Pair correlations of logarithms of complex lattice points

TL;DR

This paper investigates the statistical correlations of complex logarithms of lattice points, revealing different behaviors such as level repulsion, Poissonian distribution, and mass loss at various scalings, with applications to hyperbolic geometry.

Contribution

It establishes the existence of pair correlation functions for complex lattice points and characterizes their behavior across different scalings, including level repulsion and Poissonian limits.

Findings

Pair correlation functions exist for complex lattice points.

Level repulsion occurs at linear scaling.

Poissonian behavior appears at sublinear scaling.

Abstract

We study the correlations of pairs of complex logarithms of $\mathbb Z$-lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi orbifold $\mathrm{PSL}_2(\mathbb Z[i]) \backslash\mathbb H^3_{\mathbb R}$.