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RANDOM PLANAR METRICS
"... Abstract. A discussion regarding aspects of several quite different random planar metrics and related topics is presented. 1. ..."
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Abstract. A discussion regarding aspects of several quite different random planar metrics and related topics is presented. 1.
Percolation beyond Z^d . . .
, 2010
"... Oded Schramm (1961–2008) influenced greatly the development of percolation theory beyond the usual Z d setting, in particular the case of nonamenable lattices. Here we review some of his work in this field. ..."
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Oded Schramm (1961–2008) influenced greatly the development of percolation theory beyond the usual Z d setting, in particular the case of nonamenable lattices. Here we review some of his work in this field.
ASYMPTOTICS OF THE VISIBILITY FUNCTION IN THE BOOLEAN MODEL UMPA, By Pierre Calka ∗ MAP 5, Université Paris Descartes and
, 2011
"... Theaimofthispaperistogiveapreciseestimate onthetail probability of the visibility function in a germgrain model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle in a Boolean model. We proceed in two ormore dimensions usingcoverage ..."
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Theaimofthispaperistogiveapreciseestimate onthetail probability of the visibility function in a germgrain model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle in a Boolean model. We proceed in two ormore dimensions usingcoverage techniques.Moreover, convergence results involving a type I extreme value distribution are shown in the two particular cases of small obstacles or a large obstaclefree region. 1. Presentation of the model and results. In [19] G. Pólya introduced the question of the visibility in a forest in a discrete lattice case as well as in a random case. He first treated the problem of a person standing at the origin of the regular square lattice of R 2, when identical trees (discs with constant radius R) are situated at the other points of thelattice. In this framework he showed that in order to see at a distance r the radius R should be (asymptotically when r is large) taken as 1/r. More recently V. Janković gave in [12] an elegant proof of a detailed version of this result. The random case studied by G. Pólya was the one of the visibility in one direction: we are here interested in the global solution to this problem considering all directions simultaneously. The spherical contact distribution which can be seen as the infimum of the visibility over all directions has been intensively used