The Buffon needle problem for L\'evy distributed spacings and renewal theory

TL;DR

This paper derives the exact probability that a randomly dropped needle does not hit any points on a line with heavy-tailed spacings, extending renewal theory to power-law distributed gaps with diverging mean.

Contribution

It introduces a new theoretical framework for calculating hole probabilities in Le9vy distributed spacings, advancing renewal theory applications.

Findings

Derived exact scaling expression for hole probability

Extended renewal theory to heavy-tailed distributions

Linked results to correlation functions in statistical physics

Abstract

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.