The critical pulling force for self-avoiding walks

TL;DR

This paper proves that the critical Boltzmann weight for self-avoiding walks pulled from a surface is exactly 1, confirming a long-standing conjecture and linking free energy properties to the two-point function.

Contribution

The paper provides a simple proof that the critical point y_c equals 1 for self-avoiding walks under pulling force, confirming previous conjectures.

Findings

Critical point y_c = 1 established.

Connection between free energy and two-point function clarified.

Simplified proof of the critical point enhances understanding.

Abstract

Self-avoiding walks are a simple and well-known model of long, flexible polymers in a good solvent. Polymers being pulled away from a surface by an external agent can be modelled with self-avoiding walks in a half-space, with a Boltzmann weight $y = e^f$ associated with the pulling force. This model is known to have a critical point at a certain value $y_c$ of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value $y_c=1$ has been conjectured by several authors using numerical estimates. We provide a relatively simple proof of this result, and show that further properties of the free energy of this system can be determined by re-interpreting existing results about the two-point function of self-avoiding walks.