# Abstracts

**May 10, 2017 -- First two talks in A1.01, then B3.02:**

13:30 - 14:30. **Oriane Blondel** (Lyon): "Diffusion in kinetically constrained models"

*Abstract*: Kinetically constrained models (KCM) are interacting particle systems on Z^d which have been introduced in the physics literature in order to reproduce some aspects of the glass transition. Here we consider models with Kawasaki dynamics (conservative systems with nearest-neighbour jumps), in which a jump is allowed only if some local constraint is satisfied by the current configuration. Due to this constraint, even the ergodicity of the system at any density is non-trivial. We focus on the trajectory of a tagged particle in the system at equilibrium and show that under diffusive scaling it converges to a non-degenerate Brownian motion. This implies in particular the absence of a subdiffusive/diffusive transition, contrary to some predictions in the physics literature. Joint work with Cristina Toninelli.

14:30 - 15:30. **Richard Pymar** (Birkbeck): "Delocalising the parabolic Anderson model"

*Abstract*: The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials (including Pareto potentials) it is localised at just one point. In the talk, we discuss a natural modification of the parabolic Anderson model on Z, which exhibits a phase transition between localisation and delocalisation. This is a joint work with Stephen Muirhead and Nadia Sidorova.

16:00 - 17:00. **James Martin** (Oxford): "Percolation games"

*Abstract*: Let G be a graph (directed or undirected), and let v be some vertex of G. Two players play the following game. A token starts at v. The players take turns to move, and each move of the game consists of moving the token along an edge of the graph, to a vertex that has not yet been visited. A player who is unable to move loses the game. If the graph is finite, then one player or the other must have a winning strategy. In the case of an infinite graph, it may be that, with optimal play, the game continues for ever.

I'll focus in particular on games played on the lattice Z^d, directed or undirected, with each vertex deleted independently with some probability p. In the directed case, the question of whether draws occur is closely related to ergodicity for certain probabilistic cellular automata, and to phase transitions for the hard-core model. In the undirected case, I'll describe connections to bootstrap percolation and to maximum-cardinality matchings and independent sets.

This includes joint work with Alexander Holroyd, Irène Marcovici, Riddhipratim Basu, and Johan Wästlund.

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**March 1, 2017 -- First two talks in A1.01, then B3.02:**

13:30 - 14:30. **Christoph Hofer-Temmel** (Dutch Academy of Defence): "Disagreement percolation for Gibbs point processes"

*Abstract*: Disagreement percolation is a technique to show uniqueness of the Gibbs measure. It does so by controlling the influence of differing boundary conditions by a subcritical percolation model. If the percolation has exponentially decaying connection probabilities, then exponential decay of correlations for the Gibbs measure is also possible.

A core building block is a dependent thinning from a Poisson point process to a dominated Gibbs point process within a finite volume, where the thinning probability is related to the derivative of the free energy of the Gibbs point process. This explicit coupling permits to control the influence of differing boundary conditions explicitly.

We showcase this in two models. For the hard-sphere model, disagreement percolation establishes uniqueness and decay of correlations for a larger range of values than cluster expansion techniques are able to achieve. For the continuous random cluster model, a model with unbounded interactions, it yields the first proof of uniqueness at high temperature, thus establishing the existence of a phase transition. This includes joint work with Pierre Houdebert.

14:30 - 15:30. **Alessandra Cipriani** (Bath): "Pinning for the membrane model in higher dimensions"

*Abstract*: In this talk we are interested in studying the behavior of a d-dimensional interface around its phase transition. We will specifically discuss the case of a model of "pinning''. This problem arises when one considers an interface rewarded every time it touches the 0-hyperplane. Then there is a competition between attraction to the hyperplane and repulsion due to the decrease of entropy for interfaces pinned at 0. Tuning the strength of the attraction, two behaviors are possible: either energy wins, and the interface stays localized close to 0, or entropy wins, and the interface is repelled away from 0. We will study the effects of pinning for a particular effective interface, the membrane or Bilaplacian model, which is akin to the discrete Gaussian free field. We will draw a parallel between the two models and show how in the membrane case a positive pinning strength localises the field in higher dimensions. Joint work with E. Bolthausen (University of Zurich) and N. Kurt (TU Berlin).

