## A correlation inequality for connection events in percolation.(English)Zbl 1013.60079

Summary: It is well-known in percolation theory (and intuitively plausible) that two events of the form “there is an open path from $$s$$ to $$a$$” are positively correlated. We prove the (not intuitively obvious) fact that this is still true if we condition on an event of the form “there is no open path from $$s$$ to $$t$$”.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60C05 Combinatorial probability
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### References:

 [1] Ahlswede, R. and Daykin, D. E. (1978). An inequality for the weights of two families of sets, their unions and intersections.Wahrsch. Verw. Gebiet 43 183-185. · Zbl 0357.04011 [2] Bollobás, B. (1986). Combinatorics. Cambridge Univ. Press. · Zbl 0595.05001 [3] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89-103. · Zbl 0346.06011 [4] Grimmett, G. (1999). Percolation. Springer, New York. · Zbl 0926.60004 [5] Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13-20. · Zbl 0122.36403 [6] Sarkar, T. K. (1969). Some lower bounds of probability, Technical Report 124, Dept. Operations Research and Statistics, Stanford Univ.
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