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Large random pattern-avoiding permutations

A pattern in a permutation is a subsequence with a specific relative order. What can we say
about a typical large random permutation that avoids a particular pattern? We use a variety of
approaches. For certain classes we give a geometric description that relates these classes to other
types of well-studied objects like random walks or random trees. Using the right geometric
description we can find the the distribution of certain statistics like the number and location of
fixed points. This is based on joint work with Christopher Hoffman and Douglas Rizzolo.