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 $\{ξ_{ij}\}$ and $\{η_{ij}\}$ 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.

Figures (2)

• Figure 1: An upright path from (0,0) to (4,3). The weight of this path is $B_{1,0}\xi_{1,0}+(1-B_{1,1})\eta_{1,1}+(1-B_{1,2})\eta_{1,2}+B_{2,2}\xi_{2,2}+B_{3,2}\xi_{3,2}+(1-B_{3,3})\eta_{3,3}+B_{4,3}\xi_{4,3}$.
• Figure 2: A portion of the random walk web obtained from the $B_{ij}$'s. If $B_{ij}=0$, an edge is placed from $(i-1,j)$ to $(i,j)$; otherwise, the edge is placed from $(i,j-1)$ to $(i,j)$.

Theorems & Definitions (9)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof