# Seminars 2013/14

 07 Oct 2013 Partha Dey University of Warwick Multiple phase transitions in long-range first-passage percolation on square lattices 14 Oct 2013 Elisabetta Candellero University of Warwick Clustering phenomenon in random geometric graphs on the hyperbolic plane 21 Oct 2013 Christina Goldschmidt University of Oxford Behaviour near the extinction time in self-similar fragmentation chains 28 Oct 2013 Cécile Mailler University of Bath Smoothing Equations for Large Pólya Urns 04 Nov 2013 Paul Chleboun University of Warwick Time scale separation in the low temperature East model 11 Nov 2013 Renato dos Santos Université Lyon 1 Quenched central limit theorems for random walks in random sceneries and the energy of a charged polymer in two dimensions 18 Nov 2013 Tobias Müller Utrecht University Logic and random graphs 25 Nov 2013 Pietro Caputo Università Roma Tre Large deviations in sparse random graphs: a local weak convergence approach 02 Dec 2013 Wilfrid Kendall University of Warwick Probabilistic coupling and Nilpotent Diffusions 09 Dec 2013 Vadim Shcherbakov Royal Holloway, Univ. of London Probabilistic model of interacting populations 13 Jan 2014 Márton Balázs University of Bristol Anomalous fluctuations in one dimensional interacting systems 20 Jan 2014 Leonid Koralov University of Maryland Transition from homogenization to averaging in cellular flows 03 Feb 2014 Nicolas Broutin INRIA Paris-Rocquencourt The dual tree of a recursive lamination of the disc 10 Feb 2014 Karen Gunderson University of Bristol Random Markov processes 17 Feb 2014 Daniel Ueltschi University of Warwick A chain of Chinese restaurants 24 Feb 2014 Igor Kortchemski ENS, Paris Alien versus Predator 03 Mar 2014 Saul Jacka University of Warwick Minimising the time to shuttle a diffusion between two points 10 Mar 2014 Leif Döring ETH Zürich Self-Similarity, Voter Processes and Catalytic Branching Processes 17 Mar 2014 Amanda Turner Lancaster University Small particle limits in a regularized Laplacian random growth model 24 Mar 2014 Marcel Ortgiese Universität Münster The Interface of the Symbiotic Branching Model 31 Mar 2014 Demeter Kiss University of Cambridge Planar lattices do not recover from forest fires 07 Apr 2014 Gady Kozma Weizmann Institute of Science Random points in the metric polytope 14 Apr 2014 Istvan Redl University of Bath Skorokhod embeddings for two-sided Markov chains 02 Jun 2014 Maren Eckhoff University of Bath Long paths in first passage percolation

## Abstracts

### 7th October 2013 - Partha Dey (University of Warwick)

Multiple phase transitions in long-range first-passage percolation on square lattices
We consider a model of long-range first-passage percolation on the d-dimensional square lattice in which any two distinct vertices x, y are connected by an edge having exponentially distributed passage time with mean $|x-y|^s$, where $s>0$ is a fixed parameter and $| |$ is the $l_1$--norm on $Z^d$. We analyze the asymptotic growth rate of the set $B_t$, which consists of all $x \in Z^d$ such that the first-passage time between the origin 0 and $x$ is at most $t$, as $t\to\infty$. We show that depending on the values of $s$ there are four growth regimes: (i) instantaneous growth for $s2d+1$ like the nearest-neighbor first-passage percolation model corresponding to $s=\infty$. We find explicit growth rates and also analyze the behavior at the boundary values $s=d,2d,2d+1$.
Based on joint work with Shirshendu Chatterjee.

### 14th October 2013 - Elisabetta Candellero (University of Warwick)

Clustering phenomenon in random geometric graphs on the hyperbolic plane
In this talk we introduce the concept of random geometric graphs on the hyperbolic plane and discuss its applicability as a model for social networks. In particular, we will discuss issues that are related to clustering, which is a phenomenon that often occurs in social networks: two individuals that have a common friend are somehow more likely to be friends of each other. We give a mathematical expression of this phenomenon and explore how this depends on the parameters of our model.
(Joint work with Nikolaos Fountoulakis).

### 21st October 2013 - Christina Goldschmidt (University of Oxford)

Behaviour near the extinction time in self-similar fragmentation chains
Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly, but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual spine'' decomposition for the fragmentation, and Markov renewal theory.
This is joint work with Bénédicte Haas (Paris-Dauphine).

### 28th October 2013 - Cécile Mailler (University of Bath)

Smoothing Equations for Large Pólya Urns
This talk will focus on large two-colour Pólya urns. From the study of the asymptotic behaviour of such an urn arises a random variable denoted by W. The underying tree structure of the urn permits to see W as the solution in law of a fixed point equation, from which we can deduce information about its moments, or about the existence of a density. This work can be done on the discrete urn itself, or on its continuous time embedding. Though the two variables W (arisen from discrete or continuous time) are different, they are related by connexions, which often permit to translate results from one W to the other. This work is a collaboration with Brigitte Chauvin and Nicolas Pouyanne (Versailles, France).

### 4th November 2013 - Paul Chleboun (University of Warwick)

Time scale separation in the low temperature East model
We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbour is 1. We focus on the glassy effects caused by the kinetic constraint as the equilibrium density of 0's, given by q, tends to zero. Specifically we analyse time scale separation and dynamical heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium. For any mesoscopic length scale $L=O(q^{-\gamma})$, $\gamma<1$, we show that the characteristic time scales associated with two system sizes, $L$ and $\lambda L$, are well separated provided that $\lambda \geq 2$ is large enough. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form 111..10, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamical heterogeneity. A key result for this is a very precise computation of the relaxation time of the chain as a function of $(q,L)$, well beyond previous bounds. Finally we show that no form of time scale separation can occur on the equilibrium scale, contrary to what was previously assumed in the physical literature based on numerical simulations. Joint work with A. Faggionato and F. Martinelli.
Refs: P. Chleboun, A. Faggionato, F. Martinelli, Time scale separation and dynamic heterogeneity in the low temperature East model, arXiv:1212.2399

### 11th November 2013 - Renato dos Santos (Université Lyon 1)

Quenched central limit theorems for random walks in random sceneries and the energy of a charged polymer in two dimensions
We will discuss two simple models in random media that can be seen as energy functions of polymer models in $\Z^d$: random walks in random sceneries (RWRS), in which the interaction is between the polymer path and a random field, and the energy of a charged polymer, in which the interaction is between random charges that carried by the monomers of the polymer. As models in random media, they can be studied under different laws: the quenched law, where the random field or random charges are fixed, or the annealed law, which is the average of the previous one. Under the annealed law, these models are known to be very similar, and limit theorems have been proved for them in a variety of contexts. Quenched limit theorems are more recent; we will focus on the case of $d=2$ for both models.

### 18th November 2013 - Tobias Müller (Utrecht University)

Logic and random graphs
Random graphs have been studied for over half a century as useful mathematical models for networks and as an attractive bit of mathematics for its own sake. Almost from the very beginning of random graph theory there has been interest in studying the behaviour of graph properties that can be expressed as sentences in some logic, on random graphs. We say that a graph property is first order expressible if it can be written as a logic sentence using the universal and existential quantifiers with variables ranging over the nodes of the graph, the usual connectives AND, OR, NOT, parentheses and the relations = and ~, where x ~ y means that x and y share an edge. For example, the property that G contains a triangle can be written as Exists x,y,z : (x ~ y) AND (x ~ z) AND (y ~ z). First order expressible properties have been studied extensively on the oldest and most commonly studied model of random graphs, the Erdos-Renyi model. A number of very attractive results have been obtained, and by now we have a fairly full description of the behaviour of first order expressible properties on this model. I will describe a number of striking results that have been obtained for the Erdos-Renyi model with surprising links to other branches of math including number theory, before describing some of my own work on different models of random graphs, including the random planar graph and the random geometric graph.

### 25th November 2013 - Pietro Caputo (Università Roma Tre)

Large deviations in sparse random graphs: a local weak convergence approach
Consider the Erdös-Renyi random graph on n vertices where each edge is present independently with probability p=c/n, with c>0 fixed. For large n, a typical realization locally behaves like the Galton-Watson tree with Poisson offspring distribution with mean c. We discuss large deviations from this typical behavior, within the framework of the local weak convergence introduced by Benjamini-Schramm and Aldous-Steele. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported by trees. Along the way, we present a new configuration model which allows one to sample uniform random graphs with a given finite neighborhood distribution, provided the latter is supported by trees. We also present a new class of unimodular random trees, which generalizes the Galton-Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. This is joint work with Charles Bordenave.

### 2nd December 2013 - Wilfrid Kendall (University of Warwick)

Probabilistic coupling and Nilpotent Diffusions
Modern probability theory makes considerable use of the technique of probabilistic coupling. The idea is, to analyse a given random process by constructing two inter-dependent copies of it, defined on the same probability space, and related in such a way as to facilitate analysis. Applications include: establishing monotonicity in non-obvious situations, developing quantitative approximations to distributions of random variables, constructing gradient estimates, and even producing exact simulation algorithms for Markov chains. However the thematic question, which has driven much of the theory of probabilistic coupling, concerns whether or not one can construct the two coupled random processes so that they almost surely meet ("couple") at some future random time, and if so then whether one can construct a maximal coupling, for which the random time is smallest possible? The question is sharpened if we require the coupling to be co-adapted (also: immersed, or Markovian); this is an additional requirement that the coupling respect the underlying causal structure of the random processes, and can be viewed as implying that the coupling is easily constructable in some general sense. There is a considerable body of theory describing how to build successful co-adapted couplings for elliptic diffusions, all building on the basic reflection coupling for simple random walks or Brownian motion (very simply, the random jumps of the coupled process are arranged so far as possible to be the opposites of the random jumps of the original process). It is conjectured that successful co-adapted couplings can be built for all hypoelliptic diffusions (diffusions in d dimensions with fewer than d "directions of randomness"). In this talk I will survey the general theory of coupling, describe the known results for co-adapted couplings of hypoelliptic diffusions (in fact, Brownian motions on nilpotent Lie groups), and briefly discuss a related and very simple example which has applications to the theory of filtrations.

### 9th December 2013 - Vadim Shcherbakov (Royal Holloway, Univ. of London)

Probabilistic model of interacting populations
A birth-and-death process with non-negative integer values is a classic probabilistic model for the size of a population. We consider a finite set of such processes labelled by nodes of a connected graph. There is an interaction between neighbours (two processes/populations are neighbours, if the corresponding nodes are adjacent). Transition rates are given by functions with separating variables and this constructive feature allows to model various types of interactions. Further, it leads to certain common effects in the long term behaviour of the model in a wide range of cases. The main goal of the talk is to demonstrate these effects in the case of just two neighbouring populations and for reversible models with either constant vertex degree or star graphs.

### 13th January 2014 - Márton Balázs (University of Bristol)

Anomalous fluctuations in one dimensional interacting systems
I will describe a family of one dimensional interacting particle systems that contains the simple exclusion and the zero range processes, and many more. In the stationary distribution the current fluctuations show anomalous scalings, I will sketch parts of the proof of this phenomenon for some of our models. Meanwhile I will try to make it clear how convexity of a function of central importance leads to such unusual behaviour. The technical point that prevents us from proving anomalous scaling in great generality will also be pointed out. Our methods work with probabilistic arguments and couplings, hence it might give more intuition than alternative existing techniques of heavy combinatorics and analysis. This is joint work with Julia Komjathy and Timo Seppalainen.

### 20th January 2014 - Leonid Koralov (University of Maryland)

Transition from homogenization to averaging in cellular flows
We consider an elliptic problem that involves two parameters (the size of the domain, which may be large, and the magnitude of the diffusion coefficient, which may be small). Depending on the relation between the parameters, the asymptotics of the solution is given either by homogenization theory or by averaging. Using probabilistic techniques, we give a precise description of the transition regime, where a more intricate behavior is observed.

### 3rd February 2014 - Nicolas Broutin (INRIA Paris-Rocquencourt)

The dual tree of a recursive lamination of the disc
In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $M$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov--Hausdorff sense to a limit real tree $T$, which is encoded by $M$. We will also discuss the regularization of certain bad'' encodings of another real tree arising when inserting the chords in a different order. This is joint work with Henning Sulzbach.

### 10th February 2014 - Karen Gunderson (University of Bristol)

Random Markov processes
In 1990, Kalikow introduced the notion of random Markov processes'' to refine classifications of measure-preserving transformations. Looking at the shift operator on probability spaces of infinite sequences of some set of states, a random Markov process is a stochastic process for which there is a coupling of the sequences of states with doubly infinite sequences $(m(i))$, so that for every $i$, the distribution on the states at the $i$-th step of the process depends only on the $m(i)$ previous states. Random Markov processes are a generalization of usual Markov chains and have been used to produce counter-examples to questions about uniqueness of certain types of measure spaces and as extensions for non-periodic transformations.
In this talk, I will discuss some new results on the classification of random Markov processes on any finite number of states and extensions of these results to certain processes with infinitely many states. In addition, I will give some examples showing that further conditions are needed to extend results about finite numbers of states to those with even countably many states.
Joint work with Neal Bushaw (Arizona) and Steve Kalikow (Memphis).

### 17th February 2014 - Daniel Ueltschi (University of Warwick)

A chain of Chinese restaurants
In probability, a Chinese restaurant is a Markov process where customers enter and randomly choose to sit next to an existing customer, or they start a new table. Here, there will be many restaurants and customers are allowed to go from one place to another. The goal is to describe the invariant measure. This setting is motivated by close connections to a number of interesting situations: ideal Bose gas, zero-range process, Becker-Doring process, and spatial permutations. I will describe some results about condensation and typical size of large elements. (Joint work with Nick Ercolani and Sabine Jansen.)

### 24th February 2014 - Igor Kortchemski (École Normale Supérieure, Paris)

Alien versus Predator
We will be interested in a prey-predator dynamics on graphs, where vertices may be of three types: occupied by a prey or a predator, or unoccupied. Preys reproduce at a fixed rate and propagate only to unoccupied neighbours, while predators reproduce at another fixed rate and propagate only to prey neighbours. We will try to understand what happens if one starts with one prey and one predator: how does the system evolve? May the preys survive indefinitely? What is the final state of the system?

### 3rd March 2014 - Saul Jacka (University of Warwick)

Minimising the time to shuttle a diffusion between two points
Motivated by a problem in simulated tempering (a form of Markov chain Monte Carlo), we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel from the origin to 1 and then return. The control mechanism is that we are allowed to chose the diffusion's drift at each point in [0,1]. We consider the static and dynamic versions of this problem, where, in the dynamic version, we are only able to choose the drift at each point at the time of first visiting that point. We first consider the case where we start from 0 and then will discuss a more general case, outlining the remaining open problems.

### 10th March 2014 - Leif Döring (ETH Zürich)

Self-Similarity, Voter Processes and Catalytic Branching Processes
I will highlight a relation between self-similar processes represented by a stochastic differential equations and limits of certain stochastic partial differential equations. On the way we shall discuss some typical phenomena of interacting particle systems and interacting diffusion processes.

