## Seminars 2016/17

## Prob-L@B Internal Monthly Seminar

27 Oct 2016 | Weerapat (Pite) Satitkanitkul |
Conditioned self-similar Markov processes |

24 Nov 2016 | Alessandra Caraceni |
Self-Avoiding Walks on Random Quadrangulations |

8 Dec 2016 | Peter Gracar |
Spread of information by random walks |

2 Feb 2017 | Amitai Linker |
The contact process on evolving scale-free networks |

20 Feb 2017 | Marcel Ortgiese |
Local geometry of first passage percolation on random graphs |

30 Mar 2017 6W 1.2 |
Mathew Penrose |
Optimal cuts of random geometric graphs |

27 Apr 2017 | Sam Johnston |
The coalescent structure of continuous-time Galton-Watson trees |

25 May 2017 | Francis Lane |
The largest fragment of a homogeneous fragmentation process |

## Seminar abstracts

### 3rd October 2016 - Milton Jara (IMPA, Rio de Janeiro)

**Non-equilibrium fluctuations of one-dimensional particle systems**

We show that the density fluctuations around its hydrodynamic limit of one-dimensional particle systems converge to the solution of a non-homogeneous stochastic heat equation. The idea of the proof is to show that entropy production in Yau's relative entropy method is bounded by a constant independent of the size of the system. This control is achieved by extending the second-order Boltzmann-Gibbs principle introduced by Gonçalves-J. to the non-equilibrium setting.

### 10th October 2016 - Patricia Gonçalves (IST, Lisbon)

**The symmetric simple exclusion with slow boundaries**

In this talk I will present an exclusion process, in which particles evolve on the set of sites {1,2,...N-1}, called the bulk, according to continuous time nearest neighbor symmetric random walks and at the end points 0 and N, called the reservoirs, particles can either enter or leave the system at a rate which is slower than the rates in the bulk. The main purpose of the talk is to discuss recent results on the non-equilibrium density fluctuations for this model and to discuss the case of non-nearest neighbor jumps.

### 17th October 2016 - Gourab Ray (University of Cambridge)

**Universality of fluctutation in the dimer model**

The dimer model is a very popular model in statistical physics because of its exact solvability properties. I will try to convince you that the fluctuation in the dimer model is universal in the sense that it is more or less independent of the underlying graph and also the topology the graph is embedded in and is given by a form of Gaussian free field.

This is joint work with Nathanael Berestycki and Benoit Laslier.

### 24th October 2016 - Ryoki Fukushima (University of Kyoto)

**Quenched tail estimate for the random walk in random scenery and in random layered conductance**

We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time asymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result has an application to the tail estimates for a random walk in random conductance which has a layered structure. (Joint work with Jean-Dominique Deuschel.)

### 27th October 2016 - Weerapat (Pite) Satitkanitkul (University of Bath - PIMS seminar)

**Conditioned self-similar Markov processes**

Previously, it has been proven that, under appropriate assumptions, we can condition a positive-valued self-similar Markov processes that is naturally absorbed at 0 to avoid zero and vice versa. Recently, we have been able to extend the conditioning to the case of real-valued self-similar processes (rssMp). In my recent paper, there were two ways of conditioning presented which gives the same result. Both of the methods consider the Lamperti-Kiu representation of the rssMp.

### 31st October 2016 - Jean-Dominique Deuschel (Technical University Berlin)

**Invariance principle for random walk in time-dependent balanced random environment**

We prove a quenched central limit theorem for balanced random walks in time dependent ergodic random environments. We assume that the environment satisfies appropriate ergodicity and ellipticity conditions. The proof is based on the use of a maximum principle for parabolic difference operators. We also discuss some non-elliptic environments and the related Harnack inequality.

(Joint work with N. Berger, X.Guo and A. Ramirez)

### 7th November 2016 - Julia Komjathy (Eindhoven Technical University)

**Explosive branching processes and their applications to epidemics and distances in power-law random graphs**

In this talk I will investigate the question of explosion of branching processes, i.e., when is it possible that a BP produces infinitely many offspring in finite time. Two important cases in terms of application are age-dependent BPs and BPs arising from epidemic models where individuals are only contagious in a possibly random interval after being infected. This imposes dependencies between the birth-time of the children of an individual.

The motivation for studying the explosiveness question is to understand weighted distances in locally tree-like random graphs, such as the configuration model, in the regime where the degree distribution is a power-law with exponent between (2,3). Here, the local neighborhood of a vertex and thus the initial stages of the spreading can be the approximated by an infinite mean offspring BP. I will explain the recent results on this area.

This part is joint work with Enrico Baroni and Remco van der Hofstad.

