The universal relation between scaling exponents in first-passage percolation. (English) Zbl 1271.60101

Consider the usual \(d\)-dimensional integer lattice with nearest neighbor bonds. On the bonds, there are weights which are i.i.d. nonnegative random variables. The lenghth of a path is the sum of weights on the bonds in the path. Denote by \(T(x, y)\) the shortest path from \(x\) to \(y\), called the first-passage time. Next, denote by \(D(x, y)\) be the maximal deviation (with respect to the Euclidean distance) of the path from the straight line segment jointing \(x\) and \(y\). It has been conjectured in numerous physics papers that \[ T(x, y)-\operatorname{E}T(x, y)\sim |x-y|^{\chi} \] and \[ D(x, y)-\operatorname{E}D(x, y)\sim |x-y|^{\xi} \] for some constants \(\chi\) and \(\xi\), called fluctuation exponent and wandering exponent, respectively. Furthermore, these constants are conjectured to have the universal relation \(\chi=2\xi -1\), irrespective of the dimension.
This article presents a hard and rigorous proof of this conjecture assuming that the exponents exist in a certain sense.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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