will remain near the adsorbing plane, and so have the properties of a walk which is stretched out in $(d\minus 1)$ dimensions near $\partial\mathL^d_+$ (and compressed in the $d$-th dimension). Separating these two regimes is the adsorption critical point $a_c^+$.

The finite size scaling of the free energy $\C{F}_n(a)$ is given by $$\C{F}_n(a) \simeq \log \mu_d + (a\minus a_c^+)^{2-\alpha}\,f(n^\phi(a\minus a_c^+)),
\label{eqn24}  $$ where $f$ is a scaling function, $\alpha$ is the specific heat exponent, and $\phi$ is the finite size crossover exponent. The exponents $\alpha$ and $\phi$ are related by the hyperscaling relation $$2-\alpha=\Sfrac{1}{\phi} .
\label{eqn25}  $$

The bulk entropy contribution to $\C{F}_n(a)$ is $\log\mu_d$ in $\mathL^d_+$ (where $\mu_d$ is the growth constant of self-avoiding walks in the square lattice). Slightly redefining the scaling function, it is found that $$\C{F}_n(a) \simeq \log \mu_d + \Sfrac{1}{n}\, g(n^\phi (a\minus a_c^+)).
\label{eqn26}  $$ By plotting $n(\C{F}_n(a) \minus \log\mu_d)$ against $n^\phi(a\minus a_c^+)$, the function $g$ can be uncovered (for $n$ large and $|a\minus a_c^+|$ small).

Taking derivatives of $\C{F}_n(a)$ to $\log a$ gives the energy (density) $\C{E}_n(a)$ and specific heat $\C{C}_n(a)$ of the model. The scaling of these quantities follows directly from equation : $$\C{E}_n(a) \simeq n^{\phi-1}\,h_e(n^\phi(a\minus a_c^+)),
\q \hbox{and}\q
\C{C}_n(a) \simeq n^{\alpha\phi}\,h_c(n^\phi(a\minus a_c^+)),
