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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 $pin[0,p_c(d)]$, 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 $sigma^*(p)$.
We study space-inhomogeneous quantum walks (QWs) on the integer lattice which we assign three different coin matrices to the positive part, the negative part, and the origin, respectively. We call them two-phase QWs with one defect. They cover one-de
This paper studies the spectrum of a multi-dimensional split-step quantum walk with a defect that cannot be analysed in the previous papers. To this end, we have developed a new technique which allow us to use a spectral mapping theorem for the one-d
Let $ mathbb{L}^{d} = ( mathbb{Z}^{d},mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional hyperplane $ mathbb{
The motion of a particle in the field of dispiration (due to a wedge disclination and a screw dislocation) is studied by path integration. By gauging $SO(2) otimes T(1)$, first, we derive the metric, curvature, and torsion of the medium of dispiratio
Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric ob