Probability and Mathematical Physics Seminar

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Arbitrarily fast dispersion in long range crystals

Speaker: Mostafa Sabri, NYUAD

Location: Warren Weaver Hall 1302

Date: Friday, October 9, 2026, noon

Synopsis:

A dispersive estimate is a result of dynamical delocalization. It asserts that the supremum norm of wave packets decays like t^{-beta} as time goes on, under Schrodinger evolution. More precisely, || e^{itH} f||_infty \le t^{-beta} || f ||_1. A classical example is when H is the Laplacian on Z, where beta = 1/3. The fastest speed in one dimensional systems prior to our work was beta = 1/2, to the best of our knowledge. 

In this talk, I will discuss a result of very fast dispersion, where for any N no matter how large, we construct a 1d translation invariant graph with long range hopping where the dispersion speed is bounded by t^{-N}. The underlying Floquet function is a Weierstrass nowhere differentiable function W. As a special case we answer a question of Geman and Horowitz from 1980 regarding the existence of the local time of W for sufficiently large lacunarity.

The estimate boils down to controlling oscillatory integrals with highly irregular oscillatory phases, precluding standard van der Corput/stationary phase techniques. I will explain how we overcome this using a Dolgopyat inspired transfer operator method, which first arose in the study of Anosov flows. I will also mention a new van der Coprut estimate as well as some lower bounds on oscillatory integrals showing that our result is essentially sharp. Although the result is deterministic, the model bears analogies with series of random variables and leads to some questions on random Toeplitz matrices.

This is based on a joint work with Gaetan Leclerc and Tuomas Sahlsten.