Exact scaling form for the collapsed 2D polymer 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 . $