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Archive : Probability



Matthew Junge, Duke University Matthew Junge, Duke University Ewing 336 Title: The bullet problem with discrete speeds Abstract: A bullet is fired along the real line each second with independent uniformly random speeds from [0,1]. When two bullets collide they mutually annihilate. The still open bullet problem asks if the first bullet ever survives. We establish survival in the variant where speeds are discrete. Joint with Brittany Dygert, Christoph Kinzel, Annie Raymond, Erik Slivken, and Jennifer Zhu. 4/17/2017 5:30:00 PM4/17/2017 6:30:00 PMFalse
Subhro Ghosh, Princeton UniversitySubhro Ghosh, Princeton UniversityEwing 336Title: Large holes in particle systems : forbidden regions, large deviations and potential theory Abstract: In particles systems, a ``hole'' of size R is defined to be a ball of radius R that is devoid of particles. The study of how the probability of having such a hoe decays to 0 (as R -> infty) is an important and well-studied question in particle systems. In this talk, we ask what causes a large hole to appear ? In other words, conditioned on having a large hole, how does the configuration of particles outside the hole look like ? Surprisingly, very little is understood about this question, except in the very special case of Gaussian random matrices, where there is an accumulation of particles at the edge of the hole, and equilibrium intensity beyond. We study this question in the context of zeros of Gaussian random polynomials, and provide a complete description of the intensity profile of the outside particles. A remarkable feature that we find is the appearance of a curious ``forbidden region'' between the accumulation at the edge of the hole and the equilibrium intensity far beyond. This is in stark contrast to the case of Gaussian random matrices, and seems to be novel even in the wider setting of statistical physics models. Our methods connect to large deviation principles for random polynomials, potential theory and constrained optimization on measure spaces. These ideas can also be applied to other problems, including Jancovici-Lebowitz-Manificat laws for Coulomb systems at general temperatures, and understanding over and under-crowding phenomena for Gaussian zeros. Based on joint works with Alon Nishry. 4/10/2017 5:30:00 PM4/10/2017 6:30:00 PMFalse
Megan Bernstein, Georgia TechMegan Bernstein, Georgia TechEWG 336Title: Cutoff for the random to random shuffle Abstract: (joint work with Evita Nestoridi) The random to random shuffle on a deck of cards is given by at each step choosing a random card from the deck, removing it, and replacing it in a random location. We show an upper bound for the mixing time of the walk of 3/4n log(n) +cn steps. Together with matching lower bound of Subag (2013), this shows the walk mixes with cutoff at 3/4n log(n) steps, answering a conjecture of Diaconis. We use the diagonalization of the walk by Dieker and Saliola (2015), which relates the eigenvalues to Young tableaux. 3/20/2017 5:30:00 PM3/20/2017 6:30:00 PMFalse
Harry Crane, Rutgers UniversityHarry Crane, Rutgers UniversityEWG 336Title: Some aspects of random combinatorial structures Abstract: I will present some topics on the general theme of combinatorial stochastic processes. The discussion will be framed around observations that emerge from the study of Markov processes on arbitrary combinatorial spaces, of which partition- and graph-valued processes are special cases. I draw some connections to prior work on coalescent processes and graph limits and also highlight some potential open questions in this area. 3/13/2017 5:30:00 PM3/13/2017 6:30:00 PMFalse

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