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Interval fragmentations with choice

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Thursday, November 08, 2018
3:15 pm - 4:15 pm
Pascal Maillard (CRM (Montréal) and Université Paris-Sud)
Probability Seminar

Points fall into the unit interval according to a certain rule, splitting it up into fragments. An example rule is the following: at each step, two points are randomly drawn from the unit interval and the one that falls into the smaller (or larger) interval is discarded, while the other one is kept. This process is inspired by the so-called "power of choice" paradigm originating in the computer science literature on balanced load allocation models. The question of interest is how much the rule affects the geometry of the point cloud. With Elliot Paquette [1] we introduced a general version of this interval fragmentation model and showed that the empirical distribution of rescaled interval lengths converges almost surely to a deterministic probability measure. I will report on this work as well as on work in progress [2] where we show that the empirical measure of the points converges almost surely to the uniform distribution. The proofs involve techniques from stochastic approximation, non-linear integro-differential equations, ergodic theory for Markov processes and perturbations of semigroups on L^p spaces, amongst other things. [1] Maillard, P., & Paquette, E. (2016). Choices and intervals. Israel Journal of Mathematics, 212(1), 337–384. [2] Maillard, P., & Paquette, E. (in preparation). Interval fragmentations with choice: equidistribution and the evolution of tagged fragments