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2008

On percolation and the bunkbed conjecture

Linusson, Svante

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We study a problem on edge percolation on product graphs $G\times K_2$.

Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge.
Every edge in $G\times K_2$ is present with probability $p$ independent of other edges.
The Bunkbed conjecture states that for all $G$ and $p$ the probability that $(u,0)$ is in the same component as $(v,0)$ is greater than or equal to the probability that $(u,0)$ is in the same component as $(v,1)$ for every pair of vertices $u,v\in G$.

On percolation and the bunkbed conjecture · Around

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Theodore E. Harris, A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc
1960
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Colin McDiarmid, General Percolation and random graphs, Adv. Appl. Prob
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Richard M. Karp, The transitive closure of a random digraph, Random Structures and Algorithms
1990
Earlier work this paper cites.
Béla Bollobás and Graham Brightwell, Random walks and electrical resistances in products of graphs, Disc. Applied Math
1997
Earlier work this paper cites.
Olle Häggström, On a conjecture of Bollobás and Brightwell concerning random walks on product graphs, Combinatorics, Prob. and Computing
1998
Earlier work this paper cites.

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Then

Jacob van den Berg, Olle Häggström and Jeff Kahn, Some conditional correlation inequalities for percolation and related prcesses, Rand. Structures Algorithms
2006
Later among the works it cites.
Sven Erick Alm and Svante Linusson, A counter-intuitive correlation in a random tournament, Preprint 2009
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Sven Erick Alm and Svante Linusson, Correlations for paths in random orientations of G ⁡ ( n , p ) G(n,p) , Preprint 2009
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Svante Linusson, A note on correlations in randomly oriented graphs, Preprint 2009
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