scholarly journals The expected number of critical percolation clusters intersecting a line segment

2016 ◽  
Vol 21 (0) ◽  
Author(s):  
J. van den Berg ◽  
R.P. Conijn
Keyword(s):  
Line Segment ◽  
Critical Percolation ◽  
Expected Number ◽  
Percolation Clusters

scholarly journals Loewner driving functions for off-critical percolation clusters

2009 ◽  
Vol 80 (5) ◽  
Author(s):  
Yoichiro Kondo ◽  
Namiko Mitarai ◽  
Hiizu Nakanishi
Keyword(s):  
Critical Percolation ◽  
Percolation Clusters

scholarly journals Rényi’s parking problem revisited

2016 ◽  
Vol 16 (02) ◽  
pp. 1660006 ◽  
Author(s):  
Matthew P. Clay ◽  
Nándor J. Simányi
Keyword(s):  
Random Process ◽  
Recent Work ◽  
Computer Simulations ◽  
Line Segment ◽  
Exclusion Process ◽  
Packing Problem ◽  
Expected Number ◽  
Parking Problem ◽  
Long Line ◽  
Recursion Formulas

Rényi’s parking problem (or 1D sequential interval packing problem) dates back to 1958, when Rényi studied the following random process: Consider an interval [Formula: see text] of length [Formula: see text], and sequentially and randomly pack disjoint unit intervals in [Formula: see text] until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of [Formula: see text] is [Formula: see text], so that the ratio [Formula: see text] is the expected filling density of the random process. Following recent work by Gargano et al. [4], we studied the discretized version of the above process by considering the packing of the 1D discrete lattice interval [Formula: see text] with disjoint blocks of [Formula: see text] integers but, as opposed to the mentioned [4] result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of [Formula: see text]-gaps ([Formula: see text]) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as [Formula: see text]) is Rényi’s famous parking constant, [Formula: see text]

Traveling time and traveling length in critical percolation clusters

1999 ◽  
Vol 60 (3) ◽  
pp. 3425-3428 ◽  
Author(s):  
Youngki Lee ◽  
José S. Andrade ◽  
Sergey V. Buldyrev ◽  
Nikolay V. Dokholyan ◽  
Shlomo Havlin ◽  
...  
Keyword(s):  
Critical Percolation ◽  
Percolation Clusters
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