Organisers
Giuseppe CannizzaroLink opens in a new window and Nikolaos ZygourasLink opens in a new window
The probability seminar is held on Wednesdays 16-17 in B3.02.
Seminars in Term 1.
Oct 2 - Jon Keating (University of Oxford)
Title: Joint Moments
Abstract: I will talk about the joint moments of the characteristic polynomials of random unitary matrices and their derivatives, and will discuss briefly in this context the joint moments of the Riemann zeta-function and its derivates.
Oct 9 - Anna Ben-Hamou (LPSM Sorbonne Université)
Title: Cutoff for permuted Markov chains
Abstract: For a given finite Markov chain with uniform stationary distribution, and a given permutation on the state-space, we consider the Markov chain which alternates between random jumps according to the initial chain, and deterministic jumps according to the permutation. In this framework, Chatterjee and Diaconis (2020) showed that when the permutation satisfies some expansion condition with respect to the chain, then the mixing time is logarithmic in the size of the state space, and that this expansion condition is satisfied by almost all permutations. We will see that the mixing time can even be characterised much more precisely: for almost all permutations, the permuted chain has cutoff, at a time which only depends on the entropic rate of the initial chain.
Oct 16 - Christina Goldschmidt (University of Oxford)
Title: Trees and snakes
Abstract: Consider the following branching random walk model: individuals reproduce according to a branching process which is started from a single individual who is located at the origin. The children of a vertex receive random displacements away from the spatial location of the parent. The displacements away from a particular vertex may be dependent, with a distribution depending on the number of children. However, the set of displacements of a vertex’s children are independent of the sets of displacements associated with the children of other vertices.
We are interested in the setting where the genealogical tree is conditioned to have precisely n vertices and the offspring distribution is critical and of finite variance, so that the tree converges on rescaling distances by n^{-1/2} to Aldous’ Brownian continuum random tree. We also assume that the spatial displacements satisfy natural centring and finite variance conditions, so that, along a particular lineage, we observe something like a centred finite variance random walk. We investigate conditions under which the whole object then converges in distribution (in a suitable sense) to a Brownian motion indexed by the Brownian continuum random tree. (In order to do this, we actually make use of a standard tool known as a discrete snake, and prove convergence on rescaling to Le Gall’s Brownian snake driven by a Brownian excursion.)
Our results improve on earlier theorems of various authors including Janson and Marckert for the case where the displacements are independent of the offspring numbers, and Marckert for the globally centred, global finite variance case restricted to bounded offspring distributions. Our proof of the convergence of the finite dimensional distributions makes essential use of a discrete line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin; the tightness proof adapts a method deployed by Haas and Miermont in the context of Markov branching trees.
This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.
Oct 23 - Joost Jooritsma (Oxford)
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Oct 30 - Jeffrey Kuan (Texas A&M University)
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Nov 6 - Jimmy He (Ohio)
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Nov 13 - Ajay Chandra (Imperial College)
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Nov 20 - Majdouline Borji (Max Planck Institute, Leipsig)
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Nov 27 - Giorgio Cipolloni (University of Arizona)
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Dec 4 - Tal Orenshtein (Università Milano-Bicocca)
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