## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 55, 2007, 204-233

### A growth model in multiple dimensions and the height of a random partial order

#### Abstract

We introduce a model of a randomly growing interface in multidimensional Euclidean space. The growth model incorporates a random order model as an ingredient of its graphical construction, in a way that replicates the connection between the planar increasing sequences model and the one-dimensional Hammersley process. We prove a hydrodynamic limit for the height process, and a limit which says that certain perturbations of the random surface follow the characteristics of the macroscopic equation. By virtue of the space-time Poissonian construction, we know the macroscopic velocity function explicitly up to a constant factor.

#### Chapter information

**Source***Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007)

**Dates**

First available in Project Euclid: 4 December 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196797078

**Digital Object Identifier**

doi:10.1214/074921707000000373

**Mathematical Reviews number (MathSciNet)**

MR2459941

**Zentralblatt MATH identifier**

1181.60149

**Subjects**

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

Secondary: 82C22: Interacting particle systems [See also 60K35]

**Keywords**

characteristics growth model hydrodynamic limit increasing sequences random order second-class particle

**Rights**

Copyright © 2007, Institute of Mathematical Statistics

#### Citation

Seppäläinen, Timo. A growth model in multiple dimensions and the height of a random partial order. Asymptotics: Particles, Processes and Inverse Problems, 204--233, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000373. https://projecteuclid.org/euclid.lnms/1196797078