Proof of the Van den Berg–Kesten Conjecture

DAVID REIMER

doi:10.1017/S0963548399004113

Abstract

We prove the following conjecture of J. van den Berg and H. Kesten. For any events [Ascr ] and [Bscr ] in a product probability space, Prob([Ascr ]□[Bscr ]) [les ] Prob([Ascr ])Prob([Bscr ]), where [Ascr ]□[Bscr ] is the event that [Ascr ] and [Bscr ] occur ‘disjointly’.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Kahn, J. Saks, M. and Smyth, C. 2000. A dual version of Reimer's inequality and a proof of Rudich's conjecture. p. 98.

Gács, Peter 2001. Compatible sequences and a slow Winkler percolation. p. 694.

Smyth, Clifford 2002. Reimer's inequality and tardos' conjecture. p. 218.

van den Berg, J. and Kesten, Harry 2002. In and Out of Equilibrium. p. 1.

Csáki, Endre Révész, Pál and Shi, Zhan 2004. Large void zones and occupation times for coalescing random walks. Stochastic Processes and their Applications, Vol. 111, Issue. 1, p. 97.

Chayes, L. and Lei, H. K. 2006. Random Cluster Models on the Triangular Lattice. Journal of Statistical Physics, Vol. 122, Issue. 4, p. 647.

Balogh, József and Bollobás, Béla 2006. Bootstrap percolation on the hypercube. Probability Theory and Related Fields, Vol. 134, Issue. 4, p. 624.

HÄGGSTRÖM, OLLE 2007. Problem Solving is Often a Matter of Cooking Up an Appropriate Markov Chain*. Scandinavian Journal of Statistics, Vol. 34, Issue. 4, p. 768.

Goldstein, Larry and Rinott, Yosef 2007. Functional BKR Inequalities, and their Duals, with Applications. Journal of Theoretical Probability, Vol. 20, Issue. 2, p. 275.

Nolin, Pierre 2008. Near-critical percolation in two dimensions. Electronic Journal of Probability, Vol. 13, Issue. none,

Balogh, József Bollobás, Béla and Morris, Robert 2009. Bootstrap percolation in three dimensions. The Annals of Probability, Vol. 37, Issue. 4,

RIORDAN, OLIVER and WORMALD, NICHOLAS 2010. The Diameter of Sparse Random Graphs. Combinatorics, Probability and Computing, Vol. 19, Issue. 5-6, p. 835.

Bollobás, Béla Borgs, Christian Chayes, Jennifer and Riordan, Oliver 2010. Percolation on dense graph sequences. The Annals of Probability, Vol. 38, Issue. 1,

Binder, I. Chayes, L. and Lei, H. K. 2010. On Convergence to SLE6 I: Conformal Invariance for Certain Models of the Bond-Triangular Type. Journal of Statistical Physics, Vol. 141, Issue. 2, p. 359.

Garban, Christophe Pete, Gábor and Schramm, Oded 2010. The Fourier spectrum of critical percolation. Acta Mathematica, Vol. 205, Issue. 1, p. 19.

Markström, Klas 2010. Closure Properties and Negatively Associated Measures violating the van den Berg-Kesten Inequality. Electronic Communications in Probability, Vol. 15, Issue. none,

Schramm, Oded and Steif, Jeffrey 2010. Quantitative noise sensitivity and exceptional times for percolation. Annals of Mathematics, Vol. 171, Issue. 2, p. 619.

Garban, Christophe Pete, Gábor and Schramm*, Oded 2011. Selected Works of Oded Schramm. p. 445.

Schramm, Oded and Steif, Jeffrey E. 2011. Selected Works of Oded Schramm. p. 391.

Damron, Michael and Sapozhnikov, Artëm 2011. Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probability Theory and Related Fields, Vol. 150, Issue. 1-2, p. 257.

Download full list

Article contents

Proof of the Van den Berg–Kesten Conjecture

Abstract

Access options

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

Proof of the Van den Berg–Kesten Conjecture

Abstract

Access options

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests