Abstract:
The talk will focus on large-scale metric properties of random graphs obtained from 
by way of long range percolation. In this process, an edge between vertices 
and 
is added to the existing graph structure with probability asymptotic to 
, independently of other edges, for some positive parameters 
and~
. 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 to numbers 
and 
. In particular, the graph-theoretical distance scales sublinearly with the Euclidean distance for all 
and remains bounded for all 
.> > After an overview of known results, I will focus mainly on the regime 
. Here the graph theoretical distance between vertices at Euclidean distance 
is asymptotic to 
as 
, where 
is a known function of~ and~ and 
is a positive and bounded function subject to the log-log-periodic restriction 
for 
. While 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~ 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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Based on
https://arxiv.org/abs/2203.17207
http://arxiv.org/abs/1705.10380