We give several bounds for a graph diameter of a random graph created by a stochastic model called the long-range percolation, in which any pair of two distinct points is connected by a random bond independently. Such a problem is well studied on a finite subset of the \(d\)-dimensional square lattice, where \(d\) is a positive integer. In this manuscript, we consider the problem on a finite subset of a fractal lattice which is called the pre-Sierpinski gasket. We can observe that the Hausdorff dimension of the fractal lattice appears in the critical parameters of \(s\), where \(s\) is a value determining the order of probabilities that random bonds exist.
Long-range percolation Random graph Graph diameter Fractal lattice
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The author thanks to the referee for reading the manuscript carefully and giving kind comments for improvements.
Benjamini, I., Kesten, H., Peres, Y., Schramm, O.: Geometry of the uniform spanning forest: transitions in dimensions \(4,8,12,\cdots \). Ann. Math. 160, 465–491 (2004)CrossRefzbMATHMathSciNetGoogle Scholar