Combinatorial Theory Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
Today
14:00
Louigi Addario-Berry

Further Information: 

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

I will describe how certain recursive distributional equations can be solved by using tools from numerical analysis on the convergence of approximation schemes for PDEs. This project is joint work with Luc Devroye, Hannah Cairns, Celine Kerriou, and Rivka Maclaine Mitchell.

  • Combinatorial Theory Seminar
14 April 2020
14:00
Ron Peled

Further Information: 

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

Color each vertex of an infinite graph blue with probability $p$ and red with probability $1-p$, independently among vertices. For which values of $p$ is there an infinite connected component of blue vertices? The talk will focus on this classical percolation problem for the class of planar graphs. Recently, Itai Benjamini made several conjectures in this context, relating the percolation problem to the behavior of simple random walk on the graph. We will explain how partial answers to Benjamini's conjectures may be obtained using the theory of circle packings. Among the results is the fact that the critical percolation probability admits a universal lower bound for the class of recurrent plane triangulations. No previous knowledge on percolation or circle packings will be assumed.

  • Combinatorial Theory Seminar
21 April 2020
14:00
Agelos Georgakopoulos

Further Information: 

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

We prove that for Bernoulli bond percolation on $\mathbb{Z}^d$, $d\geq2$, the percolation density $\theta(p)$ (defined as the probability of the origin lying in an infinite cluster) is an analytic function of the parameter in the supercritical interval $(p_c,1]$. This answers a question of Kesten from 1981.

The proof involves a little bit of elementary complex analysis (Weierstrass M-test), a few well-known results from percolation theory (Aizenman-Barsky/Menshikov theorem), but above all combinatorial ideas. We used a new notion of contours, bounds on the number of partitions of an integer, and the inclusion-exclusion principle, to obtain a refinement of a classical argument of Peierls that settled the 2-dimensional case in 2018. More recently, we coupled these techniques with a renormalisation argument to handle all dimensions.

Joint work with Christoforos Panagiotis.

  • Combinatorial Theory Seminar
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