Geodesics in First Passage Percolation

We consider a wide class of ergodic first passage percolation processes on Z^2 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 Haggstrom and Pemantle, Garet and Marchand, and Hoffman who proved that first passage percolation has at least two geodesics and that coexistence is possible in a two color Richardson's growth model.Comment: 24 page

Recurrence of Simple Random Walk on Z2Z^2 is Dynamically Sensitive

Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical random walk. This is a continuous family of random walks, {S_n(t)}. Benjamini et. al. proved that if d=3 or d=4 then there is an exceptional set of t such that {S_n(t)} returns to the origin infinitely often. In this paper we consider a dynamical random walk on Z^2. We show that with probability one there exists t such that {S_n(t)} never returns to the origin. This exceptional set of times has dimension one. This proves a conjecture of Benjamini et. al

Coexistence for Richardson type competing spatial growth models

We study a large family of competing spatial growth models. In these the vertices in Z^d can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0 which are adjacent to a vertex in state 1 (or state 2) can switch to state 1 (or state 2). We think of the vertices in states 1 and 2 as infected with one of two infections while the vertices in state 0 are considered uninfected. In this way these models are variants of the Richardson model. We start the models with a single vertex in state 1 and a single vertex is in state 2. We show that with positive probability state 1 reaches an infinite number of vertices and state 2 also reaches an infinite number of vertices. This extends results and proves a conjecture of Haggstrom and Pemantle. The key tool is applying the ergodic theorem to stationary first passage percolation.Comment: 10 page

Random mass splitting and a quenched invariance principle

We will investigate a random mass splitting model and the closely related random walk in a random environment (RWRE). The heat kernel for the RWRE at time t is the mass splitting distribution at t. We prove a quenched invariance principle for the RWRE which gives us a quenched central limit theorem for the mass splitting model. Our RWRE has an environment which is changing with time. We follow the outline for proving a quenched invariant process for a random walk in a space-time random environment laid out by Rassoul-Agha and Sepp\"al\"ainen which in turn was based on the work of Kipnis and Varadhan and others.Comment: 21 page

Geodesic rays and exponents in ergodic planar first passage percolation

We study first passage percolation on the plane for a family of invariant, ergodic measures on $\mathbb{Z}^2$. We prove that for all of these models the asymptotic shape is the $\ell$-$1$ ball and that there are exactly four infinite geodesics starting at the origin a.s. In addition we determine the exponents for the variance and wandering of finite geodesics. We show that the variance and wandering exponents do not satisfy the relationship of $\chi=2\xi-1$ which is expected for independent first passage percolation.Comment: Two figures and other cosmetic changes were adde

Uniform endomorphisms which are isomorphic to a Bernoulli shift

A {\it uniformly $p$-to-one endomorphism} is a measure-preserving map with entropy log $p$ which is almost everywhere $p$-to-one and for which the conditional expectation of each preimage is precisely $1/p$. The {\it standard} example of this is a one-sided $p$-shift with uniform i.i.d. Bernoulli measure. We give a characterization of those uniformly finite-to-one endomorphisms conjugate to this standard example by a condition on the past tree of names which is analogous to {\it very weakly Bernoulli} or {\it loosely Bernoulli.} As a consequence we show that a large class of isometric extensions of the standard example are conjugate to it.Comment: 23 pages, published versio

A special set of exceptional times for dynamical random walk on Z2\Z^2

Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d > 4 there almost surely do not exist such t. Hoffman showed that for d=2 there almost surely exists t such that the random walk at time t visits the origin only finitely many times. We refine the results of Hoffman for dynamical random walk on Z^2, showing that with probability one there are times when the origin is visited only a finite number of times while other points are visited infinitely often.Comment: 29 pages (v2: Typographical fixes in abstract

Nonuniqueness for specifications in ℓ2+ϵ\ell^{2+\epsilon}

For every $p>2$, we construct a regular and continuous specification ($g$-function), which has a variation sequence that is in $l^p$ and which admits multiple Gibbs measures. Combined with a recent result of Johansson and Oberg, this determines the optimal modulus of continuity for a specification which admits multiple Gibbs measures.Comment: 16 Page

Pattern-avoiding permutations and Brownian excursion Part I: Shapes and fluctuations

Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion. Exploring this connection to Brownian excursion allows us to strengthen the recent results of Madras and Pehlivan, and Miner and Pak as well as to understand many of the interesting phenomena that had previously gone unexplained.Comment: 24 pages, The paper has been split into two parts to make the results more accessible. Part I contains results on limit shapes and fluctuations while Part II contains results on the asymptotic distribution of fixed point

Cover time for the frog model on trees

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\mu$ on the full $d$-ary tree of height $n$. If $\mu= \Omega( d^2)$, all of the vertices are visited in time $\Theta(n\log n)$ with high probability. Conversely, if $\mu = O(d)$ the cover time is $\exp(\Theta(\sqrt n))$ with high probability.Comment: 36 pages; revisions in response to referees' comments; accepted in Forum of Math Sigma, Probabilit