Random Walk Model on a Hyper-Spherical Lattice

TL;DR

This paper investigates a random walk on hyper-spherical lattices to understand critical phenomena in hyper-spherical geometries, revealing phase transition behaviors and adsorption properties relevant to statistical physics.

Contribution

It introduces a novel analysis of random walks on hyper-spheres and applies this to phase transitions and polymer adsorption in curved geometries.

Findings

Identified a phase diagram with second and first order transitions separated by a tricritical point.

Discovered exponential decay of adsorbed monomers near critical adsorption energy.

Observed crossover from exponential to linear growth in adsorption at specific energies.

Abstract

We use a one-dimensional random walk on $D$-dimensional hyper-spheres to determine the critical behavior of statistical systems in hyper-spherical geometries. First, we demonstrate the properties of such walk by studying the phase diagram of a percolation problem. We find a line of second and first order phase transitions separated by a tricritical point. Then, we analyze the adsorption-desorption transition for a polymer growing near the attractive boundary of a cylindrical cell membrane. We find that the fraction of adsorbed monomers on the boundary vanishes exponentially when the adsorption energy decreases towards its critical value. We observe a crossover phenomenon to an area of linear growth at energies of the order of the inverse cell radius.