English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Consider (independent) first-passage percolation on the sites of thetriangular lattice $\mathbb{T}$. Denote the passage time of the site $v$ in$\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by$b_{0,n}$ the passage time from 0 to the halfplane$\{v\in\mathbb{T}:\mbox{Re}(v)\geq n\}$, and by $T(0,nu)$ the passage time from0 to the nearest site to $nu$, where $|u|=1$. We prove that as$n\rightarrow\infty$, $b_{0,n}/\log n\rightarrow 1/(2\sqrt{3}\pi)$ a.s.,$E[b_{0,n}]/\log n\rightarrow 1/(2\sqrt{3}\pi)$ and Var$[b_{0,n}]/\logn\rightarrow 2/(3\sqrt{3}\pi)-1/(2\pi^2)$; $T(0,nu)/\log n\rightarrow1/(\sqrt{3}\pi)$ in probability but not a.s., $E[T(0,nu)]/\log n\rightarrow1/(\sqrt{3}\pi)$ and Var$[T(0,nu)]/\log n\rightarrow 4/(3\sqrt{3}\pi)-1/\pi^2$.This answers a question of Kesten and Zhang (1997) and improves our previouswork (2014). From this result, we derive an explicit form of the central limittheorem for $b_{0,n}$ and $T(0,nu)$. A key ingredient for the proof is themoment generating function of the conformal radii for conformal loop ensembleCLE$_6$, given by Schramm, Sheffield and Wilson (2009).

Limit theorems for critical first-passage percolation on the triangular lattice