# 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 January 2018 - Eleanora Kreacic (University of Oxford)

**The spread of fire on a random multigraph** - Note: this seminar is on a Thursday

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)

**On a branching random walk in a disastrous environment**

Consider a branching random walk on the integer lattice where each particle jumps to a random neighbor at rate kappa, and splits into two descendants at rate lambda. We add to this model a degeneracy, where for every site all occupants are simultaneously killed at rate 1. For this model we give a characterization for the parameters kappa and lambda where (quenched or annealed) survival has positive probability. Using this characterization we then discuss how the survival probability behaves as a function of the parameters. Joint work with Nina Gantert.

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

**Planar quadrangulations with a boundary and their limiting behavior**

We discuss distributional limits of large uniform random planar quadrangulations with a boundary. Depending on the asymptotic behaviour of the boundary length and of the scaling, we observe different limiting metric spaces, among them the Brownian half-plane with skewness, the infinite-volume Brownian disk and the infinite continuum random tree. We show how they are related to each other and discuss some of their properties.

Based on joint works with Grégory Miermont, Gourab Ray, and Loïc Richier.

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

**On classifying genealogies for general (diploid) exchangeable population models**

The genetic variation in a sample of individuals/genes depends on their relatedness which is described by their genealogy. In this talk we consider classifying the genealogies of exchangeable population models with fixed size N asymptotically as N tends to infinity.

In the first part of the talk we review classical results regarding the haploid Cannings model. Here, each individual is represented by one of its genes and thus each offspring (gene) has a unique parent. In each generation, the offspring vector to the N parents is exchangeable. With an appropriate rescaling the corresponding coalescence processes describing the genealogy converge to a limit process. Möhle and Sagitov (2001) classified all possible limit processes and showed that depending on the tail behavior of the offspring numbers the limit process is Kingman's coalescent with coalescence of pairs or is given by coalescents with (simultaneous) multiple mergers in which (several) larger groups may find a common ancestor at the same time.

In the second part of the talk we extend this result to diploid bi-parental analogues of the Cannings model. Here, the next generation is composed of offspring of parent pairs, which form an exchangeable (symmetric) array. Also, every individual carries two gene copies, each of which is inherited from one of its parents. Our result classifies the limiting coalescent processes describing the gene genealogies. Using this general result we determine the limiting coalescent in a number of examples of which some have been studied previously (in special cases) and some are new. We also point out connections to the theory of random graphs.

This talk is based on joint work with Matthias Birkner (Universität Mainz) and Huili Liu (Hebei Normal University).

### 15th February 2018 - Nicos Georgiou (University of Sussex)

**Order of fluctuations for the discrete Hammersley process** - Note: this seminar is on a Thursday

We discuss the order of the variance on a lattice analogue of the Hammersley process, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope.

For this model the shape function exhibits two flat edges and we study the order of the variance in directions that fall in the flat edge, in directions that approximate the edge of the flat edge, and in directions in the strictly concave section of the shape for the i.i.d. model and for the associated equilibrium model with boundaries. This is joint work with Janosch Ortmann and Federico Ciech.

### 19th February 2018 - David Prömel (University of Oxford)

**On Skorokhod Embeddings and Poisson Equations**

We discuss the Skorokhod embedding problem for a Levy process and a target probability measure, that is, the problem of finding a stopping time such that stopped Levy process is distributed as the given target probability measure. Assuming the probability measure possesses a positive density, we propose necessary and sufficient conditions for the existence of a solution to the Skorokhod embedding problem in terms of the Poisson equation associated to the adjoint operator of the Levy process. Furthermore, we give a fairly explicit construction of the stopping time. The talk is based on a joint work with Leif Döring, Lukas Gonon and Oleg Reichmann.

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

**Singularity analysis for heavy-tailed random variables** - This seminar will be held upstairs in the Bath Brew House, due to the UCU strike.

We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindel\"of integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws p(k)=cexp(−kα) and apply to logarithmic hazard functions cexp(−(logk)^\beta), \beta>2; they cover the big jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem. The talk is based on joint work with Nick Ercolani and Daniel Ueltschi.

### 16th April 2018 - Sean Ledger (University of Bristol)

**McKean—Vlasov problems with contagion effects**

I will introduce a McKean—Vlasov problem arising from a simple mean-field model of interacting neurons. The equation is nonlinear and captures the positive feedback effect of neuron spiking. This leads to a phase transition in the regularity of the solution: if the interaction is too strong then the system exhibits blow-up. We will cover the mathematical challenges in defining, constructing and proving uniqueness of solutions, as well as explaining the connection to PDEs, integral equations and mathematical finance.

