Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 3 / a

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- Solution a

Solution

The van den Berg-Kesten inequality says that for increasing cylinder events

A

and

B

under product bond percolation,

P_{p} (A □ B) \leq P_{p} (A) P_{p} (B),

where

A □ B

is the event that

A

and

B

have disjoint finite witness edge sets.

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