In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold @math for a Galton-Watson tree is strictly positive if and only if its offspring distribution @math has an exponential tail, i.e., @math , settling a conjecture by Huang and Durrett hd18 . On the random graph with degree distribution @math , we show that if @math has an exponential tail, then for small enough @math the contact process with the all-infected initial condition survives for @math -time w.h.p. (short survival), while for large enough @math it runs over @math -time w.h.p. (long survival). When @math is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed @math .
Less is known for the contact process on general Galton-Watson trees. Recently, Huang and Durrett @cite_1 proved that on Galton-Watson trees, @math if the offspring distribution @math is subexponential. Along with Theorem , we now have the complete characterization of the existence of extinction in the contact process on Galton-Watson trees.
@cite_1
2898008966 2006550558 1589906916 2545825365
The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival @math and (ii) when it is geometric( @math ) we have @math , where the @math are much smaller than previous estimates. We also study the critical value @math for "prolonged persistence" on graphs with @math vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known @math we give estimates on the rate of convergence. Physicists tell us that @math where @math is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct. A branching process with an absorbing lower barrier is considered. This is a Galton-Watson process with the condition that at any generation the number of individuals is greater than a lower barrier or it is equal to zero (i.e. all individuals in populations which are too small die and have no offspring). A necessary and sufficient condition is given for the process to become extinct with probability one. At the end of the paper there are three illustrating examples. 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. Consider a uniform random rooted tree on vertices labelled by @math , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number @math of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider @math and let @math denote the event that all @math cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if @math , we have @math , whereas if @math we have @math . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson( @math ) number of cars arrive at each vertex. Let @math be the number of cars which visit the root of the tree. Then for @math , we have @math , whereas for @math , we have @math . This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution.