## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 2 (1981), 186-193.

### The Shape of the Limit Set in Richardson's Growth Model

Richard Durrett and Thomas M. Liggett

#### Abstract

Let $C_p$ be the limiting shape of Richardson's growth model with parameter $p \in (0, 1\rbrack$. Our main result is that if $p$ is sufficiently close to one, then $C_p$ has a flat edge. This means that $\partial C_p \cap \{x \in R^2:x_1 + x_2 = 1\}$ is a nondegenerate interval. The value of $p$ at which this first occurs is shown to be equal to the critical probability for a related contact process. For $p < 1$, we show that $C_p$ is not the full diamond $\{x \in R^2:\|x\| = |x_1| + |x_2| \leq 1\}$. We also show that $C_p$ is a continuous function of $p$, and that when properly rescaled, $C_p$ converges as $p \rightarrow 0$ to the limiting shape for exponential site percolation.

#### Article information

**Source**

Ann. Probab. Volume 9, Number 2 (1981), 186-193.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994460

**Digital Object Identifier**

doi:10.1214/aop/1176994460

**Mathematical Reviews number (MathSciNet)**

MR606981

**Zentralblatt MATH identifier**

0457.60083

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60K99: None of the above, but in this section 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Richardson's model percolation processes contact processes branching random walks

#### Citation

Durrett, Richard; Liggett, Thomas M. The Shape of the Limit Set in Richardson's Growth Model. Ann. Probab. 9 (1981), no. 2, 186--193. doi:10.1214/aop/1176994460. https://projecteuclid.org/euclid.aop/1176994460