Random Models in Arithmetic and Spectral Theory

Project Information

RMAST

Grant agreement ID: 786758

DOI

10.3030/786758

Project closed

EC signature date 10 April 2018

Start date 1 February 2019

End date 31 January 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 840 625,00

EU contribution

€ 1 840 625,00

Coordinated by

TEL AVIV UNIVERSITY
Israel

Periodic Reporting for period 4 - RMAST (Random Models in Arithmetic and Spectral Theory)

Reporting period: 2023-08-01 to 2025-01-31

Summary of the context and overall objectives of the project

The proposal studies deterministic problems in the spectral theory of the Laplacian and in analytic number theory by using random models. There are a number of projects in spectral theory on this theme, all with a strong arithmetic ingredient. One is about minimal gaps between the eigenvalues of the Laplacian, where I seek a fit with the corresponding quantities for the various conjectured universality classes (Poisson/GUE/GOE). Another project is about curvature measures of nodal lines of eigenfunctions of the Laplacian, where I seek to determine the size of the curvature measures for the large eigenvalue limit. Yet another project originates in analytic number theory, on angular distribution of prime ideals in number fields, function field analogues and connections with Random Matrix Theory, where I raise new conjectures and problems on a very classical subject, and aim to resolve them at least in the function field setting

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The project studies deterministic problems in the spectral theory of the Laplacian and in analytic number theory by using random models. We have achieved some of the goals of the proposal, are continuing to work on the others and in addition have developed some of the themes of the proposal in unexpected directions.
Together with my (former) Ph.D. student Ezra Waxman, we have completed the study of angles of Gaussian primes, constructing a good function field model for these and completing numerical comparisons (publication 1), as well as ontaining rigorous results on almost all sectors with my poistdoc Bingrong Huang and with Jian Liu (publication 2).
We have made significant advances on the subject of the minimal gap distribution in arithmetic sequences, and my student Shvo Regavim is furthering this topic.
Together with my collaborator Prof. Igor Wigman we have completed a study of the curvature measures of nodal lines of random eigenfunctions of the Laplacian on the flat torus (publication 12).
Together with my postdoc Niclas Technau we have investigated the pair correlation function of random dilates of sequences of integers (publication 8), and hope to achieve a long-sought goal of showing that for almost all dilates, the pair correlation of square roots of integers in Poissonian.
An unexpected direction which we have started working on is the statistical properties of the eigenvalues of the Laplacian with Robin boundary conditions, where, together with Dr. Nadav Yesha and Prof Igor Wigman, we made significant inroads in understanding their deviations from the Neumann spectrum (publication 6).
A completely new direction is related to spatial statistics, where together with Dr. Tom Dvir and Prof Renana Peres (Business School at the Hebrew University) we have modelled decisions under uncertainty by using Voronoi tessellations (publication 11).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

1. Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions, and a temperature fixing boundary, described by Dirichlet boundary conditions. To date, most studies concentrated on the first few Robin eigenvalues, with applications in shape optimization and related isoperimetric inequalities and asymptotics of the first eigenvalues.
Our goal is very different, aiming to study the difference between high-lying Robin and Neumann eigenvalues. I expect the questions that arose in this study to significantly advance spectral theory.


2. The theory of uniform distribution of sequences modulo one has a long history, with many developments since its creation over a century ago. Much more recent is the study of ``local'' statistics of sequences, motivated by problems in quantum chaos and the theory of the Riemann zeta function. Key examples of such local statistics are the nearest neighbour spacing distribution, which is defined as the limit distribution (assuming it exists) of the gaps between neighbouring elements in the sequence, rescaled so as to have mean value unity; and the pair correlation function, which measures repulsion between pairs of levels, and is analytically the most accessible example of a ``local'' statistic.

For many sequences arising in number theory, one believes that these local statistics coincide with those of random numbers, but establishing this rigorously is notoriously difficult. One of the few examples where the level spacing distribution was understood is that of the fractional parts of square roots of integers, which was discovered to be non-random in 2004. However, if one perturbs the sequence by dilating it, then it is believed that the generic dilates will revert to having random behavior. One of the main goals for the second part of the project is to establish it for the pair correlation function, and the work I am currently doing with my former postdoc Niclas Technau holds great promise for this.

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