Universality for random nodal domains – UNIRANDOM

Nodal sets, i.e. vanishing loci of functions, are central objects in mathematics. Understanding the main features of a purely deterministic nodal set is generally out of reach, as illustrated by several celebrated open problems, such as Hilbert’s sixteenth problem or Yau’s conjecture. In order to capture the typical behavior of an object, one is thus tempted to randomize, which reduces here to consider nodal sets associated to random functions. Computing expected values, variances or else fluctuations around the mean of the considered nodal functionals, and in particular understanding their asymptotic behavior as the amount of noise goes to infinity, is then a true wealth of information about the possible deterministic behaviors. Besides, randomization of nodal sets is also strongly motivated by deep physical insights, such as the celebrated Berry’s conjecture.

In this framework, the project UNIRANDOM is mainly focused on universality results, that is, asymptotic properties of random level sets, holding regardless of the specific nature of the randomness involved. Establishing such universal properties for random zero sets allows one to manage what would be otherwise inextricable objects in a purely deterministic setting, which explains the tremendous importance of this area of research.

The project concentrates on four main multidimensional models of random nodal sets which find their roots in various and rich domains of mathematics and physics, namely: A. Nodal domains associated to random eigenfunctions on generic Riemannian compact manifolds, B. Arithmetic random waves, C. Random algebraic manifolds, D. Periodic random fields. This project is thus intrinsically multidisciplinary and brings together five young researchers with a common solid background in probability theory but also complementary domains of expertise. Furthermore, our project is particularly ambitious and innovative since the question of universality, although well understood for several models in dimension one, has hardly been investigated in multidimensional frameworks.

The estimated global cost of the project is 115000 euros, with the following repartition : 35% of the budget will be dedicated to missions/travels/invitations expenses, 35% devoted to the organization of 4 workshops and 1 international conference. Around 20% will serve as a compensation of teaching hours to enable the members to fully concentrate on the project. The remaining part of the budget will be used to buy computers, books, or other material, and management fees.

The main expected outcome of our project is a substantial progress in the understanding of universality phenomena for random nodal domains which, matter for both probability theory, differential and algebraic geometry and analysis. This progress will be mainly concretely materialized by publications in international peer-reviewed journals, oral communications in seminars and conferences, as well as active participations to diffusion of science for larger audience. Another expected outcome of our project is a strengthening of the collaborations between the institutes of the project members, as well as the increase of their visibility and competitivity on both national and international scenes.