Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The *random geometric graph* $G_\rho$ is obtained from $V_\rho$ by connecting points that have Euclidean distance smaller than one. Let us assume that $\rho$ is larger than the critical density (in fact, we will later consider $\rho\to\infty$), such that there is a unique infinite component $G_{\rho}^0$.
Let us assume w.l.o.g. that $0\in G_\rho^0$.

To each edge $e$ of $G_\rho^0$ we assign a positive i.i.d. random *delay* $\tau_e$ with pdf $P(\tau_e)$. The *first passage time* $T(x)$ for $x\in G_\rho^0$ is given by minimal total delay among all paths $\gamma$ from 0 to $x$, i.e.

$$T(x)=\min_{\gamma:0\to x}\sum_{e\in\gamma}\tau_e.$$
Let the *hop count* $H(x)$ be the number of edges in the path of shortest delay (assuming it exists and is unique).

Using *subadditive ergodic theory* (Kingman, 1973) it has been shown that under mild conditions on $P(\tau_e)$ the passage time $T(x)$ almost surely scales linearly with $|x|$ in the large $|x|$ limit. However, to my knowledge no exact values for the scaling constant, which is often called the *time constant*, are known. The same seems to be true for the *hop count constant* $h_\rho$, i.e.

$$h_\rho = \lim_{|x|\to\infty}\frac{H(x)}{|x|}.$$

So how about the limiting situation $\rho\to\infty$, in which the connectivity becomes very large? Assuming that $P(\tau_e)$ is smooth and non-zero as $\tau_e\to 0$, the first passage time $T(x)$ will go to zero (when keeping $|x|$ fixed). However, the hop count $H(x)$ is by construction bounded from below by $\lceil|x|\rceil$.

Q1: Are any properties known about the time minimizing paths $\gamma$ in the $\rho\to\infty$ limit?

Numerical experiments suggest that $h_\infty:=\lim_{\rho\to\infty}h_\rho\approx 2$ independently of the distribution $P(\tau_e)$. Below are some examples of the time-minimizing paths (in blue) for $\rho=10,20,30,40$ and $P(\tau_e)$ the exponential distribution with mean 1. In each case the blue path is roughly twice as long as the shortest path (in red).

Image:

Q2: Is $h_{\infty} =2$? Is there a simple (heuristic) explanation?

Note: So far the most relevant (recent) literature I have found is in applications to "Wireless sensor networks". Any pointers to more pure mathematics papers in which such hop counts are studied in plane graphs are most welcome.