# The survival probability and r-point functions in high dimensions

3 Citaties (Scopus)

### Uittreksel

In this paper we investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then n\theta_n\ra 2/(AV), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4+1 dimensions, and the spread-out contact process above 4+1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.
Originele taal-2 Engels Eindhoven Eurandom 16 Gepubliceerd - 2013

### Publicatie series

Naam Report Eurandom 2013004 1389-2355

### Vingerafdruk

Survival Probability
Oriented Percolation
Higher Dimensions
Lace Expansion
Super-Brownian Motion
Scale Function
Contact Process
Scaling Limit
Physical Model
Weak Convergence
Statistical Model
Error Estimates
Tail
High-dimensional
Directly proportional
Denote
Vertex of a graph
Model

### Citeer dit

Hofstad, van der, R. W., & Holmes, M. P. (2013). The survival probability and r-point functions in high dimensions. (Report Eurandom; Vol. 2013004). Eindhoven: Eurandom.
Hofstad, van der, R.W. ; Holmes, M.P. / The survival probability and r-point functions in high dimensions. Eindhoven : Eurandom, 2013. 16 blz. (Report Eurandom).
title = "The survival probability and r-point functions in high dimensions",
abstract = "In this paper we investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then n\theta_n\ra 2/(AV), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4+1 dimensions, and the spread-out contact process above 4+1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.",
author = "{Hofstad, van der}, R.W. and M.P. Holmes",
year = "2013",
language = "English",
series = "Report Eurandom",
publisher = "Eurandom",

}

Hofstad, van der, RW & Holmes, MP 2013, The survival probability and r-point functions in high dimensions. Report Eurandom, vol. 2013004, Eurandom, Eindhoven.
Eindhoven : Eurandom, 2013. 16 blz. (Report Eurandom; Vol. 2013004).

TY - BOOK

T1 - The survival probability and r-point functions in high dimensions

AU - Hofstad, van der, R.W.

AU - Holmes, M.P.

PY - 2013

Y1 - 2013

N2 - In this paper we investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then n\theta_n\ra 2/(AV), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4+1 dimensions, and the spread-out contact process above 4+1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.

AB - In this paper we investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then n\theta_n\ra 2/(AV), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4+1 dimensions, and the spread-out contact process above 4+1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.

M3 - Report

T3 - Report Eurandom

BT - The survival probability and r-point functions in high dimensions

PB - Eurandom

CY - Eindhoven

ER -

Hofstad, van der RW, Holmes MP. The survival probability and r-point functions in high dimensions. Eindhoven: Eurandom, 2013. 16 blz. (Report Eurandom).