Interaction versus entropic repulsion for low temperature Ising polymers

TL;DR

This paper demonstrates that at low temperatures, entropic repulsion dominates boundary interactions in 2D Ising model contours, ensuring boundary effects do not alter the surface tension significantly.

Contribution

It proves that entropic repulsion prevails over boundary attraction in low-temperature Ising polymers, maintaining the integrity of the contour partition function.

Findings

Entropic repulsion dominates boundary attraction at low temperatures.

Boundary interactions do not significantly modify the surface tension.

Results apply broadly to various low-temperature statistical mechanics models.

Abstract

Contours associated to many interesting low-temperature statistical mechanics models (2D Ising model, (2+1)D SOS interface model, etc) can be described as self-interacting and self-avoiding walks on $\mathbb Z^2$. When the model is defined in a finite box, the presence of the boundary induces an interaction, that can turn out to be attractive, between the contour and the boundary of the box. On the other hand, the contour cannot cross the boundary, so it feels entropic repulsion from it. In various situations of interest a crucial technical problem is to prove that entropic repulsion prevails over the pinning interaction: in particular, the contour-boundary interaction should not modify significantly the contour partition function and the related surface tension should be unchanged. Here we prove that this is indeed the case, at least at sufficiently low temperature, in a quite general framework that applies in particular to the models of interest mentioned above.