The contact process on a tree: Behavior near the first phase transition

We study the critical behavior of the contact process on a homogeneous tree. It is shown that if the degree of the tree is greater than four, then the survival probability θ(λ) behaves like (λ − λc)β with β = 1 when λ is near but above the critical point λc, and the expected infection time χ(λ) behaves like (λc − λ)−γ with γ = 1 when λ is near but below λc. Analogous results for the oriented percolation model are also obtained.