Sharp asymptotic for the chemical distance in long-range percolation

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11620%2F19%3A10403974" target="_blank" >RIV/00216208:11620/19:10403974 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=p7z4kD7eDq" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=p7z4kD7eDq</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1002/rsa.20849" target="_blank" >10.1002/rsa.20849</a>

Result language
angličtina
Original language name
Sharp asymptotic for the chemical distance in long-range percolation
Original language description
We consider instances of long-range percolation on Zd and Rd, where points at distance r get connected by an edge with probability proportional to r(-s), for s is an element of (d,2d), and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance D(x,y) between x and y in the limit as |x - y|->infinity. For the model on Zd we show that, in probability as |x|->infinity, the distance D(0,x) is squeezed between two positive multiples of (logr)Delta, where Delta:=1/log2(1/gamma) for gamma: = s/(2d). For the model on Rd we show that D(0,xr) is, in probability as r ->infinity for any nonzero x is an element of Rd, asymptotic to phi(r)(logr)Delta for phi a positive, continuous (deterministic) function obeying phi(r(gamma)) = phi(r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10103 - Statistics and probability

Project
<a href="/en/project/GA16-15238S" target="_blank" >GA16-15238S: Collective behavior of large stochastic systems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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