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5. Miscellaneous

Here we collect a couple of problems that did not quite fit under the previous topics.

    1.     Let ${\rm Ball}(t):=\{z\in\mathbb{Z}^d:T(0,z)\le t\}$ be the ball of radius $t$ in the first-passage metric.

      Add remark

      Problem 5.1.

      [Amir Dembo] Is a random walk on ${\rm Ball}(t)$ recurrent? What if the jump rate of the walker is allowed to depend on $t$?

        •     Häggström and Pemantle [MR1659548] introduced a model for competing growth as a mean to study the geodesic structure in first-passage percolation with exponential weights. In this model each vertex can be in either of three states 0, 1 or 2. A site in state 0 flips to state 1 (or 2) at a rate $\lambda_1$ (or $\lambda_2$) times the number of neighbors in state 1 (or 2). A site in state 1 or 2 remains in that state forever. In [MR1659548] it was proved that coexistence between (that is, unbounded growth of) the two types has positive probability for $\lambda_1=\lambda_2$, and conjectured that coexistence has probability zero for $\lambda_1\neq\lambda_2$. This model has come to be known as the two-type Richardson model.

          Add remark

          Problem 5.2.

          [Christopher Hoffman] Prove the Häggström-Pemantle conjecture that the two-type Richardson model has almost surely a single surviving type for $\lambda_1\neq\lambda_2$.
              The conjecture is known to be true for all but possibly a countable set of ratios between the two parameters, see [MR1794536].

              Cite this as: AimPL: First passage percolation, available at http://aimpl.org/firstpercolation.