known that $c_n^+ = \mu_d^{n+o(n)}$ [@HTW82].

If positive walks are counted with respect to the number of vertices in $\partial\mathL^d_+$ (these are *visits*), then the walks are *adsorbing walks*. For example, the walk in figure \[figure0\] is an adsorbing walk with $6$ visits.

Let $c_n^+(v)$ be the number of adsorbing walks of length $n$ from the origin in $\mathL^d_+$, with $v$ visits to $\partial \mathL^d_+$. The canonical partition function of adsorbing walks is obtained by introducing an activity $a$ conjugate to the number of visits: $$Z_n(a) = \sum_{v=0}^n c_n^+(v)\,a^v  .
\label{eqnZ}   $$ When $a$ is large, then $Z_n(a)$ is dominated by walks with a large number of visits, and if $a$ is small (but positive), then $Z_n(a)$ is dominated by walks with a small number of visits.

The *finite size free energy* of these models is computed from the partition function $Z_n(a)$ (see equation ), and is given by $$\C{F}_n(a) = \Sfrac{1}{n} \log \sum_v c_n^+(v)\, a^v .
\label{eqn23}   $$ The *limiting free energy* of the model is given by the thermodynamic limit in the model: $$\C{F}(a) = \lim_{n\to\infty} \Sfrac{1}{n} \log \sum_v c_n^+(v)\, a^v .
\label{eqn25F}  $$ This limit exists (see reference [@HTW82], and also, for example, reference [[@JvR15]]{}), and it 