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Gravitational Allocation to Poisson Points

Mathematical Physics & Probability

Speaker: Ron Peled, UC Berkeley
Location: 2112 MSB
Start time: Wed, Mar 5 2008, 4:10PM

Given a translation invariant point process in R^d of intensity 1, an allocation rule is a translation-equivariant mapping that allocates to each point in the process a set in R^d of unit volume, such that the sets allocated to different points are disjoint and their union covers almost all of R^d. In other words, we partition R^d to sets of volume 1 and match them with the point process in a translation equivariant way. Allocation rules can give a better understanding of the underlying point process, they measure in some sense how uniformly the mass is spread over space. They can also be used for obtaining so called extra head rules. In this talk we will consider the standard Poisson point process in R^d, allocation rules for this process were constructed by Hoffman, Holroyd and Peres using the Gale-Shapley stable marriage algorithm. I will describe a new allocation rule in dimensions 3 and higher, inspired by recent work of Nazarov, Tsirelson, Sodin and Volberg, that is defined by flow along the integral curves of a gravitational force field induced by the Poisson points. The main result is that this allocation is 'efficient', in the sense that the diameter of the cell allocated to a given point is a random variable with exponentially decaying tails. This is the first deterministic allocation with this property. Time permitting, I will tell also of matching lower bounds for the tail of the diameter, and bounds for other parameters of the process. This is joint work with Sourav Chatterjee, Yuval Peres and Dan Romik.