A New Finite-Size Scaling Approach to Random Walks

TL;DR

This paper introduces a novel finite-size scaling method for random walks that improves upon previous approaches, providing a consistent and exact framework applicable across all dimensions, with implications for phase diagrams and disordered systems.

Contribution

The paper develops a new finite-size scaling approach for random walks that supersedes the inconsistent renormalization group method, applicable in any dimension and based on exact correlation functions.

Findings

New finite-size scaling method validated across dimensions

Derived phase diagram for surface-bulk interacting random walks

Discussion on extensions to disordered systems

Abstract

We present a new finite-size scaling method for the random walks (RW) superseeding a previously widely used renormalization group approach, which is shown here to be inconsistent. The method is valid in any dimension and is based on the exact solution for the two-point correlation function and on finite size scaling. As an application, the phase diagram is derived for random walks with a surface-bulk interaction where the system has either a surface or a defect. Possible extensions to disordered systems are also discussed.