Analysis Of Four Particle Systems

This thesis deals with four models of stochastic dynamics on relevant large finite systems. The first one is the contact process on random graphs on n vertices with power law degree distributions. If the infection rate is [lamda], then nonrigorous mean field calculations suggest that the critical value [lamda]c of the infection rate is positive when the power [alpha] is larger than 3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs in [25]. Here, we show that the critical value [lamda]c is zero for any value of [alpha] larger than 3, and the contact process starting from all vertices infected, with a probability tending to 1 as n increases to infinity, maintains a positive density of infected vertices for time at least exp(n1[-][delta] ) for any positive [delta] . We also establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is infected with probability [rho]([lamda]). It is expected that [rho]([lamda]) is asymptotically C[lamda][beta] as [lamda] decreases to zero for some positive constants C and [beta] . Here we show that [beta] lies between [alpha] [-] 1 and 2[alpha] [-] 3, and so [beta] is larger than 2 for any [alpha] larger than 3. Thus even though the graph is locally tree-like, [beta] does not take the mean field critical value which equals 1. The second one is a model for gene regulatory networks that is a modification of Kauffmann's [30] random Boolean networks. There are three parameters: n = the number of nodes, r = the number of inputs to each node, and p = the expected fraction of 1's in the Boolean functions at each node. Following a standard practice in the physics literature, we use an appropriate threshold contact process on a random graph on n nodes, in which each node has in degree r, to approximate its dynamics. We show that if r is larger than 2 and r [MIDDLE DOT] 2p(1 [-] p) is larger than 1, then the threshold contact process persists for a long time, which corresponds to chaotic behavior of the Boolean network. We prove that the persistence time is at least exp cnb(p) with b(p) > 0 when r [MIDDLE DOT] 2p(1 [-] p) > 1, and b(p) = 1 when (r [-] 1) [MIDDLE DOT] 2p(1 [-] p) > 1. The third one is related to a gossip process defined by Aldous [3]. In this process, space is a discrete N x N torus, and the state of the process at time t is the set of individuals who know the information. Information spreads from a vertex to its nearest neighbors at rate 1/4 each and at rate N [-][alpha] to a vertex chosen at random from the torus. We will be interested in the case in which [alpha] is smaller than 3, where the long range transmissions significantly accelerate the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically (2 [-] 2[alpha]/3)N [alpha]/3 log N . After an appropriate random centering and scaling by N [alpha]/3 , the fraction of informed population is almost a deterministic function which satisfies an integro-differential equation. The fourth and the final one is about the discrete time threshold-two contact process on a random r-regular graph on n vertices. In this process, a vertex with at least two occupied neighbors at time t will be occupied at time t + 1 with probability p, and vacant otherwise. We use a suitable isoperimetric inequality to show that if r is larger than 3 and p is close enough to 1, then starting from all vertices occupied, there is a positive density of occupied vertices up to time exp(c(p)n) for some positive constant c(p). In the other direction, another appropriate isoperimetric inequality allows us to show that there is a decreasing function 2 ( p) so that if the number of occupied vertices in the initial configu- ration is smaller than 2 (p)n, then with high probability all vertices are vacant at time 2 log n/ log(2/(1 + p)). These two conclusions imply that the density of occupied vertices in the quasi-stationary distribution (defined in Chapter 5) is discontinuous at the critical probability pc ∈ (0, 1)