Licea, C.; Newman, C. M.; Piza, M. S. T. Superdiffusivity in first-passage percolation. (English) Zbl 0870.60096 Probab. Theory Relat. Fields 106, No. 4, 559-591 (1996). 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) Cited in 1 ReviewCited in 24 Documents 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 Keywords:transverse fluctuations; passage times; nearest neighbour edges; critical value PDFBibTeX XMLCite \textit{C. Licea} et al., Probab. Theory Relat. Fields 106, No. 4, 559--591 (1996; Zbl 0870.60096) Full Text: DOI