## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 2 (1993), 296-338.

### On the Speed of Convergence in First-Passage Percolation

#### Abstract

We consider the standard first-passage percolation problem on $\mathbb{Z}^d: \{t(e): e \text{an edge of} \mathbb{Z}^d\}$ is an i.i.d. family of random variables with common distribution $F, a_{0,n} := \inf\{\sum^k_1 t(e_1): (e_1, \cdots, e_k)$ a path on $\mathbb{Z}^d$ from 0 to $n \xi_1\}$, where $\xi_1$ is the first coordinate vector. We show that $\sigma^2(a_{0,n}) \leq C_1 n$ and that $P\{|a_{0,n} - Ea_{0,n}| \geq x\sqrt{n}\} \leq C_2 \exp(-C_3 x)$ for $x \leq C_4 n$ and for some constants $0 < C_i < \infty$. It is known that $\mu := \lim(1/n)Ea_{0,n}$ exists. We show also that $C_5 n^{-1} \leq Ea_{0,n} - n\mu \leq C_6 n^{5/6}(\log n)^{1/3}$. There are corresponding statements for the roughness of the boundary of the set $\tilde{B}(t) = \{\nu: \nu$ a vertex of $\mathbb{Z}^d$ that can be reached from the origin by a path $(e_1, \cdots, e_k)$ with $\sum t(e_i) \leq t\}$.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 2 (1993), 296-338.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005426

**Digital Object Identifier**

doi:10.1214/aoap/1177005426

**Mathematical Reviews number (MathSciNet)**

MR1221154

**Zentralblatt MATH identifier**

0783.60103

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

**Keywords**

First-passage percolation speed of convergence asymptotic shape roughness of boundary Eden model method of bounded differences

#### Citation

Kesten, Harry. On the Speed of Convergence in First-Passage Percolation. Ann. Appl. Probab. 3 (1993), no. 2, 296--338. doi:10.1214/aoap/1177005426. https://projecteuclid.org/euclid.aoap/1177005426