Threshold $theta geq 2$ contact processes on homogeneous trees

Luiz Renato Fontes; Roberto H. Schonmann

paper

Threshold $t h e t a g e q 2$ contact processes on homogeneous trees

arXiv:math/0603109

Abstract

We study the threshold $t h e t a g e q 2$ contact process on a homogeneous tree $T_{b}$ of degree $k a pp a = b + 1$ , with infection parameter $l amb d a g e q 0$ and started from a product measure with density $p$ . The corresponding mean-field model displays a discontinuous transition at a critical point $l amb d a_{c}^{M F} (k a pp a, t h e t a)$ and for $l amb d a g e q l amb d a_{c}^{M F} (k a pp a, t h e t a)$ it survives iff $p g e q p_{c}^{M F} (k a pp a, t h e t a, l amb d a)$ , where this critical density satisfies $0 < p_{c}^{M F} (k a pp a, t h e t a, l amb d a) < 1$ , $l i m_{l amb d a t o in f t y} p_{c}^{M F} (k a pp a, t h e t a, l amb d a) = 0$ . For large $b$ , we show that the process on $T_{b}$ has a qualitatively similar behavior when $l amb d a$ is small, including the behavior at and close to the critical point $l amb d a_{c} (T_{b}, t h e t a)$ . In contrast, for large $l amb d a$ the behavior of the process on $T_{b}$ is qualitatively distinct from that of the mean-field model in that the critical density has $p_{c} (T_{b}, t h e t a, in f t y) := l i m_{l amb d a t o in f t y} p_{c} (T_{b}, t h e t a, l amb d a) > 0$ . We also show that $l i m_{b t o in f t y} b l amb d a_{c} (T_{b}, t h e t a) = P h i_{t h e t a}$ , where $1 < P h i_{2} < P h i_{3} < ...$ , $l i m_{t h e t a t o in f t y} P h i_{t h e t a} = in f t y$ , and $0 < l imin f_{b t o in f t y} b^{t h e t a (t h e t a - 1)} p_{c} (T_{b}, t h e t a, in f t y) l e q l im s u p_{b t o in f t y} b^{t h e t a / (t h e t a - 1)} p_{c} (T_{b}, t h e t a, in f t y) < in f t y$ .

27 pages

Threshold thetageq2 contact processes on homogeneous trees

Abstract

Threshold $t h e t a g e q 2$ contact processes on homogeneous trees