The Lovász Local Lemma is Not About Probability

Given a collection of independent events each of which has strictly positive probability, the probability that all of them occur is also strictly positive. The Lov\'asz local lemma (LLL) asserts that this remains true if the events are not too strongly negatively correlated. The formulation of the lemma involves a graph with one vertex per event, with edges indicating potential negative dependence. The word "Local" in LLL reflects that the condition for the negative correlation can be expressed solely in terms of the neighborhood of each vertex. In contrast to this local view, Shearer developed an exact criterion for the avoidance probability to be strictly positive, but it involves summing over all independent sets of the graph. In this work we make two contributions. The first is to develop a hierarchy of increasingly powerful, increasingly non-local lemmata for bounding the avoidance probability from below, each lemma associated with a different set of walks in the graph. Already, at its second level, our hierarchy is stronger than all known local lemmata. To demonstrate its power we prove new bounds for the negative-fugacity singularity of the hard-core model on several lattices, a central problem in statistical physics. Our second contribution is to prove that Shearer's connection between the probabilistic setting and the independent set polynomial holds for \emph{arbitrary supermodular} functions, not just probability measures. This means that all LLL machinery can be employed to bound from below an arbitrary supermodular function, based only on information regarding its value at singleton sets and partial information regarding their interactions. We show that this readily implies both the quantum LLL of Ambainis, Kempe, and Sattath~[JACM 2012], and the quantum Shearer criterion of Sattath, Morampudi, Laumann, and Moessner~[PNAS 2016].