Exact scaling form for the collapsed 2D polymer phase

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 $\gamma .$