## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 1 (2017), 278-376.

### A criterion for convergence to super-Brownian motion on path space

Remco van der Hofstad, Mark Holmes, and Edwin A. Perkins

#### Abstract

We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the $r$-point functions for $r=2,\ldots,5$. The $r$-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the $r$-point functions, but the lack of tightness has been an obstruction to proving convergence on path space.

We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 1 (2017), 278-376.

**Dates**

Received: April 2014

Revised: June 2014

First available in Project Euclid: 26 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1485421334

**Digital Object Identifier**

doi:10.1214/14-AOP953

**Mathematical Reviews number (MathSciNet)**

MR3601651

**Zentralblatt MATH identifier**

1364.82025

**Subjects**

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60F17: Functional limit theorems; invariance principles 60G68 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 05C05: Trees

**Keywords**

Lattice trees super-Brownian motion functional limit theorem

#### Citation

van der Hofstad, Remco; Holmes, Mark; Perkins, Edwin A. A criterion for convergence to super-Brownian motion on path space. Ann. Probab. 45 (2017), no. 1, 278--376. doi:10.1214/14-AOP953. https://projecteuclid.org/euclid.aop/1485421334