16:00 - 17:00. **Ilya Chevyrev** (Oxford): "Recent developments in càdlàg rough paths"

*Abstract*: In this talk, we will present the main features of rough paths theory in the càdlàg setting. We will discuss several notions of solutions to RDEs with discontinuous drivers, introduce a convenient metric on the space of càdlàg rough paths which renders the solution map continuous, and present an (enhanced) BDG inequality for lifts of càdlàg local martingales. Time permitted, we will demonstrate two applications: the first is a robust method to obtain general Wong-Zakai-type theorems in the spirit of Kurtz-Pardoux-Protter, the second is an application to stochastic flows which extends a classical result of Kunita. Joint work with Peter Friz.

**January 25, 2017 -- All talks in MS.04:**

13:30 - 14:30. **Paul Chleboun** (Oxford): "Static length scales and glassy dynamics in spin plaquette models"

14:30 - 15:30. **Daniela Bertacchi** (University of Milan, Bicocca): "Galton-Watson processes in varying environment and accessibility percolation"

*Abstract*: We introduce a generalization of the classic Galton-Watson process by considering a varying environment; more precisely, the offspring distributions may depend on time. We provide sufficient conditions for survival or extinction which rely only on the first and second moments of the offspring distributions. These results are then applied to branching processes in varying environment with selection where every particle has a real-valued label and labels can only increase along genealogical lineages; we obtain analogous conditions for survival or extinction. These last results can be interpreted in terms of accessibility percolation on Galton-Watson trees, which represents a relevant tool for modeling the evolution of biological populations. This is a joint work with P.M.Rodriguez and F.Zucca.

16:00 - 17:00. **Fabio Zucca** (Milan University of Technology): "Generating functions and extinction probabilities for Branching Random Walks"

*Abstract*: It is well-known that many basic properties of a branching process can be established by looking at the generating function and its fixed points. Similarly, survival probabilities of a branching random walk (BRW) can be seen as fixed points of a (possibly infinite-dimensional) generating function. If the BRW is irreducible and the space is finite then there are at most two fixed points. It has been believed, since the late 80's, that this result could hold for every irreducible BRW. We prove that it may be false if the space is infinite.

This leads to many interesting relations between global survival and local survival probabilities. In particular, for a generic continuous-time branching random walk, the so-called "strong local survival", unlike local and global survival, is not a monotone property with respect to the reproduction rates. At the end we discuss some questions and examples. This is a joint work with D.Bertacchi.

**November 30, 2016 -- First two talks in A1.01, then B3.02:**

13:30 - 14:30. **Andrew Wade** (Durham): "Invariance principle for non-homogeneous random walks"

*Abstract*: We consider a class of spatially non-homogeneous random walks in multidimensional Euclidean space with zero drift, which in any dimension (two or higher) can be recurrent or transient depending on the details of the walk. These walks satisfy an invariance principle, and have as their scaling limits a class of martingale diffusions, with law determined uniquely by an SDE with discontinuous coefficients at the origin. Furthermore, pathwise uniqueness of this SDE may fail. The radial coordinate of the diffusion is a Bessel process of dimension greater than 1. Unique characterization of the law of the diffusion, which must start at the origin, is natural via excursions built around the Bessel process; each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining appropriately the Riemannian metric g on the sphere S allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Brownian motion with drift on the Riemannian manifold (S,g). In particular, this provides a multidimensional generalisation of the Pitman-Yor representation of the excursions of Bessel process with dimension between one and two.

Furthermore, the density of the stationary law of the angular component with respect to the volume element of g can be characterised by a linear PDE involving the Laplace-Beltrami operator and the divergence under the metric g.

(This is joint work with Nicholas Georgiou and Aleksandar Mijatovic.)

14:30 - 15:30. **Jordan Stoyanov** (Sofia): "Recent Developments in Moment Determinacy of Probability Distributions"

*Abstract:* We deal with distributions, one-dimensional or multi-dimensional, with all moments finite.

It is well-known that any such a distribution is either uniquely determined by its moment (M-determinate) or it is non-unique (M-indeterminate). The main discussion will be on some new questions about existing and new conditions for M-determinacy and/or M-indeterminacy of functional transformations of random data. New results will be reported with hints for their proof. Examples and counterexamples will be given and also open questions and conjectures outlined.

