We are looking for a model non-integrable dynamical system where the diffusion rate in the stochastic region is practically independent of the value of the Lyapunov Characteristic Number, $\lambda$. We select as a model the perturbed pendulum and we study how $\lambda$ as well as the width, $w$, of the stochastic layer around the separatrix of the primary resonance depend on the values of the perturbation ($\epsilon$) and the length ($l$) of the pendulum. We find that the functions $\lambda(l)$, $w(l)$, $\lambda(\epsilon)$ and $w(\epsilon)$ follow power laws. In particular both $\lambda (l)$ and $w (l)$ scale as the Lyapunov exponent and the width of the resonance of the unperturbed system, i.e. as $l^{\mathrm{-1/2}}$ and $l^{\mathrm{3/2}}$ respectively. Therefore, the width of the stochastic layer is inversely proportional to the cube of $\lambda$ and, for sufficiently small values of $l$, diffusion is restricted to a thin layer, although $\lambda$ takes large values.