Phase diagram of inhomogeneous percolation with a defect plane

Abstract

Let ${\mathbb{L}}$ be the $d$-dimensional hypercubic lattice and let ${\mathbb{L}}_0$ be an $s$-dimensional sublattice, with $2 \leq s < d$. We consider a model of inhomogeneous bond percolation on ${\mathbb{L}}$ at densities $p$ and $\sigma$, in which edges in ${\mathbb{L}}\setminus {\mathbb{L}}_0$ are open with probability $p$, and edges in ${\mathbb{L}}_0$ open with probability $\sigma$. We generalizee several classical results of (homogeneous) bond percolation to this inhomogeneous model. The phase diagram of the model is presented, and it is shown that there is a subcritical regime for $\sigma< \sigma^*(p)$ and $p<p_c(d)$ (where $p_c(d)$ is the critical probability for homogeneous percolation in ${\mathbb{L}}$), a bulk supercritical regime for $p>p_c(d)$, and a surface supercritical regime for $p<p_c(d)$ and $\sigma>\sigma^*(p)$. We show that $\sigma^*(p)$ is a strictly decreasing function for $p\in[0,p_c(d)]$, with a jump discontinuity at $p_c(d)$. We extend the Aizenman-Barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase. We prove that the cluster size distribution decays exponentially in the subcritical phase, and sub-exponentially in the supercritical phases. For a model of lattice animals with a defect plane, the free energy is related to functions of the inhomogeneous percolation model, and we show how the percolation transition implies a non-analyticity in the free energy of the animal model. Finally, we present simulation estimates of the critical curve $\sigma^*(p)$.