## 5. Miscellaneous

Here we collect a couple of problems that did not quite fit under the previous topics.-
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.
#### 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*.#### Problem 5.2.

[Christopher Hoffman] Prove the Häggström-Pemantle conjecture that the two-type Richardson model has almost surely a single surviving type for $\lambda_1\neq\lambda_2$.

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