The seminar is organized by the Mathematics and Statistics departments and covers topics in pure and applied probability.

Organizers: Guillaume Barraquand, Ivan Corwin, Julien Dubedat, Ioannis Karatzas, Jeffrey Kuan, Marcel Nutz, Philip Protter, Hao Shen, Yi Sun, and Li-Cheng Tsai.

Meeting Time: Wednesday 5:30-6:30 pm and Friday 12:00-1:00 pm

Location: Math 520 (Directions to the Mathematics Department)

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Contact: probability_seminar@math.columbia.edu

Date and Time | Speaker | Title |
---|---|---|

Friday, February 3 12-1 pm |
Allan Sly (Princeton) | First passage percolation on rotationally invariant fieldsI will discuss new results on continuum models of first passage percolation which are rotationally invariant. For such models we give a multi-scale argument showing that the variance grows as O(n^{1-\epsilon}). Joint work with Riddhipratim Basu and Vladas Sidoravicius |

Friday, February 10 12-1 pm |
Firas Rassoul-Agha (Utah) | KPZ wandering exponent for random walk in i.i.d. dynamic Beta random environmentWe condition random walk in an i.i.d. dynamic Beta random environment to escape at an atypical velocity. The conditioned process converges to another random walk in random environment (RWRE). The new environment is a Doob transform of the original one by a harmonic function that is a Busemann type limit and solves a variational formula for the quenched large deviation rate function. Along the way we construct the stationary Beta polymer and prove fluctuation bounds for it. The Doob conditioned RWRE is in duality with this polymer and as a result it has a KPZ wandering exponent of 2/3. |

Wednesday, February 15 5:30-6:30 pm |
Praveen Kolli (CMU) | Fluctuations of diffusions interacting through the ranksWe study the fluctuations of a system of diffusions interacting through the ranks when the number of diffusions goes to infinity. It is known that the empirical cumulative distribution function of such diffusions converges to a non-random limiting cumulative distribution function which satisfies the porous medium PDE. We show that the fluctuations of the empirical cumulative distribution function around its limit are governed by a suitable SPDE. (Joint work with Misha Shkolnikov) |

Friday, February 17 9am-1pm 413 Kent |
Columbia University Symposium on Probability & Society | |

Wednesday, February 22 5:30-6:30 pm |
Tobias Johnson (NYU) | Galton-Watson fixed points, tree automata, and interpretationsConsider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman. |

Friday, March 3 12-1 pm |
Boris Hanin (MIT) | Airy scaling of random Hermite functions at the causticRandom Hermite functions are eigenfunctions of fixed energy for the isotropic harmonic oscillator in R^d. They therefore transition from highly oscillatory to exponentially damped at the caustic, the analog of the edge of the spectrum in random matrix theory. The purpose of this talk is to explain how the Airy kernel and its relatives appear in the scaling limit of random Hermite functions around the caustic as Planck's constant h goes to 0. Rather than being the kernel of the determinantal process, the Airy kernel will appear as the covariance function of a limit Gaussian field. This is joint work with Steve Zelditch and Peng Zhou. |

Wednesday, March 8 5:30-6:30 pm |
Lisa Hartung (NYU) | The Structure of Extreme Level Sets in Branching Brownian MotionBranching Brownian motion (BBM) is a classical process in probability, describing a population of particles performing independent Brownian motion and branching according to a Galton Watson process. Arguin et al. and Aidekon et al. proved the convergence of the extremal process. In the talk we discuss how one can obtain finer results on the extremal level sets by using a random walk-like representation of the extremal particles. We establish among others the asymptotic density of extremal particles at a given distance from the maximum and the upper tail probabilities for the distance between the maximum and the second maximum (joint work with Aser Cortines and Oren Louidor). |

Thursday, March 231:10-2:10 pm 303 MuddSpecial time/location |
Ivan Corwin (Columbia) | Beyond the Gaussian Universality Class (Applied Probability and Risk Seminar)The Gaussian distribution describes many basic systems arising across mathematics, science and society. However, many complex random systems related to interface growth, big data, stochastic optimization, traffic / queuing flow, and stochastic PDEs do not follow fall into this universality class. The talk will explain how these and other important types of real world systems fall into a different universality class -- the so called Kardar-Parisi-Zhang class. This talk will be aimed at a very general audience and will feature almost no equations and lots of videos and examples. |

Friday, March 24 12-1 pm |
Haya Kaspi (Technion) | An infinite-Dimensional Skorohod Map and Continuous Parameter PrioritiesThe Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this lecture I’ll introduce a measure-valued analog of this map that transforms each element of a certain class of càdlàg paths that take values in the space of signed measures on the positive half line to a càdlàg path that takes values in the space of non-negative measures on that space. This is done in a way that for each point \(x > 0\), and a signed measure valued process \((m_t)\) the path \(t \mapsto m_t[0, x]\) is transformed via the classical Skorokhod map on the half-line, and the regulating functions for different \(x > 0\) are coupled. We show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well known models are the EDF -- earliest-deadline-first, the SJF -- shortest-job-first and the SRPT -- shortest-remaining-processing-time scheduling policies. Concentrating in this talk on the EDF, I’ll show how the map provides a framework within which one forms fluid model equations, proves uniqueness of solutions to these equations and establishs convergence of scaled state processes to the fluid model. In particular, for this model, our approach leads to new convergence results in time-inhomogeneous settings, which appear to fall outside the scope of existing approaches. Joint work with Rami Atar, Anup Biswas and Kavita Ramanan. |

Friday, March 31 | Columbia-Princeton Probability Day | |

Friday, April 7 12-1 pm |
Vadim Gorin (MIT) | Height fluctuations through Schur generating functionsFor a large number of stochastic systems of 2d statistical mechanics and asymptotic representation theory the smoothed global fluctuations remain finite as the size of the system grows. Their asymptotic is governed by (typically log-correlated) Gaussian fields. I will present a new approach for studying such fluctuations through Schur generating functions of the underlying measures. The approach produces in a unified way the asymptotic theorems for many systems, including random domino and lozenge tilings, non-intersecting random walks, decompositions of tensor products, quantum random walks on irreducible representations. |

Wednesday, April 19 4:30-7:15 pm |
Columbia-Courant Probability Seminar Jean Bertoin (Zurich) 4:30-5:30 pm (colloquium) Zhou Fan (Stanford) 5:45-6:30 pm Cyril Labbé (Paris-Dauphine) 6:30-7:15 pm |
TBA |

Friday, April 21 | Applied Probability Day |