# Dimension improvement in Dhar's refutation of the Eden conjecture

## TL;DR

This paper improves bounds on the Eden model's infection spread in high dimensions, providing evidence that the Eden conjecture about the limit shape being a Euclidean ball does not hold above dimension 22.

## Contribution

The authors extend Dhar's computations with modern tools to significantly lower the known dimension where the Eden conjecture fails.

## Key findings

- Eden conjecture does not hold in dimensions greater than 22
- Improved computational bounds on infection speed in high-dimensional Eden models
- Lowered the known dimension threshold from 35 to 22 for the conjecture's failure

## Abstract

We consider the Eden model on the d-dimensional hypercubical unoriented lattice , for large d. Initially, every lattice point is healthy, except the origin which is infected. Then, each infected lattice point contaminates any of its neighbours with rate 1. The Eden model is equivalent to first passage percolation, with exponential passage times on edges. The Eden conjecture states that the limit shape of the Eden model is a Euclidean ball. By putting the computations of Dhar [Dha88] a little further with modern computers and efficient implementation we obtain improved bounds for the speed of infection. This shows that the Eden conjecture does not hold in dimension superior to 22 (the lower known dimension was 35).