arXiv subjects

Christopher Lutsko

Publications and source records attributed to Christopher Lutsko.

25 records · Page 2Linked to original sources

arXiv2020

Long-Range Correlations of Sequences Modulo 1

In this paper we consider the fractional parts of a general sequence, for example the sequence $α\sqrt{n}$ or $αn^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $αn^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations.

math.DS↗

arXiv2020

Directions in Orbits of Geometrically Finite Hyperbolic Subgroups

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case, orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather, they may equidistribute on a fractal subset. Understanding the behaviour of these orbits near the boundary is central to Patterson-Sullivan theory and much further work. Our theorem characterizes the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish an formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which was not known previously even in the lattice case.

math.DS↗

arXiv2020

Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

We prove the invariance principle for a \emph{random Lorentz-gas} particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius $r$, placed according to a Poisson point process of density $\varrho$, in the limit $\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order $T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969), Spohn (1978) and Boldrighini-Bunimovich-Sinai (1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski-Ryzhik (2006), respectively, Erd\H os-Salmhofer-Yau (2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2)The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation -- expressed in terms of the kinetic time scale -- is much longer and fully explicit.

math.PR↗

arXiv2020

Gap Statistics of the Sequence $\{α\sqrt{n}\}$

The gaps in the sequence $\{\sqrt{n}\}$ were shown by Elkies-McMullen (2004) to have a limiting distribution which is not the exponential distribution. However it is conjectured that the distribution of gaps in the sequence $\{α\sqrt{n}\}$ is exponential, provided $α^2$ is irrational. For almost all values of $α$, we prove an important step in this direction. In particular, we show that all the correlations are Poissonian along a subsequence. Therefore, our result implies that the gap distribution converges to the exponential distribution along the same subsequence.

math.DS↗

arXiv2019

Invariance Principle for the Random Wind-Tree Process

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes - the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912). We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.

math.PR↗

arXiv2019

Farey Sequences for Thin Groups

The classical Farey sequence of height $Q$ is the set of rational numbers in reduced form with denominator less than $Q$. In this paper we introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete (and in particular thin) subgroups, they can be used to study interesting number theoretic sequences - for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute, that the gap distribution converges, and we answer an associated problem in Diophantine approximation with Fuchsian groups. Moreover, for one specific example, we use a sparse Ford configuration construction to write down an explicit formula for the gap distribution. Finally for this example, we construct the analogue of the Gauss measure in this context which we show is ergodic for the Gauss map. This allows us to prove a theorem about the Gauss-Kuzmin statistics of the sequence.

math.DS↗

arXiv2011

A microscopic approach to nonlinear Reaction-Diffusion: the case of morphogen gradient formation

We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the $n$-th order annihilation reaction $A+A+A+...+A\rightarrow 0$, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior (for $n>α$) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for $n<α<n+1$) with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.

cond-mat.stat-mech↗