Probability laboratory at Bath

Seminar abstracts

Our seminars are usually held on Mondays at 12.15pm in room 4W 1.7. If you wish to find out more, please contact Alexandre Stauffer or one of the other members of the group.

2nd October 2017 - Jean-Francois Marckert (Université de Bordeaux)

Some news concerning coalescing processes
Consider at time 0, a family of n particles with mass 1. These masses coalesce according to the following rule: each pair of masses (m,m') coalesces when a random clock following an exponential distribution with parameter K(m,m') rings, where the kernel K is the main parameter of the model. When coalescence occurs the pair of mass (m,m') is replaced by a single mass m+m', and the clocks are updated.
When the kernel is additive (K(m,m')=m+m') or multiplicative (K(m,m')=mm') these processes possesses some combinatorial representations which allows one to prove scaling limit for the evolution of mass distribution when n goes to +infinity (Aldous, Pitman, Chassaing, Louchard, Bertoin...)
The aim of this talk is to present "old" combinatorial encodings of these processes, as well as new ones. We will see that minor modifications of the encodings allows to get much richer combinatorial environments, allowing to get new asymptotic results.
This is based on joint works with Nicolas Broutin and Minmin Wang.

9th October 2017 - Thierry Bodineau (École Polytechnique)

Large time asymptotics of a hard sphere gas close to equilibrium
We consider a tagged particle in a diluted gas of hard spheres. Starting from the hamiltonian dynamics of particles in the Boltzmann-Grad limit, we will show that the tagged particle follows a Brownian motion after an appropriate rescaling. We also discuss the convergence of the motion of a big tagged particle towards a Langevin process.

16th October 2017 - Ioan Manolescu (Université de Fribourg)

First order phase transition for the Random Cluster model with q > 4
This talk aims to prove that the phrase transition of the planar random cluster model (and that of the associated Potts model) is discontinuous when q > 4. The result is obtained by computing rigorously the correlation length of the critical RCM using a correspondence with the six vertex model. The latter may be expressed using the transfer matrix formalism; the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix may then be computed using the Bethe ansatz.

23rd October 2017 - Denis Villemonais (École des Mines de Nancy)

Non-failable approximation method for conditioned distributions
We present a new approximation method for conditional distributions, which is non-failable and, under mild conditions, which converges uniformly in time. Our results are illustrated by their application to a neutron transport model. This is a joint work with William Oçafrain.

30th October 2017 - Timothy Budd (Université Paris-Saclay)

Nesting of loops versus winding of walks
Random planar maps with a boundary coupled to an O(n) loop model are expected, upon uniformization, to scale towards Liouville Quantum Gravity (LQG) in the unit disk together with an independent Conformal Loop Ensemble (CLE). This expectation was corroborated by work of Borot, Bouttier & Duplantier in which they found that the asymptotic nesting statistics of loops on the planar map side was related through a Knizhnik-Polyakov-Zamolodchikov (KPZ) relation to the equivalent for CLE obtained by Miller, Watson & Wilson. In this talk I will show that the nesting on the planar map side is closely related, even at the combinatorial level, to another random process in the plane: the winding of two-dimensional random walks around the origin. Inspection of the scaling limit of the latter in terms of Brownian motion sheds some light on the appearance of the KPZ relation.

6th November 2017 - Henning Sulzbach (University of Birmingham)

Galton-Watson trees and Apollonian networks
In the analysis of critical Galton-Watson trees conditional on their sizes, two complementary approaches have proved fruitful over the last decades. First, the so-called size-biased tree allows the study of local properties such as node degrees. Second, the global scaling limit, the Continuum Random tree, provides access to the study of average or extreme node depths. In this talk, I will discuss the approximation of the tree by subtrees with bounded node degrees. Here, the so-called "heavy subtrees" play an important role. An application will be given in terms of uniform Apollonian networks. The talk is based on joint work with Luc Devroye (McGill, Montreal) and Cecilia Holmgren (Uppsala).

