In the high temperature (or small $a$) regime, the partition function scales as $$Z_n(a) \simeq B_\lambda n^{\gamma_t -1} h_\lambda(n^\phi |t|)\,
\kappa_-^{n\,|t|^{1/\phi}},\q
\hbox{if $a< a_c^+$},
\label{eqn3}   $$ where $t = (a\minus a_c^+)$ and $\lambda$ denotes the high temperature regime, and where $h_\lambda(x) \simeq |x|^{(\gamma_1-\gamma_t)/\phi}$ and $\log \kappa_- \simeq |a\minus a_c^+|^{-1/\phi}\log \mu_d$. Putting $a=1$, for example, and adsorbing constants and functions of $t$ into $B_\lambda$, give $$Z_n(1) \simeq B_\lambda n^{\gamma_1-1} \mu_d^n, 
\q\hbox{since $\C{F}(a) = \log \mu_d$ if $a<a_c^+$},$$ and the exponent $\gamma_1$ is the entropic exponent of half-space walks, namely $c_n^+ \sim n^{\gamma_1-1} \mu_d^n$.

At the critical adsorption point $a_c^+$, the above scaling is modified to $$Z_n(a_c^+) \simeq B_c n^{\gamma_t -1}\,\mu_d^n ,\q
\hbox{if $a= a_c^+$},
\label{eqn15}  $$ where $\gamma_t$ is the entropic exponent associated with adsorbing walks at the critical adsorption point. The ensemble of half-space walks at the critical point has associated *surface entropic exponent* $\gamma_s$, and this is related to $\gamma_t$ by $\gamma_t
= \gamma_s$ (see for example section 9.1.3 in reference [@JvR15]).

The scaling in the adsorbed phase is similar to the above, but now with different exponents $$Z_n(a) \simeq B_{\tau_0} n^{\gamma_t -1} h_{\taus0} (n^\phi |t|)\,
\kappa_+^{n\,|t|^{1/\phi}},\q
\hbox{if $a> a_c^+$},$$ where $h_{\taus0} (x) \simeq |x|^{(\gamma_+ -\gamma_t)/\phi}$, and where the 