### 23rd April 2018 - Dario Spanò (University of Warwick)

**Duality and fixation for Xi-Fleming-Viot models with selection in a random environment**

Xi-Fleming-Viot models are the most general class of measure-valued processes arising as scaling limit of a population genetics model with exchangeable offspring distribution (Cannings models). They are jump-diffusion "allele-frequency" processes characterised by a dual coalescent genealogy which may allow for simultaneous multiple mergers. In this talk we will introduce a generalisation of Cannings models allowing for several types of selection, thus implying a partially exchangeable offspring distribution. We shall focus on the behaviour of a two-type model, where the two types have different fitness strength, which fluctuates randomly with time. We will derive its scaling limit frequency process and the corresponding dual ancestral process, the latter resulting in a branching- coalescing chain with random environment and arbitrarily-sized jumps. We will focus on the fate of the selectively weak allele and, using duality, we will investigate in which cases the selective type goes to fixation with probability one.

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

**Convergence of percolation on uniform quadrangulations** - This seminar will be held in 6W 1.1.

Let Q be a uniformly random quadrangulation with simple boundary decorated by a critical (p=3/4) face percolation configuration. We prove that the chordal percolation exploration path on Q between two marked boundary edges converges in the scaling limit to SLE(6) on the Brownian disk (equivalently, a Liouville quantum gravity surface). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. Our method of proof is robust and, up to certain technical steps, extends to any percolation model on a random planar map which can be explored via peeling. (joint work with E. Gwynne)

### 14th May 2018 - Vitalii Konarovskyi (University of Leipzig)

**A particle model for Wasserstein type diffusion**

In the talk, a family of sticky reflecting particle system on the real line with strictly local interaction will be presented. The model is a physical improvement of the classical Arratia flow, but now particles can split up and they transfer a mass that influence their motion. The particle system can be also interpreted as an infinite dimensional version of sticky reflecting dynamics on a simplicial complex. In the talk, an infinite dimensional SDE with discontinuous coefficients which describes the particle model will be presented. We will discuss the existence of its weak solution, using a finite particle approximation. We will also consider a reversible case, where the construction is based on a new family of measures on the set of real increasing functions as reference measures for naturally associated Dirichlet forms. In this case, the intrinsic metric leads to a Varadhan formula for the short time asymptotics with the Wasserstein metric for the associated measure valued diffusion. Joint work with Max von Renesse.

### 17th May 2018 - Costanza Benassi (University of Warwick)

**Classical spin systems and random loop models** - Note: this seminar is on a Thursday

Random loop models appear in a great variety of situations in both the probability and mathematical physics literature. We focus on random loop representations for O(n) models in classical statistical mechanics. We review the Brydges-Fröhlich-Spencer representation and propose a generalised random current representation for this class of systems. We focus on these loop models and investigate some striking conjectures which have been recently suggested: macroscopic loops should appear and the distribution of their lengths is expected to be a member of the Poisson Dirichlet distribution family.

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

**Modelling evolution in a fluctuating environment**

Since the pioneering work of Fisher, Haldane and Wright at the beginning of the 20th Century, mathematics has played a central role in theoretical population genetics. In turn, population genetics has provided the motivation both for important classes of probabilistic models, such as coalescent processes, and for deterministic models, such as the celebrated Fisher-KPP equation. Whereas coalescent models capture `relatedness’ between genes, the Fisher KPP equation captures something of the interaction between natural selection and spatial structure. What has proven remarkably difficult is to combine the two, at least in the biologically relevant setting of a two-dimensional spatial continuum. Moreover, whereas the Fisher-KPP equation assumes that the selective advantage of a particular genetic type is constant in time and space, it has long been recognised that fitnesses of different genetic types will fluctuate in space and time, driven by temporal and spatial fluctuations in the environment. We briefly discuss some problems and (fewer) solutions related to modelling populations undergoing fluctuating selection.

### 24th May 2018 - Tyler Helmuth (University of Bristol)

**Recurrence of the vertex-reinforced jump process in two dimensions** - Note: this seminar is on a Thursday

The vertex-reinforced jump process (VRJP) is a linearly reinforced random walk in continuous time. Roughly speaking, the VRJP prefers to jump to vertices it has visited in the past. The model contains a parameter which controls the strength of the reinforcement.

Sabot and Tarres have shown that the VRJP is related to the H22 supersymmetric hyperbolic spin model, a model that originated in the study of random band matrices. By making use of results for the H22 model they were able to prove recurrence of the VRJP for strong reinforcement in all dimensions. I will present a new and direct connection between the VRJP and hyperbolic spin models (both supersymmetric and classical), and show how this connection can be used to prove that the VRJP is recurrent in two dimensions for any reinforcement strength.

Based on joint work with R. Bauerschmidt and A. Swan.

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**Past seminars**

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**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**

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