subscript $\taus0$ denotes the low temperature (and large $a$) scaling. Since $\sfrac{1}{n} \log Z_n(a) = \C{F}(a)\,(1\plus o(1))$, it follows that $\log \kappa_+  \simeq |a\minus a_c^+|^{-1/\phi} \C{F}(a)$. This, in particular, gives the scaling $$\C{F}_s(a) \sim |a \minus a_c^+|^{1/\phi}$$ for the singular part of the free energy in the adsorbed phase (consistent with the hyperscaling relation ). The scaling of $Z_n(a)$ simplifies here to $$Z_n(a) \simeq B_{\tau_0} n^{\gamma_+-1} \; e^{n\thin\C{F}(a)} ,\q
\hbox{if $a>a_c^+$}.$$ The exponent $\gamma_+$ should be that of adsorbed walks, and so given by $\gamma_+ = \gamma^{(d-1)}$, the entropic exponent of walks in one dimension lower.

  Function                   Definition
  -------------------------- ------------------------------------------------------------------------------------------
  $\mathL^d$                 The hypercubic lattice
  $\mathL^d_+$               The half-hypercubic lattice
  $\partial\mathL^d_+$       The boundary of $\mathL^d_+$ (it is isomorphic is $\mathL^{d-1}$)
  $\mu_d$                    The growth constant of self-avoiding walks in $\mathL^d$
  $c_n^+(v)$                 The number of positive walks of length $n$ and $v$ visits from $\vec{0}$ in $\mathL^d_+$
  $Z_n(a)$                   Partition function of positive adsorbing walks of length $n$ and activity $a$
  $\C{F}_n(a)$               Finite size free energy: $\displaystyle F_n(a) = \sfrac{1}{n} \log Z_n(a)$
  $\C{F}(a)$                 Limiting free energy: $\displaystyle \C{F}(a) = \lim_{n\to\infty} \C{F}_n(a)$
  $\C{E}_n(a)$, $\C{E}(a)$   Energy density and limiting energy density
  $\C{C}_n(a)$, $\C{C}(a)$   Specific heat and limiting specific heat
  $H_n$                      The mean height of the endpoint of 