### 17thMarch 2014 - Amanda Turner (Lancaster University)

Small particle limits in a regularized Laplacian random growth model
In 1998 Hastings and Levitov proposed a one-parameter family of models for planar random growth in which clusters are represented as compositions of conformal mappings. This family includes physically occurring processes such as diffusion-limited aggregation (DLA), dielectric breakdown and the Eden model for biological cell growth. In the simplest case of the model (corresponding to the parameter alpha=0), James Norris and I showed how the Brownian web arises in the limit resulting from small particle size and rapid aggregation. In particular this implies that beyond a certain time, all newly aggregating particles share a single common ancestor. I shall show how small changes in alpha result in the emergence of branching structures within the model so that, beyond a certain time, the number of common ancestors is a random number whose distribution can be obtained. This is based on joint work with Fredrik Johansson Viklund (Uppsala) and Alan Sola (Cambridge).

### 24th March 2014 - Marcel Ortgiese (Universität Münster)

The Interface of the Symbiotic Branching Model
The symbiotic branching model describes two interacting spatial populations that move diffusively on the real line and branch at a rate proportional to the product of the number of particles present at a site. More formally, the system is described by two interacting stochastic partial differential equations with correlated driving noises. Starting the model from two complementary Heaviside functions (corresponding to two spatially separated populations), one can consider the interface between the two populations, defined as the region where both populations are present. We investigate the interface by showing that the diffusively rescaled system converges and that there is an interesting limit corresponding to spatially separated populations. This is based on joint work with Jochen Blath and Matthias Hammer (both TU Berlin).

### 31st March 2014 - Demeter Kiss (University of Cambridge)

Planar lattices do not recover from forest fires
Consider supercritical (p > p_c) site percolation on Z^2. Destroy the infinite open cluster, that is, make all its vertices closed. Then open the closed vertices independently from each other with probability delta. Let theta(p,delta) denote the probability that the origin is in an infinite cluster in the configuration thus obtained. We show that there is a positive delta such that theta(p, delta) = 0 for all p > p_c. This is a joint work with Ioan Manolescu and Vladas Sidoravicius.

### 7th April 2014 - Gady Kozma (Weizmann Institute of Science)

Random points in the metric polytope
We investigate a random metric space on n points constrained to have all distances smaller than 2, or in other words, we take a random point from the Lebesgue measure on the intersection of the so-called metric polytope with the cube [0,2]^(n(n-1)/2). We find that, to a good precision, the distances behave simply like i.i.d. numbers between 1 and 2. The proof uses an interesting mix of entropy methods and the Szemeredi regularity lemma (all terms will be explained in the talk). Joint work with Tom Meyerovitch, Ron Peled and Wojciech Samtoij.

### 14th April 2014 - Istvan Redl (University of Bath)

Skorokhod embeddings for two-sided Markov chains
Let $(X_n \colon n\in\Z)$ be a two-sided recurrent Markov chain with fixed initial state $X_0$ and let $\nu$ be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time T such that $(X_{T+n} \colon n\in\Z)$ has the law of the same Markov chain with initial distribution $\nu$. In the case when our criterion is satisfied we give an explicit solution, which is a stopping time, and we show that this solution is optimal in the sense that no other solution has more finite moments. Our results generalise previous results by Liggett (2001) and Holroyd and Peres (2006) and also complement those of Last, Morters and Thorisson (2013). This is a joint work with Peter Morters.

### 2nd June 2014 - Maren Eckhoff (University of Bath)

Long paths in first passage percolation
Consider the complete graph with i.i.d. edge weights that have a heavy tail at zero. Using coupling and branching process techniques, we investigate properties of the smallest-weight path between two vertices. Moreover, we show that the smallest-weight tree of a vertex converges locally to the invasion percolation cluster on the Poisson-weighted infinite tree, and discuss connections to the minimum spanning tree. The talk is based on joint work with Jesse Goodman, Remco van der Hofstad and Francesca Nardi.