Descripción del proyecto
Nuevos enfoques para describir los ceros de los polinomios aleatorios
Los polinomios aleatorios atraen la atención de los físicos porque pueden describir el comportamiento de los gases de Coulomb o ciertos gases de Bose con mucha precisión. El proyecto financiado con fondos europeos RandPol se propone estudiar los ceros de los polinomios aleatorios, uno de los problemas matemáticos más antiguos y fundamentales. Las cuestiones que suelen surgir a escala macroscópica son las siguientes: ¿Dónde se pueden ubicar los ceros? ¿Existen descripciones exhaustivas del comportamiento de los ceros aislados? Los investigadores también se proponen comprender el comportamiento de los polinomios aleatorios a escala microscópica. En particular, RandPol relacionará la distribución local de los ceros y la interacción entre ceros adyacentes con la energía renormalizada, clave para describir el comportamiento local de las partículas en un gas de Coulomb.
Objetivo
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 lead 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 physiqucs).
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.
Ámbito científico
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Régimen de financiación
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinador
1211 Geneve
Suiza