Random polynomials appeared in the 1930's with the pioneering works of Bloch, Polya and Kac. The study of random polynomials gained a strong interest of the physicists since the 1990's due to their connexion with interaction particle systems such as electron gases (also called Coulomb gases). Random polynomials were first considered as a toy model for more complicated systems, but in the 2000's they appeared to exactly describe the behavior of certain Bose Gases. We want to understand the behavior of random polynomials at the macroscopic and microscopic scale. At the macroscopic scale, several questions naturally arise: Where can we locate the zeros? Are there isolated zeros far away from the rest of them? Can we describe the behavior of those isolated zeros? Very recent breakthrough (2019) were made in both localisation and isolated zeros for specific models of random pomynomials with some symmetry constraints. Our goal is to obtain general results on these questions. We plan to develop new ideas to tackle these questions, without relying on the symmetry structure. At the microscopic scale, we want to understand the local repartition of the zeros and the interaction between neighboors. This question leads to adapt concepts from mathematical physics such as the microscopic renormalized energy to the study of zeros of random polynomials. Recent studies also linked the local behavior of zeros of random polynomials to the zeros of Random Analytic Functions, which have a lot of connexions to several domains in mathematics (Analysis, Image processing,random matrix theory, mathematical physics). The concept of renormaliazed energy is very recent (2017), and was introduced by Leblé and Serfaty to understand the local behavior of particles from a Coulomb gas. We think that this approach can lead to a new understanding of the behavior of zeros of random polynomials. Objectives : The main objective is to build a bridge between the study of zeros of random polynomials and an a specifial class of Coulomb gases: the determinantal planar jellium. The first goal is to understand the behavior of the particules outside of their equilibirum set, if they exist, and the second is to understand the microscopic behavior of the particules when we zoom inside the equilibrium set. Impact: Our results are mostly interesting for mathematicians (in probability and mathematical physics), and physicists (statistical physics). Understanding very precisely a two dimensional integrable model may not be immediatly useful for practical use, but it allows us to develop tools and intuitions on what could happen for real systems. This project allowed us to build bridges between mathematical theories, and to enlarge the scope of available techniques to solve problems.

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