Exact scaling form for the collapsed 2D polymer phase

TL;DR

The paper discusses the exact scaling form of the partition function for collapsed 2D polymers, highlighting the universality of the configuration exponent and the specific form of the partition function in the low-temperature phase.

Contribution

It provides an exact scaling form for the partition function of collapsed 2D polymers and conjectures the precise value of the configuration exponent.

Findings

Partition function follows a specific scaling form involving bulk and perimeter fugacities.

The configuration exponent  is conjectured to be exactly known in 2D.

Supports the universality of the configuration exponent  in the collapsed phase.

Abstract

It has been recently argued that interacting self-avoiding walks (ISAW) of length $ \ell , $ in their low temperature phase (i.e. below the $ \Theta $-point) should have a partition function of the form: $$ Q_{\ell} \sim \mu^{ \ell}_ 0\mu^{ \ell^ \sigma}_ 1\ell^{ \gamma -1}\ , \eqno $$ where $ \mu_ 0(T) $ and $ \mu_ 1(T) $ are respectively bulk and perimeter monomer fugacities, both depending on the temperature $ T. $ In $ d $ dimensions the exponent $ \sigma $ could be close to $ (d-1)/d, $ corresponding to a $ (d-1) $-dimensional interface, while the configuration exponent $ \gamma $ should be universal in the whole collapsed phase. This was supported by a numerical study of 2D partially {\sl directed\/} SAWs for which $ \sigma \simeq 1/2 $ was found. I point out here that formula (1) already appeared at several places in the two-dimensional case for which $ \sigma =1/2, $ and for which one can even conjecture the exact value of $ \gamma . $