Probabilistic combinatorics and lattice gas estimates

The talk gives an application of lattice gas theory to probabilistic combinatorics, following a recent paper of Scott and Sokal. There is a large collection of bad events; the problem is to show that there is some non-zero probability that none of them occur. The result is that this probability is bounded below by the partition function of the dependency graph of the events. A subsequent talk will give Dobrushin’s explicit lower bound for this partition function. 1 The inclusion-exclusion principle The inclusion-exclusion principle says that if f(S) = ∑ T :S⊂T g(T ). (1) then g(T ) = ∑ S:T⊂S (−1)|S|−|T |f(S). (2) Here is a proof. Fix T and compute ∑ S:T⊂S (−1)|S|−|T |f(S) = ∑ S:T⊂S ∑ T ′:S⊂T ′ (−1)|S|−|T |g(T ′) = ∑ T ′:T⊂T ′ g(T ′) ∑ S:T⊂S⊂T ′ (−1)|S|−|T |. (3) However ∑ S:T⊂S⊂T ′ (−1)|S|−|T | = ∏ x∈T ′\T (1− 1) = δT T ′ . (4) So ∑ S:T⊂S (−1)|S|−|T |f(S) = ∑ T ′:T⊂T ′ g(T )δT T ′ = g(T ). (5)