## Electronic Journal of Probability

### Cube Root Fluctuations for the Corner Growth Model Associated to the Exclusion Process

#### Abstract

We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order $t^{2/3}$. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order $t^{1/3}$, and also that the transversal fluctuations of the maximal path have order $t^{2/3}$. We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 42, 1094-1132.

Dates
Accepted: 29 November 2006
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.ejp/1464730577

Digital Object Identifier
doi:10.1214/EJP.v11-366

Mathematical Reviews number (MathSciNet)
MR2268539

Zentralblatt MATH identifier
1139.60046

Rights

#### Citation

Balazs, Marton; Cator, Eric; Seppalainen, Timo. Cube Root Fluctuations for the Corner Growth Model Associated to the Exclusion Process. Electron. J. Probab. 11 (2006), paper no. 42, 1094--1132. doi:10.1214/EJP.v11-366. https://projecteuclid.org/euclid.ejp/1464730577

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