arXiv subjects

Sinziana M. Eckner

Publications and source records attributed to Sinziana M. Eckner.

2 recordsLinked to original sources

arXiv2015

Fixation to Consensus on Tree-related Graphs

We study a continuous time Markov process whose state space consists of an assignment of +1 or -1 to each vertex of a graph G. The graphs that we treat are related to homogeneous trees of degree K $\geq$ 3, such as finite or infinite stacks of such trees. The initial spin configuration is chosen from a Bernoulli product measure with density $θ$ of +1 spins. The system evolves according to an agreement inducing dynamics: each vertex, at rate 1, changes its spin value to agree with the majority of its neighbors. We study the long time behavior of this system and prove that, if $θ$ is close enough to 1, the system reaches fixation to consensus. The geometric percolation-type arguments introduced here may be of independent interest.

math.PR↗

arXiv2014

Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices

We study Markov processes in which $\pm 1$-valued random variables $σ_x(t), x\in \mathbb{Z}^d$, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density $ρ^+$ (resp., $ρ^-$), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $ρ^+ >0$ and $ρ^- =0$, all sites are fixed plus, while for $ρ^+ >0$ and $ρ^-$ very small (compared to $ρ^+$), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

math.PR↗