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Supercritical Case

The supercritical case is where $\alpha<1$. In this case we have that longer edges are preferred over shorter ones; in particular, $t^{-1}\mathcal{B}_t$ can extend outside $B_1$. Furthermore, since any vertex $\mathbf{x}$ has probability $p$ of being reached in the optimal time $||\mathbf{x}||^{\alpha}$ we will not have the same result as in the critical case.

In this case our simulations suggested that the limiting shape of $t^{-\frac{1}{\alpha}}\mathcal{B}^{(\alpha)}_t$ should coincide approximately with the $l^1$ ball $B_1$. The figure below shows one quadrant from a simulation where the edge weights have distribution

\[f(x)=p\chi_{\{1\}}(x)+(1-p)e^{1-x}\chi_{\{(1,\infty)\}}(x) \]

for $p=0.7$. This is a distribution with an atom at $\{1\}$ and an exponential tail. The white region shows the unoccupied space, the black regions shows the standard ball $B_1$ and the scaled ball $t^{\frac{1}{\alpha}}B_{1}$, the dark grey region shows the active vertices under the LRFPP model and the light grey points are those which are active in both the nearest neighbour and long-range models. For this simulation the nearest neighbour model has been coupled with the long range one so that the nearest neighbour growth set cannot extend further than the corresponding long range one. The growth set for the long-range model has small fluctuations around a flat piece which extends over the entire edge of the ball. This suggests that the limiting shape of $t^{-\frac{1}{\alpha}}\mathcal{B}^{(\alpha)}_t$ exists and is $B_1$.

Flat piece for supercritical long range model

Our main result for the subcritical case is indeed exactly what the simulations have suggested in that the limiting shape of the scaled growth set is the unit $l^1$ ball.

Theorem 2:

If $\alpha<1, p \in (0,1]$ we have that for $\mu \in \mathcal{M}_p$ and any $\varepsilon>0$ that:

\[\mathbb{P}\left[B_{1-\varepsilon} \subseteq t^{-\frac{1}{\alpha}}\mathcal{B}^{(\alpha)}_t \subseteq B_{1+\varepsilon} \; for \; large \; t\right]=1 \]