# The acceptance profile of invasion percolation at $p_c$ in two dimensions

## TL;DR

This paper investigates the behavior of the acceptance profile of invasion percolation at the critical point in two dimensions, revealing it remains bounded away from zero and one as the process grows.

## Contribution

It provides the first analysis of the acceptance profile at criticality in two dimensions, showing it does not converge to trivial limits.

## Key findings

- Acceptance profile at $p_c$ stays bounded away from 0 and 1
- Confirms non-trivial behavior of invasion percolation at criticality
- Extends understanding of invasion percolation in 2D

## Abstract

Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of $\mathbb{Z}^d$, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the `acceptance profile' of the invasion: for a given $p \in [0,1]$, it is the ratio of the expected number of invaded edges until time $n$ with weight in $[p,p+\text{d}p]$ to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile $a_n(p)$ converges to one for $p<p_c$ and to zero for $p>p_c$. In this paper, we consider $a_n(p)$ at the critical point $p=p_c$ in two dimensions and show that it is bounded away from zero and one as $n \to \infty$.