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Richard Kenyon : Random maps from Z2 to Z

One of the most basic objects in probability theory is the simple random walk, which one can think of as a random map from Z to Z mapping adjacent points to adjacent points. A similar theory for random maps from Z2 to Z had until recently remained elusive to mathematicians, despite being known (non-rigorously) to physicists. In this talk we discuss some natural families of random maps from Z2 to Z. We can explicitly compute both the local and the large-scale behavior of these maps. In particular we construct a "scaling limit" for these maps, in a similar sense in which Brownian motion is a scaling limit for the simple random walk. The results are in accord with physics.

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