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Zeros of random polynomials

Periodic Reporting for period 1 - RandPol (Zeros of random polynomials)

Reporting period: 2020-09-01 to 2022-08-31

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.

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.
The fact that zeros of random polynomials cluster on a certain part of space, called the equilibrium set, has been studied intensively since the 1970's. Recently, it has been shown that this phenomenon is a common feature of all known models of random polynomials. Our primary goal was to study the existence of zeros outside the equilibirum set: we call them outliers. The second objective was to study outliers in another mathematical physics model, the jellium, which describes the position of electrons attracted by a continuous positive charge distribution. This model the best electrostatic candidate to model the behavior of zeros of random polynomials, and our goal was to understand zeros of random polynomials and jellium jointly.
We managed to obtain a full description on the locations of zeros of random polynomials and the jellium outside the support of their equilibirum measure.
Two situations can appear: first, if we are looking in a simply connected hole of the equilibrium set (a hole which has no hole), then the outliers of zeros of random polynomials and the jellium are described by a universal point process, call the Bergman point process. This means that the outliers do not depend on the specificity of the model and follow a universal behavior. To our knowledge, this is the first model which presents such outlier structure. Second, if the hole is not simply connected (for instance, an annulus), we made a very surprising discovery: the outliers do not converge, but can be described by a continuous family of weighted Bergman point processes. Furthermore, numerical simulations tend to show that these point processes are very close one from another, and it seems very hard tp distinguish them. This is the first model of mathematical physics for which such behavior arises, and it opens many perspectives for the study of general Coulomb gases.
The result for the jellium is slightly more general than the result for zeros of random polynomials. Zeros of random polynomials are a very complicated object to study and we reach the limits of the available theoretical tools, which forced us to do some extra assumptions.
Scientific progress and potential impact:
The outcome of this project is the paper "Universality for outliers in weakly confined Coulomb-type systems", made in collaboration with David Garcia-Zelada, Alon Nishry and Aron Wennman. In the paper, we discover a new behavior for the particles of a Coulomb gas outside the support of the equilibrium measure. For a specific class of Coulomb gases, called jellium, we show that the outliers are described by a new universal process called the Bergman process. In addition, we also establish this behavior for zeros of random polynomials. This is a success for this research project as the goal was to obtain new results by comparing Coulomb gases and zeros of random polynomials. In this paper, we develop, in the case of Coulomb gases, a new technique to study the asymptotic properties of orthogonal polynomials in dimension 2, and techniques to understand reproducing kernels associated to orthogonal polynomials. In addition, we later realized that this techniques could be adapted to zeros of random polynomials.

I hope that this result will lead to many other interesting research projects in the same field, as it opens many new perspectives for both Coulomb gases and random polynomials. In the Coulomb gas litterature, the outliers were usually not studied and the focus was made on outliers-free systems. Now, outliers can be understood and studied for their own interest.

Socio economic impact:
The results of this project deal with high energy statistical physics. In theory, there exist physical systems which can be build to observe the results of this project. Unfortunately, it is highly unrealistic to expect a physical realization of these systems: the amount of electric energy needed to observe the results of this project is exponential with the number of particles and would require unreasonable amounts of energy.
The results of this project are part of a broad scientific trend, with the growing interest of the understanding of outliers in many particles systems (random matrices, coulomb gases, random polynomials), and asymptotic properties of zeros of random polynomials in signal processing. The results obtained here may not be directly used in applied problems, but I am convinced that the techniques that we developed will diffuse and influence other researchers in the areas mentionned previously.
Zeros of Kac random polynomials