A recent theorem of Bissacot, et al. proved using results about the cluster expansion in statistical mechanics extends the LovÃ¡sz Local Lemma by weakening the conditions under which its conclusionâ€¦ (More)

We prove game-theoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Local Lemma which can dramaticallyâ€¦ (More)

Given an instance of the preferential attachment graph Gn = ([n], En), we would like to find vertex 1, using only â€˜localâ€™ information about the graph; that is, by exploring the neighborhoods of smallâ€¦ (More)

We prove that the set of quadratic growths attainable by integervalued superharmonic functions on the lattice Z2 has the structure of an Apollonian circle packing. This completely characterizes theâ€¦ (More)

We state a conjecture relating integer-valued superharmonic functions on Z2 to an Apollonian circle packing of R2. The conjecture is motivated by the Abelian sandpile process, which evolvesâ€¦ (More)

Proceedings of the National Academy of Sciencesâ€¦

2017

We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of theâ€¦ (More)

We design and analyze a protocol for dividing a state into districts, where parties take turns proposing a division, and freezing a district from the other partyâ€™s proposed division. We show that ourâ€¦ (More)

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for aâ€¦ (More)

Shearer gave a general theorem characterizing the family L of dependency graphs labeled with probabilities pv which have the property that for any family of events with a dependency graph from Lâ€¦ (More)