# Tight Lower Bounds on the Contact Distance Distribution in Poisson Hole Process

## TL;DR

This paper establishes new lower bounds on the contact distance distribution in the Poisson Hole Process, providing insights into spatial properties relevant for wireless networks and stochastic geometry.

## Contribution

It introduces a novel method to derive lower bounds on the contact distance CDF in PHP, accounting for the effect of holes, which was previously challenging.

## Key findings

- Derived lower bounds for contact distance CDF in PHP
- Applicable to both random points and hole centers
- Introduced a tractable bounding technique for PHP properties

## Abstract

In this letter, we derive new lower bounds on the cumulative distribution function (CDF) of the contact distance in the Poisson Hole Process (PHP) for two cases: (i) reference point is selected uniformly at random from $\mathbb{R}^2$ independently of the PHP, and (ii) reference point is located at the center of a hole selected uniformly at random from the PHP. While one can derive upper bounds on the CDF of contact distance by simply ignoring the effect of holes, deriving lower bounds is known to be relatively more challenging. As a part of our proof, we introduce a tractable way of bounding the effect of all the holes in a PHP, which can be used to study other properties of a PHP as well.