![]() ![]() In our case, this maximum grows faster than linear in time. for the position of the rightmost particle. The result on the speed gives a limit theorem for the maximum of the branching random walk, i.e. Interpreting the random variables Y v as displacements of the ospring from the parent, (Y v ) v2 describes a branching random walk. the denition of the speed depends on the distribution of Y v. In contrast to the classical case where the random variables Y v have nite exponential moments, the normalization in. In the case where the upper tail of the distribution of Y v is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. : We consider an innite Galton-Watson tree and label the vertices v with a collection of i.i.d. ![]() Several, more specific, versions of the model described will also be considered, and a cleaner, more simplified set of sharp conditions will be established for each case. Our results include sharp conditions with respect to the sequence of random variables and the sequence of drifts, that determine whether the model is transient (meaning the probability infinitely many frogs return to the origin is $0$) or non-transient. particles) at positive integer points is a sequence of independent random variables which is increasing in terms of the standard stochastic order, and that the sequence of leftward drifts associated with frogs originating at these points is decreasing. Additional conditions that we impose on our model include that the number of frogs (i.e. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Once activated, the trajectories of distinct particles are independent. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point posses equal leftward drift. We examine a system of interacting random walks with leftward drift on $\mathbb$, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. ![]()
0 Comments
Leave a Reply. |