arXiv subjects

Enrique D. Andjel

Publications and source records attributed to Enrique D. Andjel.

3 recordsLinked to original sources

arXiv2013

First passage percolation and escape strategies

Consider first passage percolation on $\mathbb{Z}^d$ with passage times given by i.i.d. random variables with common distribution $F$. Let $t_π(u,v)$ be the time from $u$ to $v$ for a path $π$ and $t(u,v)$ the minimal time among all paths from $u$ to $v$. We ask whether or not there exist points $x,y \in \mathbb{Z}^d$ and a semi-infinite path $π=(y_0=y,y_1,\dots)$ such that $t_π(y, y_{n+1})<t(x,y_n)$ for all $n$. Necessary and sufficient conditions on $F$ are given for this to occur. When the support of $F$ is unbounded, we also obtain results on the number of edges with large passage time used by geodesics.

math.PR↗

arXiv2006

Long-range exclusion processes, generator and invariant measures

We show that if $μ$ is an invariant measure for the long range exclusion process putting no mass on the full configuration, $L$ is the formal generator of that process and $f$ is a cylinder function, then $Lf\in\mathbf{L}^1(dμ)$ and $\int Lf dμ=0$. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on $\mathbb{Z}^d$ when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on $\mathbb{Z}$ when the underlying random walk is irreducible and either has zero mean or allows jumps only to the nearest-neighbors.

math.PR↗

arXiv2000

Convergence to the maximal invariant measure for a zero-range process with random rates

We consider a one-dimensional totally asymmetric nearest-neighbor zero-range process with site-dependent jump-rates - an environment. For each environment p we prove that the set of all invariant measures is the convex hull of a set of product measures with geometric marginals. As a consequence we show that for environments p satisfying certain asymptotic property, there are no invariant measures concentrating on configurations with critical density bigger than $ρ^*(p)$, a critical value. If $ρ^*(p)$ is finite we say that there is phase-transition on the density. In this case we prove that if the initial configuration has asymptotic density strictly above $ρ^*(p)$, then the process converges to the maximal invariant measure.

math.PR↗