Journal Issue10.1002/RSA.V26:3
Percolation in Voronoi tilings
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TL;DR: It is proved that for large d, 2-d(9d log d)-1 ≤ pc(d) ≤ C2-d$\sqrt{d}$ log d, and the existence of a phase transition as the proportion p of open cells is varied is proved.
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Abstract: We consider a percolation process on a random tiling of Rd into Voronoi cells based on points of a realization of a Poisson process. We prove the existence of a phase transition as the proportion p of open cells is varied and provide estimates for the critical probability pc. Specifically, we prove that for large d, 2-d(9d log d)-1 ≤ pc(d) ≤ C2-d$\sqrt{d}$ log d. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005
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Citations
Introduction to random graphs
Alan Frieze,Michał Karoński +1 more
- 01 Jan 2016
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
A combinatorial characterization of the testable graph properties: it's all about regularity
Noga Alon,Eldar Fischer,Ilan Newman,Asaf Shapira +3 more
- 21 May 2006
TL;DR: One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.
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Algorithmic and Analysis Techniques in Property Testing
Dana Ron
- 27 Jan 2010
TL;DR: This monograph surveys results in property testing, where the emphasis is on common analysis and algorithmic techniques.
A Combinatorial Characterization of the Testable Graph Properties: It's All About Regularity
TL;DR: This paper shows that in some sense, testing for Szemeredi-partitions is as hard as testing any testable graph property, and gives an intuitive explanation as to what makes a graph property testable.
A characterization of the (natural) graph properties testable with one-sided error
Noga Alon,Asaf Shapira +1 more
- 23 Oct 2005
TL;DR: It is shown that a graph property P has an oblivious one-sided error tester, if and only if P is (semi) hereditary, and infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one- sided error.
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Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions
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Itai Benjamini,Oded Schramm +1 more
TL;DR: In this article, the authors studied percolation in the hyperbolic plane and in transitive planar graphs that are quasi-isometric to the Hyperbolic Plane.
Regular Languages are Testable with a Constant Number of Queries
TL;DR: This paper discusses the testability of more complex languages and shows, in particular, that the query complexity required for testing context-free languages cannot be bounded by any function of $\epsilon$.
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Erdõs on Graphs : His Legacy of Unsolved Problems
Fan Chung,Ron Graham +1 more
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TL;DR: In this paper, the authors introduce the concept of extreme graph theory and graph enumeration for color, packing, and covering, and hypergraphs, which is an extension of Graph Enumeration.
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