Common trends in the critical behavior of the Ising and directed walk models

TL;DR

This paper explores the intrinsic connection between layered 2D Ising and directed walk models, revealing that their critical behaviors are related and providing exact critical exponents for various coupling distributions.

Contribution

It demonstrates the relationship between the Ising and directed walk models and derives exact critical exponents for hierarchical and aperiodic couplings using renormalization.

Findings

Critical exponents are obtained exactly for several distributions.

The thermodynamical properties of the Ising model are encoded in the directed walk transfer matrix.

The models are inherently related through their transfer matrices.

Abstract

We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained into the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization.