Shape fluctuations are different in different directions

Abstract

We consider the first passage percolation model on ${\mathbf{Z}}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. Let $T(u,v)$ be the passage time from $u$ to $v$. In this paper, we show that, whenever $F(0)<p_c$, ${\sigma }^2(T((0,0),(n,0)))\ge C\log n$ for all $n\ge 1$. Note that if $F$ satisfies an additional special condition, $\inf \mathrm{supp}(F)=r>0$ and $F(r)>{\vec{p}}_c$, it is known that there exists $M$ such that for all $n$, ${\sigma }^2(T((0,0),(n,n)))\le M$. These results tell us that shape fluctuations not only depend on distribution $F$, but also on direction. When showing this result, we find the following interesting geometrical property. With the special distribution above, any long piece with $r$-edges in an optimal path from $(0,0)$ to $(n,0)$ has to be very circuitous.

Published in at http://dx.doi.org/10.1214/009117907000000213 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Singularity points for first passage percolation

Cited by in corpus (6)

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• Limit theorems for maximum flows on a lattice
• Upper bounds on the non-random fluctuations in first passage percolation with low moment conditions
• Shape curvatures and transversal fluctuations in the first passage percolation model
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle