Interpolation percolation

TL;DR

This paper investigates conditions under which one can find continuous functions that interpolate given random sets at countably infinite points, addressing a probabilistic percolation problem with implications for stochastic processes.

Contribution

It provides criteria for the almost sure existence of regular functions interpolating independent stationary random sets at infinitely many points, advancing understanding of interpolation in stochastic environments.

Findings

Criteria for almost sure existence of interpolating functions

Analysis of stationary random sets like Poisson processes

Open questions on interpolation in probabilistic percolation

Abstract

Let X be a countably infinite set of real numbers and let Y_x, x \in X, be an independent family of stationary random subsets of the real numbers, e.g. homogeneous Poisson point processes. We give criteria for the a.s. existence of various "regular" functions f with the property that f(x) \in Y_x for all x \in X. Several open questions are posed.