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Local Limit of the Fixed Point Forest

Mathematical Physics & Probability

Speaker: Erik Slivken, UC Davis
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Location: 1147 MSB
Start time: Wed, Feb 15 2017, 4:10PM

Consider the following partial "sorting algorithm" on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size n, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this "fixed point forest" exhibits a rich structure. We consider the fixed point forest in the limit n -> \infty and show using Stein's method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We explore various statistics associated with this limiting tree.

This talk combines joint work with Anne Schilling and Tobias Johnson and joint work with Samuel Regan.