Publications [#377350] of Richard T. Durrett

Papers Published

1. Cristali, I; Junge, M; Durrett, R, Poisson percolation on the square lattice, Alea, vol. 16 no. 1 (January, 2019), pp. 429-437 [doi]
   (last updated on 2024/08/31)

   Abstract:
   Suppose that on the square lattice the edge with midpoint x becomes open at rate ∥x∥σ-1 . Let ρ(x, t) be the probability that the corresponding edge is open at time t and let n(p; t) be the distance at which edges are open with probability p at time t. We show that with probability tending to 1 as t → σ: (i) the open cluster containing the origin ℂ0(t) is contained in the square of radius n(pc-∈, t), and (ii) the cluster fills the square of radius n(pc+∈, t) with the density of points near x being close to θ(ρ(x, t)) where θ(p) is the percolation probability when bonds are open with probability p on ℤ2. Results of Nolin suggest that if N = n(pc, t) then the boundary uctuations of ℂ0(t) are of size N4/7.