Phase diagram of inhomogeneous percolation with a defect plane

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 $sigma^*(p)$.

The entropic pressure of a lattice polygon

The entropic pressure in the vicinity of a two dimensional square lattice polygon is examined as a model of the entropic pressure near a planar ring polymer. The scaling of the pressure as a function of distance from the polygon and length of the polygon is determined and tested numerically.

Adsorbed self-avoiding walks subject to a force

We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that the appropriate limiting free energy exists where there is an applied force and a surface potential term, and prove that this free energy is convex in appropriate variables. We then derive an expression for the limiting free energy in terms of the free energy without a force and the free energy with no surface interaction. Finally we show that there is a phase boundary between the adsorbed phase and the desorbed phase in the presence of a force, prove some qualitative properties of this boundary and derive bounds on the location of the boundary.

Minimal knotted polygons in cubic lattices

An implementation of BFACF-style algorithms on knotted polygons in the simple cubic, face centered cubic and body centered cubic lattice is used to estimate the statistics and writhe of minimal length knotted polygons in each of the lattices. Data are collected and analysed on minimal length knotted polygons, their entropy, and their lattice curvature and writhe.

On the universality of knot probability ratios

Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is $p_n(K)/p_n$ and is known to decay exponentially with length. Little is known rigorously about the asymptotics of $p_n(K)$, but there is substantial numerical evidence that $p_n(K)$ grows as $p_n(K) simeq , C_K , mu_emptyset^n , n^{alpha-3+N_K}$, as $n to infty$, where $N_K$ is the number of prime components of the knot type $K$. It is believed that the entropic exponent, $alpha$, is universal, while the exponential growth rate, $mu_emptyset$, is independent of the knot type but varies with the lattice. The amplitude, $C_K$, depends on both the lattice and the knot type. The above asymptotic form implies that the relative probability of a random polygon of length $n$ having prime knot type $K$ over prime knot type $L$ is $frac{p_n(K)/p_n}{p_n(L)/p_n} = frac{p_n(K)}{p_n(L)} simeq [ frac{C_K}{C_L} ]$. In the thermodynamic limit this probability ratio becomes an amplitude ratio; it should be universal and depend only on the knot types $K$ and $L$. In this letter we examine the universality of these probability ratios for polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices. Our results support the hypothesis that these are universal quantities. For example, we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure-eight, independent of the underlying lattice, giving an estimate of the intrinsic entropy associated with knot types in closed curves.