  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 a certain self-repellence condition is 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.
