From exponential counting to pair correlations

TL;DR

This paper establishes a general theoretical framework for understanding pair correlations in exponentially growing discrete sets, demonstrating Poissonian behavior under certain conditions, with applications to geodesics in negatively curved spaces.

Contribution

It introduces an abstract result linking exponential growth properties to pair correlation distributions, extending previous work to geometric and graph settings.

Findings

Distribution of differences is exponential with parameter δ/2

Pair correlation exhibits Poissonian behavior under certain conditions

Application to geodesics confirms theoretical predictions

Abstract

We prove an abstract result on the correlations of pairs of elements in an exponentially growing discrete subset $\mathcal E$ of $[0,+\infty[\,$ endowed with a weight function. Assume that there exist $\alpha\in\mathbb R$, $c,\delta>0$ such that, as $t\to+\infty$, the weighted number $\widetilde\omega(t)$ of elements of $\mathcal E$ that are not greater than $t$ is equivalent to $c\,t^\alpha e^{\delta t}$. We prove that the distribution function of the unscaled differences of elements of $\mathcal E$ is $t\mapsto\frac\delta 2\,e^{-|t|}$, and that, under an error term assumption on $\widetilde\omega(t)$, the pair correlation with a scaling with polynomial growth exhibits a Poissonian behaviour. We apply this result to answer a question of Pollicott and Sharp on the pair correlations of closed geodesics and common perpendiculars in negatively curved manifolds and metric graphs.