Asymptotics for $2D$ Critical First Passage Percolation

Understand

We consider first-passage percolation on $\mathbb{Z}^2$ with i.i.d.

• weights, whose distribution function satisfies $F(0) = p_c = 1/2$.
• This is sometimes known as the "critical case" because large clusters of zero-weight edges force passage times to grow at most logarithmically, giving zero time constant.
• Denote $T(\mathbf{0}, \partial B(n))$ as the passage time from the origin to the boundary of the box $[-n,n] \times [-n,n]$.