### 14th November 2016 - Luciano Campi (London School of Economics)

**N-player games and mean field games with absorption**

We consider a symmetric N-player game with weakly interacting diffusions and an absorbing set. We study the existence of Nash equilibria of the limiting mean-field game and establish, under a non-degeneracy condition of the diffusion coefficient, that the latter provides nearly optimal strategies for the N-player game. Moreover, we provide an example of a mean-field game with absorption whose Nash equilibrium is not a good approximation of the pre-limit game.

This talk is based on a joint work with Markus Fischer (Padua University).

### 24th November 2016 - Alessandra Caraceni (University of Bath - PIMS seminar)

**Self-Avoiding Walks on Random Quadrangulations**

The local limit of random quadrangulations (UIPQ) and the local limit of quadrangulations with a simple boundary (the simple boundary UIHPQ) are two very well studied objects. We shall see how the simple boundary UIHPQ relates to an annealed model of self-avoiding walks on random quadrangulations, and how metric information obtained for the UIHPQ enables us to quantify statistics such as the displacement of the self-avoiding walk from the origin, as well as how the biasing of random quadrangulations by the number of their self-avoiding walks affects their local limit.

### 28th November 2016 - David Croydon (University of Warwick)

**Scaling limits of stochastic processes via the resistance metric**

The connections between electricity and probability are deep, and have provided many tools for understanding the behaviour of stochastic processes. In this talk, I will describe a recent result in this direction, which states that if a sequence of spaces equipped with so-called "resistance metrics" and measures converge with respect to the Gromov-Hausdorff-vague topology, and a certain non-explosion condition is satisfied, then the associated stochastic processes also converge. This result generalises previous work on trees, fractals, and various models of random graphs. I further conjecture that it will be applicable to the random walk on the incipient infinite cluster of critical bond percolation on the high-dimensional integer lattice.

### 5th December 2016 - Florian Vollering (University of Bath)

**Fix points for random motions in dynamic random sceneries**

Consider a random motion on Z, given by a stationary sequence of increments. Along the path of this motion it observes values of a random scenery, which is evolving in time as well. This string of observations provides a new sequence of increments, which can be used to construct the law of a new random motion. Iterating this procedure, we find that for each law of the scenery process, there is a unique fix point. Furthermore, under fairly general conditions on the scenery, this fix point motion is super-diffusive with a scaling of t^{2/3} compared to the t^{1/2} scaling of Brownian motion.

### 8th December 2016 - Peter Gracar (University of Bath - PIMS seminar)

**Spread of information by random walks**

A conductance graph on Z^d is a nearest-neighbour graph where all of the edges have positive weights assigned to them. In this talk, I will consider the spread of information between particles performing continuous time simple random walks on a conductance graph. I do this by developing a general multi-scale percolation argument using Lipschitz surfaces that can also be used to answer other questions of this nature. Joint work with Alexandre Stauffer.

### 12th December 2016 - Cristina Toninelli (UPMC, Paris VI)

**Bootstrap percolation and kinetically constrained spin models: critical time and lengths scales**

Kinetically constrained spin models (KCSM) are a class of interacting particle systems with Glauber dynamics which are used by physicists to model liquid-glass transition. The key feature of KCSM is that a move occurs only if the configuration satisfies a local constraint requiring a minimal number of empty sites in a certain neighborhood. I will discuss the connection of KCSM to monotone cellular automata of bootstrap percolation type and recall some universality results for bootstrap percolation models in two dimensions.

I will then present our results which establish the critical growth of the random infection time of the origin for certain KCSM when the equilibrium density of empty sites goes to zero. I will conclude by discussing some universality conjectures for time scales for a generic choice of the constraints motivated by the bootstrap universality results.

### 9th January 2017 - Ellen Powell (University of Cambridge)

**Critical branching diffusions in bounded domains**

I will discuss branching diffusions in a bounded domain D in which particles are killed upon hitting the boundary. It is known that any such process undergoes a phase transition when the branching rate reaches a critical value: the first eigenvalue of the generator of the diffusion. I will consider various properties of the critical system, including the structure of the associated genealogical tree. When the system is conditioned to survive for a long time, it turns out that this tree converges to the Brownian CRT.

### 16th January 2017 - Clément Foucart (Paris 13)

**Continuous-state branching processes, extremal processes and super-individuals**

Consider a branching population model given by a flow of continuous-state branching processes (as Bertoin and Le Gall 2000). We characterize its long-term behaviour through subordinators and extremal processes. The extremal processes arise in the case of supercritical processes with infinite mean and of subcritical processes with infinite variation. The jumps of these extremal processes are interpreted as specific initial individuals whose progenies overwhelm the population. These individuals, which correspond to the records of a certain Poisson point process embedded in the flow, are called super-individuals. They radically increase the growth rate to infinity in the supercritical case, and slow down the rate of extinction in the subcritical one. This notion of super-individuals allows us in particular to recover the so-called Eve property (defined in Duquesne and Labbé 2014).

