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Subcritical Case

The subcritical case is where $\alpha>1$. In this case we have that shorter edges are preferred over the longer ones. Although any path to a given vertex in optimal time must only consist of nearest neighbour edges, very few longer edges can still be used in a path to the flat piece due to the way in which the growth set is scaled. In this case it is less obvious that a limiting shape exists. There is an obvious upper bound formed by the unit ball in $l^1$ and the limiting shape in the nearest neighbour case forms an obvious lower bound. We have shown that there is indeed a limiting shape and these upper and lower bounds immediately give us that the limiting shape has a flat piece of at least the length in the nearest neighbour case.

In this case our simulations suggested that the limiting shape of $t^{-1}\mathcal{B}^{(\alpha)}_t$ exists and should coincide with the limiting shape in the nearest neighbour case along the flat piece but deviate from both the unit $l^1$ ball and the asymptotic shape from the nearest neighbour model away from this region. 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 $p=0.55$. This is a distribution with an atom at $\{1\}$ and an exponential tail. The white region shows the unoccupied space, the black region shows the standard ball $B_1$, 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.

Flat piece for subcritical long range model

The main result that we have proven for the subcritical case is the following theorem which shows that a limiting shape exists and gives the exact form in terms of the asymptotic speed.

Theorem 3:
If $\alpha \geq 1, \varepsilon>0$ we have that $\exists A_\mu^{(\alpha)}$ deterministic such that

\[ \mathbb{P}[(1-\varepsilon)A_\mu^{(\alpha)}\subseteq t^{-1}\mathcal{B}^{(\alpha)}_t \subseteq (1+\varepsilon)A_\mu^{(\alpha)} \quad for \; large \; t]=1\]

Furthermore $A_\mu^{(\alpha)}=\{x \in \mathbb{R}^2: \varphi^{(\alpha)}\leq 1\}$ where

\[\varphi^{(\alpha)}(x):=\lim_{n \rightarrow \infty}\frac{T^{(\alpha)}(0,nx)}{n}\]

which exists almost surely and in $L^1$, moreover the convergence is uniform on compact sets.

We have also proven that in the specific case that the distribution $\mu$ has bounded support then the flat piece of the long-range model coincides precisely with that of the nearest neighbour model. Based on this result, our simulation and heuristic arguments we conjecture that the flat piece for the long-range model coincides with that of the nearest neighbour model for any $\mu \in \mathcal{M}_p$ in the subcritical regime.