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Rigidity and tolerance for perturbed lattices


Speaker: Yuval Peres
Location: 1147 MSB
Start time: Wed, Nov 6 2019, 3:10PM

Consider a perturbed lattice {v+Y_v: v in Z^d} obtained by adding Gaussian variables Y_v to the lattice points in Z^d. Suppose that one point, say Y_0, is removed from this perturbed lattice; Is it possible for an observer, who sees just the remaining points, to detect that a point is missing? In one and two dimensions, the answer is positive: the two point processes (before and after Y_0 is removed) can be distinguished using smooth statistics, (cf. Holroyd and Soo (2011)). The situation in higher dimensions is more delicate; our solution depends on a game-theoretic idea, in one direction, and on the unpredictable paths constructed by Benjamini, Pemantle and the speaker (1998), in the other. I will also describe a conjectured Kosterlitz-Thouless type transition in 2D. (Joint work with Allan Sly).

Rescheduled from the previous week