The survival probability and r-point functions in high dimensions

Abstract

In this paper we investigate the survival probability, θn, in high-dimensional statistical physical models, where θn 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 nθn → 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 (which 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.

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

“…Together with the results in [32] and the identification of the survival probability in high-dimensions in [23] (see also [21,22] for sharper results in the context of oriented percolation), these results prove the convergence of the finitedimensional distributions to those of the canonical measure of super-Brownian motion (CSBM), thus showing that CSBM is the only possible limiting càdlàg process in these models. Proving tightness and hence a full statement of weak convergence on path space in these settings has remained a major open problem.…”

“…Convergence of the survival probabilities for branching random walk reduces to a statement about Galton-Watson branching processes and is a well-known result due to Kolmogorov [35]; see [18], Section I.10 or [43], Theorem II.1.1. The corresponding property (2.3) for lattice trees, oriented percolation and the contact process is a very recent result [21][22][23]. Due to this fact, the weak convergence in (1.8) can easily be translated into a statement about convergence of the corresponding (conditional) probability measures on the Polish space D,…”

“…For critical models of branching random walk, lattice trees, oriented percolation and the contact process (for sufficiently spread-out kernels above the respective critical dimensions), it is known (see, e.g., [23,43]) that there exist modeldependent positive constants A and V so that the survival probabilities satisfy…”

“…The scaling of time and space by n and n −1/2 , respectively, is the scaling under which random walk converges to Brownian motion. The scaling of the measure by n −1 occurs since typically (see [23]) if the population is alive at time n, it is of size n. However, since the population survives until time n with probability of order 1/n by (1.3), we would see nothing at time t = 1 in the limit as n → ∞. Therefore to get a nontrivial scaling limit we define a sequence of finite (but nonprobability) measures μ n on the Skorokhod space D(M F (R d Clearly this sequence of measures cannot converge to a finite measure.…”

“…For oriented percolation and the contact process above 4 dimensions, convergence of the finite-dimensional distributions has been proved, due to the survival probability and r-point function asymptotics provided in [21][22][23] and [25,27,30], respectively. Tightness remains an open problem in each case.…”