Probability seminar: The survival probability in high dimensions
Remco van der Hofstad, Eindhoven University of Technology and Eurandom
SM3, School of Mathematics
A branching process is a simple population model where individuals have a random number of children, independently of one another. Branching processes have a phase transition. Indeed, when the mean offspring is at most 1, the branching process dies out almost surely, while for mean offspring larger than one, the branching process survives with positive probability. Branching processes with mean offspring equal to one are called critical.
A classical result by Kolmogorov from 1938 states that the probability \theta_n that a critical branching process with finite-variance offspring survives to time n decays like 2/(n\gamma), where \gamma is the variance of the offspring distribution. A related result by Yaglom from 1945 states that, conditionally on survival to time n, the number of individuals alive scales like n times an exponential random variable.
In this talk, we extend such results to high-dimensional statistical physical models. Models to which our results apply are oriented percolation above 4+1 dimensions, the contact process above 4+1 dimensions, and lattice trees above 8 dimensions. We give general conditions to show that Kolmogorov's and Yaglom's theorem hold. In the case of oriented percolation, this reproves a result with den Hollander and Slade (that was proved using a 100 page lace-expansion argument), at the cost of losing explicit error estimates.
[This is joint work with Mark Holmes, building on work with Gordon Slade, Frank den Hollander and Akira Sakai.]