Sublinear variance in first-passage percolation for general distributions
The paper studies the variance of the first-passage time \(\tau(0,x)\) from \(0\) to \(x\in\mathbb Z^d\) in the percolation model on the lattice \(\mathbb Z^d\), defined by \[ \tau(0,x)=\inf_{\gamma:0\rightarrow x}\sum_{e\in\gamma} t_e. \] In the above definition, \(\gamma=(v_0=0,e_0,v_1,\dots,e_N,v_N=x)\) is the lattice path that joins \(0\) to \(x\) and \((t_e)\) is a collection of independent random variables with common distribution \(\mu\) on the set of nearest-neighbor edges. For \(d=1\), \(\tau(0,x)\) becomes a sum of i.i.d.\ random variables and the variance of \(\tau(0,x)\) is of order \(\| x\|_1\), while for \(d\geq 2\), \(\tau(0,x)\) is the minimum over correlated sums of i.i.d.\ random variables. In the remarkable paper [Ann. Probab. 31, No. 4, 1970--1978 (2003; Zbl 1087.60070)], \textit{I. Benjamini} et al. proved that, in the case when the edge-weight distribution \(\mu\) is uniform on the set of two positive values (=\,weights) \(a\) and \(b\), the variance is sublinear in distance, i.e., \[ \operatorname{Var}\tau(0,x)\leq C(a,b)\frac{\| x\|_1}{\log\| x\|_1}, \] for all \(d\geq 2\). The present paper extends the above sublinear result to measures \(\mu\) obeying some moment condition.
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