square lattice}; \\
1.306 \pm 0.007, & \hbox{in the cubic lattice}.
}$$ These results are then used to estimate the crossover exponent $\phi$ associated with the adsorption transition, giving $$\phi =
\cases{
0.496 \pm 0.009, & \hbox{in two dimensions}; \\
0.505 \pm 0.006, & \hbox{in three dimensions}.
}$$ The microcanonical density function of adsorbing walks is determined as well, and shown to have properties consistent with the location of the critical points above. In addition, the specific heat of the model is determined, and it is found that it has scaling behaviour consistent with the estimates of the critical points above.

It is also shown that the adsorption transition is seen in a change in the metric scaling of walks at the critical point. The desorbed phase is a phase of positive walks with the scaling properties of self-avoiding walks in a good solvent, while the adsorbed walk has the metric properties of a walk in one dimension less; that is, of walks adhering to the adsorbing surface.

Scaling of the generating and partition functions are found to be consistent with the exact values of critical exponents determined elsewhere in two dimensions [@BEG89; @BY95], and with the value $\phi=\shalf$ in 