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                * Princeton Discrete Math Seminar *
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Speaker: Colin Defant, Harvard

Thursday 20th February, 3:00 in Fine Hall 224.

Title: Random subwords, pipe dreams, and billiards

Fix a probability p in (0,1). Let s_i denote the transposition in the symmetric
group S_n that swaps i and i+1. Let w be a finite word over the alphabet
{s_1,...,s_{n-1}}. Let sub_p(w) be the random subword of w obtained by deleting
each letter independently with probability 1-p. Let v_p(w) be the permutation
represented by sub_p(w). For a large class of words w, we study the distribution
of v_p(w) and obtain an asymptotic formula for its expected number of
inversions. In one special case, this allows us to resolve a conjecture of
Morales, Panova, Petrov, and Yeliussizov about random pipe dreams. We then
discuss a natural generalization of this setup in which the symmetric group is
replaced by an arbitrary Coxeter group W, focusing primarily on the case where W
is an affine Weyl group. In this case, for various choices of w, we can view
v_p(w) as the location after a certain amount of time of a random billiard
trajectory that, upon hitting a hyperplane in the Coxeter arrangement of W,
reflects off of the hyperplane with probability 1-p. We prove a central limit
theorem for the distribution of v_p(w) and obtain a remarkably simple formula
for the associated covariance matrix. This yields a precise asymptotic formula
for the expected Coxeter length of v_p(w). The part of this talk focused on
affine Weyl groups is based on joint work with Pakawut Jiradilok and Elchanan
Mossel. 

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