Percolation and isoperimetry on roughly transitive graphs

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When G is transitive, the fact that p_c < 1 was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov's theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.Comment: 32 pages, 2 figure

Coupling of Brownian motions in Banach spaces

Consider a separable Banach space $\mathcal{W}$ supporting a non-trivial Gaussian measure $\mu$. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two $\mathcal{W}$-valued Brownian motions $\mathbf{B}$ and $\widetilde{\mathbf{B}}$ begun at starting points $\mathbf{B}(0)$ and $\widetilde{\mathbf{B}}(0)$ if and only if the difference $\mathbf{B}(0)-\widetilde{\mathbf{B}}(0)$ of their initial positions belongs to the Cameron-Martin space $\mathcal{H}_{\mu}$ of $\mathcal{W}$ corresponding to $\mu$. For more general starting points, can there be a "coupling at time $\infty$", such that almost surely $\|\mathbf{B}(t)-\widetilde{\mathbf{B}}(t)\|_{\mathcal{W}} \to 0$ as $t\to\infty$? Such couplings exist if there exists a Schauder basis of $\mathcal{W}$ which is also a $\mathcal{H}_{\mu}$-orthonormal basis of $\mathcal{H}_{\mu}$. We propose (and discuss some partial answers to) the question, to what extent can one express the probabilistic Banach space property "Brownian coupling at time $\infty$ is always possible" purely in terms of Banach space geometry?Comment: 12 page

Coexistence of competing first passage percolation on hyperbolic graphs

We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two first-passage percolation processes $\text{FPP}_1$ and $\text{FPP}_\lambda$, spreading with rates $1$ and $\lambda>0$ respectively, on a graph $G$. $\text{FPP}_1$ starts from a single vertex at the origin $o$, while the initial configuration of $\text{FPP}_\lambda$ consists of infinitely many \emph{seeds} distributed according to a product of Bernoulli measures of parameter $\mu>0$ on $V(G)\setminus \{o\}$. $\text{FPP}_1$ starts spreading from time 0, while each seed of $\text{FPP}_\lambda$ only starts spreading after it has been reached by either $\text{FPP}_1$ or $\text{FPP}_\lambda$. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probability. We show that this is the case when $G$ is vertex transitive, non-amenable and hyperbolic, in particular, for any $\lambda>0$ there is a $\mu_0=\mu_0(G,\lambda)>0$ such that for all $\mu\in(0,\mu_0)$ the two processes coexist with positive probability. This is the first non-trivial instance where coexistence is established for this model. We also show that $\text{FPP}_\lambda$ produces an infinite cluster almost surely for any positive $\lambda,\mu$, establishing fundamental differences with the behavior of such processes on $\mathbb{Z}^d$.Comment: 53 pages, 13 figure

Branching Random Walks on Free Products of Groups

We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter $\lambda$ is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by $\Phi(\lambda)$. The main result states that the function $\Phi(\lambda)$ has only one point of discontinuity which is at $\lambda_{c}=R$ where $R$ is the radius of convergence of the Green function of the underlying random walk. Furthermore, $\Phi(R)$ is bounded by one half the Hausdorff dimension of the boundary of the original Cayley graph and the behaviour of $\Phi(R)-\Phi(\lambda)$ as $\lambda \uparrow R$ is classified. In the case of free products of infinite groups the end-boundary can be decomposed into words of finite and words of infinite length. We prove the existence of a phase transition such that if $\lambda\leq \tilde\lambda_{c}$ the end boundary of the trace consists only of infinite words and if $\lambda>\tilde\lambda_{c}$ it also contains finite words. In the last case, the Hausdorff dimension of the set of ends (of the trace and the original graph) induced by finite words is strictly smaller than the one of the ends induced by infinite words.Comment: 39 pages, 4 figures; final version, accepted for publication in the Proceedings of LM

Bootstrap percolation and the geometry of complex networks

On a geometric model for complex networks (introduced by Krioukov et al.) we investigate the bootstrap percolation process. This model consists of random geometric graphs on the hyperbolic plane having N vertices, a dependent version of the Chung-Lu model. The process starts with infection rate p=p(N). Each uninfected vertex with at least View the MathML source infected neighbors becomes infected, remaining so forever. We identify a function pc(N)=o(1) such that a.a.s. when p≫pc(N) the infection spreads to a positive fraction of vertices, whereas when p≪pc(N) the process cannot evolve. Moreover, this behavior is “robust” under random deletions of edges

The number of ends of critical branching random walks

We investigate the number of topological ends of the trace of branching random walk (BRW) on a graph, giving a sufficient condition for the trace to have infinitely many ends. We then describe some interesting examples of non-symmetric BRWs with just one end

Martin boundaries and asymptotic behavior of branching random walks

Let $G$ be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between $t$-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph $U$ we study two aspects of branching random walks on $U$: when the trajectories visit $U$ infinitely often (survival) and when they stay inside $U$ forever (persistence). We show that there are cases, when $U$ is not connected, where the branching random walk does not survive in $U$, but the random walk on $G$ converges to the boundary of $U$ with positive probability. In contrast, the branching random walk can survive in $U$ even though the random walk eventually exits $U$ almost surely. We provide several examples and counterexamples.Comment: 25 page