is a convex function of $\log a$ with a singular point at $a=a_c^+$ (which is the *adsorption critical point* in the model). For $a<a_c^+$ the model is in a *desorbed state*, and for $a>a_c^+$ the model is in an *adsorbed state*. It is known that $a_c^+>1$ [@JvR98], and $a_c^+ < \sfrac{\mu_d}{\mu_{d-1}}$ [@HTW82], and $$\C{F}(a) \cases{
= \log \mu_d, & if $a\leq a_c^+$; \\
> \log \mu_d, & if $a>a_c^+$.
}$$ Since $\C{F}(a)$ is a convex function of $\log a$, it is differentiable for almost all $a>0$, and it follows that the density of visits to the adsorbing plane is $a\sfrac{d}{da}\C{F}(a) = 0$ if $a<a_c^+$ (this is the *desorbed phase*), and $a\sfrac{d}{da} \C{F}(a) >0$ if $a>a_c^+$ $\alsu$ (whenever $\C{F}(a)$ is differentiable). This is the *adsorbed phase*. In the desorbed phase the walk tends to make few returns to the adsorbing plane (walks of length $n$ will return, on average, $o(n)$ times to the adsorbing plane in the desorbed phase). Thus, a desorbed walk will tend to move away from the boundary into the bulk of $\mathL^d_+$. An adsorbed walk, on the other hand, is expected to have a positive density of returns to the adsorbing plane. This implies that the walk 