*15:30 - 16:00. Coffee break in the Statistics Common Room (Room C0.06)*

16:00 - 17:00. **Adrian Gonzales Casanova** (TU Berlin): "Fixation and duality in a Xi coalescent model with selection"

*Abstract*: In this talk we will introduce a generalisation of the Wright Fisher model, for a population with finite size and non-overlapping generations, allowing for several types of selection as well as simultaneous multiple mergers. The construction provides an almost sure dual relation between its frequency process and its ancestral process. The latter can be interpreted as a discrete analogue to the celebrated ancestral selection graph. We will also study a two type frequency process with general selection and general coalescent mechanism, and investigates in which cases the selective type goes to fixation with probability one. (This is talk is based on a joint project with Dario Spano)

**November 2, 2016 -- Room MS.03:**

13:30 - 14:30.** Alessandra Caraceni** (Bath): "A stroll around Random Infinite Quadrangulations of the Plane"

*Abstract*: The investigation of large random combinatorial objects has spawned a very active research field in the study of local and scaling limits of random planar maps. After a general introduction, we shall focus on an object named the Uniform Infinite Quadrangulation of the Half-Plane (UIHPQ), which arises as a local limit of random quadrangulations with a boundary whose size and perimeter are sent to infinity. We present a wide range of results obtained in joint work with N. Curien, and discuss their consequences concerning an annealed model of self-avoiding walks on large random quadrangulations.

14:30 - 15:30.** Márton Balázs** (Bristol): "How to initialise a second class particle?"

*Abstract*: This talk will be on interacting particle systems. One of the best known models in the field is the simple exclusion process where every site has 0 or 1 particles. It has long been established that under certain rescaling procedure this process converges to solutions of a deterministic nonlinear PDE (Burgers' equation). Particular types of solutions, called rarefaction fans, arise from decreasing step initial data.

Second class particles are probabilistic objects that come from coupling two interacting particle systems. They are very useful and their behaviour is highly nontrivial.

The beautiful paper of P. A. Ferrari and C. Kipnis connects the above: they proved that the second class particle of simple exclusion chooses a uniform random velocity when started in a rarefaction fan. The extremely elegant proof

is based, among other ideas, on the fact that increasing the mean of a Bernoulli distribution can be done by adding or not adding 1 to the random variable.

For a long time simple exclusion was the only model with an established large scale behaviour of the second class particle in its rarefaction fan. I will explain how this is done in the Ferrari-Kipnis paper, then show how to do this for other models that allow more than one particles per site. The main issue is that most families of distributions are not as nice as Bernoulli in terms of increasing their parameter by just adding or not adding 1. To overcome this we use a signed, non-probabilistic coupling measure that nevertheless points out a canonical initial probability distribution for the second class particle. We can then use this initial distribution to greatly generalize the Ferrari-Kipnis argument. I will conclude with an example where the second class particle velocity has a mixed discrete and continuous distribution. (Joint work with Attila László Nagy.)

**2 March 2016**

1:30-2:30 **Balazs Rath** (Budapest) : Multiplicative coalescent with linear deletion: a rigid representation

We consider a modification of the dynamical Erdős-Rényi random graph model, where connected components are removed ("frozen") with a rate linearly proportional to their size. One may also view the time evolution of the list of component sizes of the graph as a multiplicative coalescent process with linear deletion (MCLD). We discuss a somewhat surprising "rigid representation" of the MCLD, similar to recent results of Broutin and Marckert (2015, PTRF) and Limic (2016). Joint work in progress with James Martin.

2:30-3:30 **Khalil Chouk** (TU Berlin) : Random operator with singular potential

4:00-5:00 **Codina Cotar** (UCL) : Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

We introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge- and to vertex-reinforced random walks with super-linear reinforcement on arbitrary in infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs. For edge-reinforced random walks, we complete the results of Limic and Tarres [Ann. Probab. (2007)] and we settle a conjecture of Sellke [Technical Report 94-26, Purdue University (1994)] by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times.

For vertex-reinforced random walks, we extend results previously obtained on Z by Volkov [Ann. Probab. (2001)] and by Basdevant, Schapira and Singh [Ann. Probab. (2014)], and on complete graphs by Benaim, Raimond and Schapira [ALEA (2013)]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times.

(This is joint work with Debleena Thacker)

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**3 February 2016**

1:30-2:30 **Elie Aidekon** (UPMC Paris) : Scaling limit of the recurrent biased random walk on a Galton-Watson tree

We consider a biased random walk on a Galton-Watson tree. This Markov chain is null recurrent for a critical value of the bias. In that case, Peres and Zeitouni proved that the height of the Markov chain properly rescaled converges in law to a reflected Brownian motion B. We show here that the trace of this Markov chain converges in law to the Brownian forest encoded by B. Joint work with Loïc de Raphélis.

