Directions in Orbits of Geometrically Finite Hyperbolic Subgroups¹:
Consider a discrete subgroup of the isometry group of half-space (for example a lattice). Then take the orbit of a point by said subgroup. Consider now another point in the space which we call the observer. The goal of this project is to understand the distribution of the directions of the points of the orbit in larger and larger balls around the observer (where the ‘direction’ of a point is the vector pointing along the geodesic connecting the observer to the point). We are able to fully describe the limiting distribution for so-called geometrically finite subgroups.
The problem has been addressed previously by Marklof and Vinogradov (for lattices) and by Zhang (for certain other examples). Although the set up may seem rather contrived the problem is in fact rather natural. Geometrically finite subgroups are of increasing interest due to their connection to circle packings, Schottky groups and other objects in hyperbolic geometry. Recently Mohammadi, Oh, Shah and others have proven results about the asymptotic number of orbit points in larger and larger balls. Which is a generalization of lattice point counting. Our work can be viewed as a fine-scale analysis of the same dynamics.
[This project has been completed and submitted to the Arxiv]
The Random Lorentz Gas – Going Beyond Boltzmann-Grad: (joint with Bálint Tóth)
Consider an array of spherical scatterers distributed throughout the plane on the points of a Poisson process. Consider a point particle moving in straight lines and colliding elastically with the scatterers. This is the so-called Lorentz Gas, introduced by Henrik Lorentz in 1905. Gallavotti (1970) and later Spohn (1978) proved that under the correct scaling (Boltzmann-Grad limit) as the radius of the scatterers goes to zero, for finite time, the Lorentz gas converges in distribution to a random flight process (Gallavotti showed that the limiting process obeys a linear Boltzmann equation while Sphon proved the convergence and more).
Our goal in this project is to extend this picture. By using probabilistic coupling, one can jointly realize both the Lorentz gas and the Markovian flight process for times proportional to the radius of the scatterers to the power -2. Hence we are looking at infinite times. We can prove the necessary convergence which allows us to prove an invariance principle and central limit theorem.
It has long been of great interest to study this problem for infinite times. As we need to take the joint limit of radius to zero and time to infinity, this is a big step in that direction.
[This project is in the process of being written up]
¹ The image on the left is taken from the fantastic and beautiful book Indra’s Pearls: The Vision of Felix Klein – Mumford, Series and Wright. I highly recommend checking it out, it is written for any mathematical background and gives fantastic background to the theory of Schottky groups, circle packings and, in general, symmetry.