Limit shape formulas for a generalized Seppäläinen–Johansson model

Abstract

We consider a simplified model of first-passage percolation, involving two families of i.i.d. random variables $\{{\mathrm{\xi }}_{ij}\}$ and $\{{\mathrm{\eta }}_{ij}\}$ corresponding to the weights of the horizontal and vertical edges respectively. We obtain an explicit formula for the limiting shape of the first-passage distance expressed in terms of the corresponding limit shapes of the two sets of weights for the Seppäläinen–Johansson model. We also study the limiting fluctuations of this model when at least one of the sets of weights is Bernoulli distributed.