A Problem in Last-Passage Percolation

TL;DR

This paper investigates the asymptotic behavior of the number of directed paths in a last-passage percolation model with Bernoulli-like weights, establishing key properties of their exponential growth rate.

Contribution

It introduces a new analysis of path counts in a last-passage percolation model with specific random weights, deriving limit properties of their exponential growth.

Findings

Established the limit of the exponential growth rate of path counts.

Derived properties of the growth rate function in the model.

Provided insights into the structure of optimal paths in the percolation setting.

Abstract

Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We consider paths $\pi \subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin and with the last coordinate increasing along the path, and of length $n$. Define for such paths $W(\pi) = \text{number of vertices $\pi_i, 1 \le i \le n$, with}X(\pi_i) = e^b$. Finally let $N_n(\al) = \text{number of paths $\pi$ of length $n$ starting at $\pi_0 = \bold 0$ and with $W(\pi) \ge \al n$.}$ We establish several properties of $\lim_{n \to \infty} [N_n]^{1/n}$.