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Superdiffusivity in first-passage percolation. (English) Zbl 0870.60096

The main results of the paper are concerning lower bounds for the exponent \(\xi\) describing transverse fluctuations of the growing surface of the random set \(B(t)= \{v\in\mathbb{Z}^d: T(0,v)\leq t\}\) in standard first passage (undirected bond) percolation on \(\mathbb{Z}^d\) \((d\geq 2)\). Here, the indices of the i.i.d. passage times \(\tau(e)\geq 0\) are nearest neighbour edges \(e=\{u,v\}\), and \(T(u,v)\) is denoting the passage time between sites \(u,v\in\mathbb{Z}^d\). One (of several possible!) definitions of \(\xi\) adopted in the paper is the following point-to-point definition: For some vector \(x\in\mathbb{R}^d\) \((x\neq 0)\) define the line \(L_x\) to be \(\{\alpha x:\alpha\in\mathbb{R}\}\), and let \({\mathcal C}(x,w)\) denote the closed cylinder in \(\mathbb{R}^d\) of radius \(w>0\) and symmetry axis \(L_x\). For sites \(u,v\in\mathbb{Z}^d\) put \({\mathcal M}(u,v)=\{r:r\) is a path from \(u\) to \(v\) with passage time \(T(u,v)\}\). Saying that \({\mathcal M}(u,v)\) is in a set \(A\subset\mathbb{R}^d\) means that every \(r\) in \({\mathcal M}(u,v)\) only touches sites in \(A\). Then, \(\xi\) is defined by \(\xi:=\sup\{\gamma\geq 0:\limsup_{|v|\to\infty} P({\mathcal M}(0,v)\) is in \({\mathcal C}(v,|v|^\gamma))<1\}\). One of the main results then says that if the bottom of the support of the common distribution of the \(\tau(e)\)’s equals zero and if \(P(\tau(e)=0)<p_c\) (the critical value for \(\mathbb{Z}^d\)) and \(E[(\tau(e))^2]<\infty\), then, for all \(d\geq 2\), \(\xi\geq 1/(d+1)\).
Reviewer: K.Schürger (Bonn)

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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