In this article, we discuss several properties of the basic contact process on hexagonal lattice H, showing that it behaves quite similar to the process on d-dimensional lattice Zd in many aspects. Firstly, we construct a coupling between the contact process on hexagonal lattice and the oriented percolation, and prove an equivalent finite space-time condition for the survival of the process. Secondly, we show the complete convergence theorem and the polynomial growth hold for the contact process on hexagonal lattice. Finally, we prove exponential bounds in the supercritical case and exponential decay rates in the subcritical case of the process.
We prove that two independent continuous-time simple random walks on the infinite open cluster of a Bernoulli bond percolation in the lattice Z2 meet each other infinitely many times.An application to the voter model is also discussed.
Let N = (G, c) be a random electrical network obtained by assigning a certain resistance for each edge in a random graph G ∈ G(n, p) and the potentials on the boundary vertices. In this paper, we prove that with high probability the potential distribution of all vertices of G is very close to a constant.
We prove that a C2 unimodal interval map with critical order not greater than 2 has the decay of geometry property, by showing that all the cross-ratio estimates needed in the previous proof for the C3 case remain true.
SHEN WeiXiao1,2 1 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
