Coexistence in three type last passage percolation model
نویسنده
چکیده
A three types competition model governed by directed last passage percolation on N 2 is considered. We prove that coexistence of the three types, i.e. the sets of vertices of the three types are simultaneously unbounded, occurs with positive probability. Moreover, the asymptotic angles formed by the two competition interfaces with the horizontal axis are determined and their probability of being different is positive. As a key step, a stochastic domination between subtrees of the last passage percolation tree is obtained.
منابع مشابه
Stochastic domination in the last passage percolation tree
A three colors competition model on (Z) governed by directed last passage percolation is considered. A stochastic domination argument between subtrees of the last passage percolation tree is put forward. Applied to the case of exponential random times, it allows us to prove that coexistence is possible, i.e. three unbounded colored areas occur with positive probability. Furthermore, asymptotic ...
متن کاملCoexistence in Two-type First-passage Percolation Models by Olivier Garet
We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Zd or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Zd , there is a strictly positive probability that {z ∈ Zd ;d(y, z) < d(x, z)} and {z ∈ Zd ;d(y, z) > d(x, z)} are both infinite sets. We...
متن کاملCoexistence in two-type first-passage percolation models
We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Z or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Z, there is a strictly positive probability that {z ∈ Z; d(y, z) < d(x, z)} and {z ∈ Z; d(y, z) > d(x, z)} are both infinite set...
متن کاملGeodesics in First-Passage Percolation
We consider a wide class of ergodic first passage percolation processes on Z2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four color Richardson’s growth model. This improves earlier results of Häggström and Pemantle [9], Garet and Marchand [7] and Hoffman [11] who proved that first passage percolation...
متن کاملSublinear variance for directed last-passage percolation
A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions. Corresponding results are obtained for the ground state of the ‘directed polymers in a ran...
متن کامل