Daniel Valesin

Phase transition of the contact process on random regular graphs

We consider the contact process with infection rate lambda on a random (d + 1)-regular graph with n vertices, G(n). We study the extinction time tau(Gn) (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether lambda is smaller or larger than lambda(1) (T-d), the lower critical value for the contact process on the infinite, (d + 1) -regular tree: if lambda lambda(1) (T-d), it grows exponentially with n. This result differs from the situation where, instead of G(n), the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on T-d.

Tightness for the interface of the one-dimensional contact process

We consider a symmetric, finite-range contact process with two types of infection; both have the same (supercritical) infection rate and heal at rate 1, but sites infected by Infection I are immune to Infection 2. We take the initial configuration where sites in (-infinity, 0] have Infection I and sites in [1, infinity) have Infection 2, then consider the process rho(t) defined as the size of the interface area between the two infections at time t. We show that the distribution of rho(t) is tight, thus proving a conjecture posed by Cox and Durrett in [Bernoulli 1 (1995) 343-370].

Extinction time for the contact process on general graphs

We consider the contact process on finite and connected graphs and study the behavior of the extinction time, that is, the amount of time that it takes for the infection to disappear in the process started from full occupancy. We prove, without any restriction on the graph G, that if the infection rate

Monotonicity and phase diagram for multirange percolation on oriented trees

\lambda

Improved asymptotic estimates for the contact process with stirring

λ is larger than the critical rate of the one-dimensional process, then the extinction time grows faster than

From survival to extinction of the contact process by the removal of a single edge

\exp \{|G|/(\log |G|)^\kappa \}