Critical first-passage percolation starting on the boundary

Abstract

We consider first-passage percolation on the two-dimensional triangular lattice $\mathcal{T}$. Each site $v\in \mathcal{T}$ is assigned independently a passage time of either $0$ or $1$ with probability $1/2$. Denote by $B^{+}(0,n)$ the upper half-disk with radius $n$ centered at $0$, and by $c_n^{+}$ the first-passage time in $B^{+}(0,n)$ from $0$ to the half-circular boundary of $B^{+}(0,n)$. We prove \[\lim_{n\rightarrow\infty}\frac{c_n^+}{\log n}=\frac{\sqrt{3}}{2π}~ a.s.,~\lim_{n\rightarrow\infty}\frac{E c_n^+}{\log n}=\frac{\sqrt{3}}{2π},~\lim_{n\rightarrow\infty}\frac{\mathrm{Var}(c_n^+)}{\log n}=\frac{2\sqrt{3}}π-\frac{9}{π^2}.\] These results enable us to prove limit theorems with explicit constants for any first-passage time between boundary points of Jordan domains. In particular, we find the explicit limit theorems for the cylinder point to point and cylinder point to line first-passage times.

16 pages, revision after the referee's report, to appear in Stochastic Processes and their Applications

References in corpus (2)

• Conformal radii for conformal loop ensembles
• Limit theorems for critical first-passage percolation on the triangular lattice

Cited by in corpus (3)

• Measurement-driven entanglement transition in hybrid quantum circuits
• Conformal invariance and quantum non-locality in critical hybrid circuits
• Entanglement phase transitions in random stabilizer tensor networks