Critical values in an anisotropic percolation on $\mathbb{Z}^2$

In this thesis, we consider an anisotropic finite-range bond percolation model on $\mathbb{Z}^2$. On each horizontal layer $\{(x,i):x\in\mathbb{Z}\}$ for $i\in\mathbb{Z}$, we have edges $\langle(x,i),(y,i)\rangle$ for $1\leq|x-y|\leq N$ with $N\in\mathbb{N}$. There are also vertical edges connecting two nearest neighbor vertices on distinct layers $\langle(x,i),(x,i+1)\rangle$ for $x,i\in\mathbb{Z}$. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability $\lambda/(2N)$ with $\lambda\geq 1$, while vertical edges are open with probability $\epsilon$ to be suitably tuned as $N$ grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature.

We first deal with the critical case when $\lambda=1$. If $\epsilon=\kappa N^{-2/5}$, we see a phase transition in $\kappa$: positive and finite constants $C_1,C_2$ exist so that there is no percolation if $\kappaC_2$. The derivation of the scaling limit is inspired by works on the long range contact process. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process, which is interesting by itself. A renormalization argument is used for the percolative regime.

We then deal with the supercritical case when $\lambda>1$. If $\epsilon=e^{-\kappa N}$, we can also see a phase transition in $\kappa$. The horizontal and vertical edges can be discovered through subordinate process in each regime. The proof is based on the analysis of supercritical branching random walk but several levels of attritions are introduced to make sure the independent structure. The comparison between our original percolation and the percolation on the inhomogeneous square lattice is used in the renormalization scheme.