Average-height criterion for the order of polymer adsorption transitions

Let $\mathcal{C}$ be an infinite subclass of self-avoiding walks in an upper half-plane, with starting point on the surface. For $\omega\in\mathcal{C}$, let $h(\omega)$ be the maximum height above the surface reached by $\omega$, and let $\langle h_n\rangle$ be the average of $h(\omega)$ over walks $\omega\in\mathcal{C}$ of length $n$. Average-height criterion. If walks in $\mathcal{C}$, equipped with a surface interaction associated with steps along the surface, exhibit an adsorption phase transition, then the transition is first-order if

$$\langle h_n\rangle=\Theta(n),$$

and second-order if

$$\langle h_n\rangle=O(n^{3/4}).$$

This is presented as a rough conjectural explanation for why prudent-walk models have first-order transitions, contrasting with models whose average heights grow sublinearly. The criterion is not proved in the paper.