9th November 2017 - Wioletta Ruszel (Technical University Delft)

Odometers and bi-Laplacian fields - Note: this seminar is on a Thursday in 6W 1.1
The divisible sandpile model is a special case of the class of continuous fixed energy sandpile models on some lattice or graph where the initial configuration is random and the evolution deterministic. One question which arises is under which conditions the model will stabilize or not. The amount of mass u(x) emitted from a certain vertex x during stabilization is called the odometer function. In this talk we will construct the scaling limit of the odometer function of a divisible sandpile model on a torus and show that it it converges to a continuum bi-Laplacian field.
This is joint work with A. Cipriani (U Bath) and R. Hazra (ISI Kolkatta).

13th November 2017 - Fredrik Viklund (KTH, Sweden)

Loop-erased walk and natural parametrization
Loop-erased random walk (LERW) is the random self-avoiding walk one gets after erasing the loops in the order they form from a simple random walk. Lawler, Schramm and Werner proved that LERW in 2D converges in the scaling limit to the Schramm-Loewner evolution with parameter 2 (SLE(2)), as curves viewed up to reparametrization. It is however more natural to view the discrete curve as parametrized by (renormalized) length and it has been believed for some time that one then has convergence to SLE(2) equipped with the so-called natural parametrization, which in this case is the same as 5/4-dimensional Minkowski content. I will discuss recent joint works with Greg Lawler (Chicago) and Christian Benes (CUNY) and Greg Lawler that prove this stronger convergence, focusing on explaining the main ingredients and ideas of the argument.

27th November 2017 - Giuseppe Cannizzaro (Imperial College London)

The KPZ equation: universality and discretizations
The KPZ class is the given of a family of stochastically growing interfaces in 1+1 dimensions whose fluctuations under the 1:2:3 scaling are presumed to be universal. The KPZ equation is nothing but one instance of this class, playing though a distinguished role that turns it into a universal object itself.
In this talk, we will explain how this equation was originally derived, its connection with the KPZ class and clarify in what sense it is universal. In support of this latter claim, we will consider a family of space-time discrete systems and show that, when suitably rescaled, they converge to its solution. The talk is based on a joint project with K. Matetski.

4th December 2017 - Pietro Siorpaes (Imperial College London)

Structure of martingale transports in finite dimensions
Martingale optimal transport, a variant of the classical optimal transport problem where a martingale constraint is imposed on the coupling, was born to study model-independent pricing. In a recent paper, Beiglböck, Nutz and Touzi show that in dimension one there is no duality gap and that the dual problem admits an optimizer. A key step towards this achievement is the characterization of the polar sets of the family of all martingale couplings. Here we aim to extend this characterization to arbitrary finite dimension through a deeper study of the convex order.

11th December 2017 - Eleanora Kreacic (University of Oxford)

The spread of fire on a random multigraph
We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge-lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability 1/2. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having n vertices of degree 3 and alpha(n) vertices of degree 4, where alpha(n)=o(n), with i.i.d. standard exponential edge-lengths. Depending on whether alpha(n) >> \sqrt(n) or alpha(n)=O(\sqrt(n)), we prove that as n tends to infinity these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. This is joint work with Christina Goldschmidt.

15th January 2018 - Stefan Junk (TU Munich)

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22nd January 2018 - Erich Baur (Bern University of Applied Sciences)

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5th February 2018 - Nicos Georgiou (University of Sussex)

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12th February 2018 - Anja Sturm (University of Goettingen)

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19th February 2018 - David Promel (University of Oxford)

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26th February 2018 - Julien Berestycki (University of Oxford)

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5th March 2018 - Costanza Benassi (University of Warwick)

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12th March 2018 - Sabine Jansen (University of Munich)

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19th March 2018 - Tyler Helmuth (University of Bristol)

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9th April 2018 - Jonathan Hermon (University of Cambridge)

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30th April 2018 - Jason Miller (University of Cambridge)

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14th May 2018 - Marielle Simon (Inria Lille)

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21st May 2018 - Alison Etheridge (University of Oxford)

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Past seminars

2016/17: Winter and Spring
2015/16: Winter and Spring
2014/15: Winter and Spring
2013/14: Winter and Spring
2012/13: Winter and Spring
2011/12: Winter and Spring
2010/11: Winter, Spring
2009/10: Winter, Spring
2008/9: Winter, Spring
2007/8: Winter, Spring
2006/7: Winter, Spring