# Universality of the time constant for $2D$ critical first-passage percolation

## TL;DR

This paper proves that in 2D critical first-passage percolation, the normalized first-passage time grows logarithmically with distance, with a universal constant depending only on the distribution's median, and establishes the variance's exact limit.

## Contribution

It establishes the universality of the time constant and variance in 2D critical FPP, providing explicit formulas and extending results to other lattices.

## Key findings

- Time constant exists and equals I/(2√3π) times log n.
- Normalized variance converges to an explicit constant depending on I.
- Universality holds across different 2D lattices under certain conditions.

## Abstract

We consider first-passage percolation (FPP) on the triangular lattice with vertex weights $(t_v)$ whose common distribution function $F$ satisfies $F(0)=1/2$. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by $T(0,\partial B(n))$ the first-passage time from $0$ to $\{x : \|x\|_\infty = n\}$, we show existence of the "time constant'' and find its exact value to be \[ \lim_{n \to \infty} \frac{T(0,\partial B(n))}{\log n} = \frac{I}{2\sqrt{3}\pi} \text{ almost surely}, \] where $I = \inf\{x > 0 : F(x) > 1/2\}$ and $F$ is any critical distribution for $t_v$. This result shows that the time constant is universal and depends only on the value of $I$. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of $I$, under the optimal moment condition on $F$. The proof method also shows an analogous universality on other two-dimensional lattices, assuming the time constant exists.