2:30-3:30 **Antal Jarai** (Bath) : Inequalities for critical exponents in d-dimensional sandpiles

In the first part of this talk I will introduce the Abelian sandpile model and discuss bijections due to physicists Majumdar & Dhar and Ivashkevich, Ktitarev & Priezzhev that are useful in studying its dynamics. In the second part, I will present results based on these bijections as well as analysis of uniform spanning forests, that yield rigorous inequalities for some of the critical exponents. These are: the toppling probability, the radius and the size exponent for avalanches. (Joint work with Jack Hanson and Sandeep Bhupatiraju.)

4:00-5:00 **Vincent Vargas** (ENS Paris) : Liouville conformal field theory

Liouville quantum field theory (LQFT) is a family of conformal field theories (CFT) which arise as a building block of Polyakov's quantum gravity. In LQFT, one can define correlation functions and random measures (which are conjectured to be the scaling limit of large planar maps properly embedded in the sphere). In a seminal paper of theoretical physics, Belavin, Polyakov and Zamolodchikov (BPZ) introduced a general formalism, the so-called BPZ formalism, to study a CFT and in particular to compute the correlation functions. I will present recent progress in giving a rigorous mathematical meaning to the BPZ formalism in the context of LQFT. Based on joint work with A. Kupiainen and R. Rhodes.

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Stephen Connor (York): Wednesday, 25 November 2015 at 1:30-2:30, Lecture room A1.01.

**Perfect simulation for the M/G/c queue**

*Abstract: *Unlike Markov chain Monte Carlo, perfect simulation algorithms produce a sample from the exact equilibrium distribution of a Markov chain, but at the expense of a random run-time. I'll give a short introduction to these algorithms for beginners, before talking about some recent work, jointly with Wilfrid Kendall (Warwick), on designing perfect simulation algorithms for M/G/c queues.

Sergey Foss (Maxwell Institute and Heriot-Watt University): Wednesday, 25 November 2015 at 2:30-3:30, Lecture room A1.01.

**Large deviations for the distribution of the stationary sojourn time in single- and multi-server queues**

*Abstract: *First, I show that there are 5 typical ``scenarios'' for the stationary waiting/sojourn time in a GI/GI/1 queue to be large. Then consider the large deviations for the multi-server queue in teh presence of heavy tails. I also formulate open problems and hypotheses.

Paul Chleboun (Warwick) : Wednesday, 25 November 2015 at 16-17, Lecture room B3.02.

**Relaxation and mixing of kinetically constrained models.**

*Abstract*: We study the relaxation and out-of-equilibrium dynamics of a family of kinetically constrained models (KCMs) called the d-dimensional East-like processes. KCMs are spin systems on integer lattices, where each vertex is labelled either 0 or 1, which evolve according to a very simple rule: i) with rate one and independently for each vertex, a new value 1/0 is proposed with probability 1-q and q respectively; ii) the proposed value is accepted if and only if the neighbouring spins satisfy a certain constraint. Despite of their apparent simplicity, KCMs pose very challenging and interesting problems due to the hardness of the constraints and lack of monotonicity. The out-of-equilibrium dynamics are extremely rich and display many of the key features of the dynamics of real glasses, such as; an ergodicity breaking transition at some critical value, huge relaxation times close to the critical point, and dynamic heterogeneity (non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium). We discuss recent advances on the out-of-equilibrium dynamics of the East-like processes, including the dependence of the relaxation and mixing time on the system size, density, and dimension. We also look at simulations which motivate some interesting limit shape conjectures.

This is joint work with Alessandra Faggionato and Fabio Martinelli.

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Alexandre Stauffer (Bath): Wednesday, 28 October 2015 at 1:30-2:30, Lecture room MS.03.

**Random lattice triangulations**

*Abstract: *I will give an introduction to random lattice triangulations. These are triangulations of the integer points inside the square [0,n] x [0,n] where each triangulation T is obtained with probability proportional to \lambda^|T|, with \lambda being a positive real parameter and |T| being the total length of the edges in T.

I will discuss structural and dynamical properties of such triangulations, such as phase transitions with respect to \lambda, decay of correlations, local limits, and mixing time of Glauber dynamics. This is based on joint works with Pietro Caputo, Fabio Martinelli and Alistair Sinclair.

Igor Kortchemski (École polytechnique, Paris): Wednesday, 28 October 2015 at 2:30-3:30, Lecture room MS.03.

**Self-similar scaling limits of Markov chains on the positive integers**

*Abstract: *We will be interested in the scaling limits of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. We identify three different regimes (loosely speaking the transient, the recurrent and the positive-recurrent regimes) in which the scaling limit exhibits different behavior. This has for instance applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks and to the structure of large random planar maps. This is based on joint works with Jean Bertoin and Nicolas Curien.