2. The time constant and asymptotic shape
In the following, let $\mu(z)$ denote the time constant, that is the almost sure limit $\mu(z):=\lim_{n\to\infty}\frac{1}{n}T(0,nz)$. The time constant is well-known to form a norm on $\mathbb{R}^d$, and that ${\rm Ball}$ coincides with the unit ball $\{x\in\mathbb{R}^d:\mu(x)\le1\}$ in this norm. The following questions relate to the properties of the time constant and the shape.-
By convexity, the asymptotic shape has to contain a ‘diamond’ and be contained in a ‘square’. It is strongly believed that it should not equal either of the two.
Problem 2.1.
[Antonio Auffinger] For $d=2$, show that $\text{Ball}$ is not a square. -
Problem 2.2.
[Yu Zhang] Consider Bernoulli distributed edge weights with parameter $p$. Let $g(p)$ denote the time constant in the coordinate direction for this weight distribution. Is $g(p)$ differentiable as a function of $p$? -
Problem 2.3.
[Antonio Auffinger] How does $\mu({\bf e}_1)$ behave as a function of $d$? -
In greater generality we may assume that the edge weights are not independent, but obtained from some stationary and ergodic measure.
Problem 2.4.
[Jean-Christophe Mourrat] Assume that the edge weights $\{\omega_e\}$ are bounded and that $1/\omega_e$ has polynomial tails. Prove that $\mu({\bf e}_1)\neq0$, or find a counterexample.
Cite this as: AimPL: First passage percolation, available at http://aimpl.org/firstpercolation.