Marek Biskup: Asymptotic metric properties of long-range percolation graphs


The talk will focus on large-scale metric properties of random graphs obtained from \mathbb Z^d by way of long range percolation. In this process, an edge between vertices x and y is added to the existing graph structure with probability asymptotic to \beta |x-y|^{-s}, independently of other edges, for some positive parameters s and~\beta. Earlier work identified five different regimes of the asymptotic scaling of the graph-theoretical distance with the Euclidean distance depending on the relative position of the exponent s to  numbers d and 2d. In particular, the graph-theoretical distance scales sublinearly with the Euclidean distance for all s\le 2d and remains bounded for all s<d.> >    After an overview of known results, I will focus mainly on the regime d<s<2d. Here the graph theoretical distance between vertices at Euclidean distance r is asymptotic to \phi_\beta(r)(\log r)^\Delta as r\to\infty, where \Delta is a known function of~s and~d and \phi_\beta is a positive and bounded function subject to the log-log-periodic restriction \phi_\beta(r)=2\phi_\beta(r^\gamma) for \gamma:=s/(2d). While \phi_\beta appears to arise from the method of proof and its reliance on subadditivity along doubly-exponential sequences of scales, I will show that it is actually non-constant, at least for~\beta sufficiently large. The resulting arithmetic oscillations (on top of the polylogarithmic growth) are caused by the shortest paths having a dyadic hierarchical structure all the way up to unit scales.> >    The talk is based on a sequence of papers involving, in recent collaborations, Jeff Lin and Andrew Krieger. >


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