Subcritical contact process seen from the edge: convergence to quasi-equilibrium

Abstract

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractThe subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.

Nucleation and growth in two dimensions

“…The proof now follows exactly as before: by Lemmas 3.1 and 4.1, a nucleation in S grows, with high probability, to fill the whole of S within time 2 and C was chosen sufficiently large. It follows that…”

“…After time Θ( √ n), a nucleation will appear at distance = Θ( √ n) from the origin; this infection will then spread sideways at rate 1, reaching the origin after time ± O( √ ). It follows easily that = Θ( √ n) with high probability, 2 and moreover that there is no sharp threshold for . In two dimensions the situation is substantially more complicated, so let us begin by considering the growth of a single nucleation, which produces (after some time) a (roughly) square "droplet" growing around it.…”

On laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivity

“…iii. If ξ (x) sT (J m ) > 0 for some (random) s ≥ τ (1) , then choose some y ∈ ξ where ⌈s⌉ here denotes the smallest integer greater than or equal to s. Let us observe that, by construction, there exists a (random) k ∈ N such that V (2) j ≤ ξ (x) (k+j)T (B) for all j ∈ N. By a similar argument than the one carried out for V (1) , it is possible to couple V (2) with a Galton-Watson process Z (2) which is independent of Z (1) but has the same distribution, in such a way that V jT must tend to infinity as j → +∞. If not, then one can repeat this procedure to obtain a branching process Z (3) and so on.…”

“…Whenever the convergence in (1) is understood in the L 2 -sense, all previous results so far require the branching process ξ to be λ-positive. Essentially, λ-positivity means that the motion of a certain spine describing the genealogy of the branching process (which is sometimes referred to in the literature as the immortal particle) is positive recurrent, a property which proves crucial in all of the approaches developed until now.…”

Scaling limit of subcritical contact process

“…Using Proposition 1.1 and controlling the statistical effect of picking the rightmost infected site, it is shown in [AEGR15] that, for any infinite initial configuration A ⊆ −N, the subcritical contact process seen from the rightmost point converges in distribution to ν.…”