\label{eqn24EC}  $$ for some scaling functions $h_e$ and $h_c$. In the limit as $n\to\infty$, $\C{E}_n(a) \to \C{E}(a)$ (the limiting energy density) and $\C{C}_n(a)\to\C{C}(a)$ (the limiting specific heat). Existence of these limits (almost everywhere) is a consequence of the convexity properties of the limiting free energy (see for example reference [@JvR15]). Physically, $\C{E}(a)$ is the density of visits per unit length, and $\C{C}(a)$ is the rate of change in $\C{E}(a)$ as a function of changes in $\log a$ (it has a maximum at $a_c^+$).

For adsorbing walks it is thought that $\phi=\sfrac{1}{2}$ in all dimensions $d\geq 2$ [@BEG89; @BY95], and numerical evidence supporting this in dimensions lower than $d=4$ (the upper critical dimension) are available in references [@LM88A; @ML88A; @BWO99; @JvRR04; @KPHMBS13]. If $\phi = \sfrac{1}{2}$, then $\alpha = 0$, so, for example, the specific heat has scaling $\C{C}_n(a) = h_c(n^\phi(a\minus a_c^+))$, and plotting measurements of $\C{C}_n(a)$ against the rescaled variable $\tau=n^\phi(a\minus a_c^+)$ for small values of $\tau$ should collapse the curves to a limiting curve (with some finite size corrections to scaling), exposing the scaling function $h_c$.

The partition function has a more complicated scaling law (see, for example, equation (23) in reference [@JvRR04], or section 4.2.2 in reference [@JvR15]). 