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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.
    1.     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 $d$?
                  The above has been determined for exponential edge weights by Dhar [MR0949856], and for general weights by Auffinger and Tang [arXiv:1601.07898].
                •     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.