This is based on a joint work with Chunhua Ma (Nankai university).

### 2nd February 2017 - Amitai Linker (CMM / University of Bath - PIMS seminar)

**The contact process on evolving scale-free networks**

In this talk we study the contact process, commonly used to model the spread of disease, running on a class of evolving scale-free networks where each node updates its connections at independent random times, possibly depending on the degree of the node. The creation and destruction of stars,as we will see, generates a behaviour which is different from the one seen in static scale-free networks, increasing the rate at which the infection spreads and decreasing the time it spends on metastable states. As a result there is a phase transition between a phase where there is slow extinction for all infection rates, and a phase where there is quick extinction if the infection rate is small enough. Playing with the parameters of the model will also reveal different mechanisms used by the process to survive for a long time.

Joint work with Peter Mörters and Emmanuel Jacob.

### 6th February 2017 - Sarah Penington (University of Oxford)

**Branching Brownian motion, mean curvature flow and the motion of hybrid zones**

Hybrid zones are interfaces between populations which occur when two genetically distinct groups can interbreed, but hybrid offspring have a lower evolutionary fitness. We model their behaviour using the spatial Lambda-Fleming-Viot process. I will discuss a result on the motion of hybrid zones and also a related probabilistic proof of a known PDE result connecting the Allen-Cahn equation and mean curvature flow. The proofs rely on duality relations with a branching and coalescing random walk and a branching Brownian motion.

Joint work with Alison Etheridge and Nic Freeman.

### 20th February 2017 - Marcel Ortgiese (University of Bath - PIMS seminar)

**Local geometry of first passage percolation on random graphs**

First passage percolation is a simple model for the spread of a rumour on a network. We will answer the following question: Imagine that you can only see the graph structure up to a fixed depth around you. If you know that a rumour is started somewhere on a distant vertex of the graph, can you predict from where the rumour will reach you? In other words, we will study the local geometry of geodesics near a target vertex. To formalize this question we use a notion of local convergence based on the work of Benjamini and Schramm. Joint work with Steffen Dereich (Münster).

### 23rd February 2017 - Malwina Luczak (Queen Mary)

**SIR epidemics on graphs with given degrees**

We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model, infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove that there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak may occur. We prove that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. In contrast to earlier results for this model, our results only require basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex. (Joint work with Svante Janson and Peter Windridge.)

We also study the regime just above the threshold: we determine the probability that a large epidemic occurs and the size of a large epidemic. This is joint work with Thomas House, Svante Janson, Peter Windridge.

### 27th February 2017 - Roland Bauerschmidt (University of Cambridge)

**Local Kesten-McKay law for random regular graphs of fixed degree**

I will discuss recent results concerning the spectral properties of random regular graphs of large but fixed degree. We prove random matrix type estimates for the delocalisation of the eigenvectors and the concentration of the spectral measure at small scales. Our approach combines the almost deterministic tree-like structure of random regular graphs at small distances with intuition from random matrix theory for large distances. We estimate the Green's function by a resampling of the boundary edges of large balls in the graphs. This is joint work with J. Huang and H.-T. Yau.

### 6th March 2017 - Henry Panti (UADY - Yucatan)

**On the recurrent extensions of real self similar Markov processes**

In this talk, we obtain necessary and sufficient conditions for the existence of recurrent extensions of real self-similar Markov processes. In doing so, we solve an old problem originally posed by Lamperti for positive self-similar Markov processes. Our main result ensures that a real self-similar Markov process with a finite hitting time of the point zero has a recurrent extension that leaves 0 continuously if and only if the Markov Additive Process associated, via Lamperti transformation, satisfies the Cramér's condition. This is a joint work with Juan Carlos Pardo and Víctor Rivero.

### 13th March 2017 - Martin Keller-Ressel (TU Dresden)

**Generalized distance (multi-)covariance: Measuring stochastic dependency beyond correlation**

Distance covariance was introduced by Székely, Rizzo and Bakirov in 2007 as a measure of (non-linear) stochastic dependency between two random variables. On a finite sample space, distance covariance can be computed by forming the Euclidian distance matrices between all sample points of each variable and calculating the empirical covariance between these two distance matrices. We generalize distance covariance in two aspects: First, we show that Euclidian distance can be replaced by any distance defined through a non-degenerate negative definite function, such as Minkowski distances, while retaining most theoretical properties of distance covariance. Second, we generalize distance covariance to a measure of dependency between an arbitrary number of random variables. Here the distinction between pairwise (in)dependence and mutual (in)dependence plays a crucial role. In the accompanying mathematical theory, Lévy processes, characteristic functions and Gaussian processes are playing major roles. The talk is based on joint work with Björn Böttcher and Rene Schilling (both TU Dresden).

### 20th March 2017 - Perla Sousi (University of Cambridge)

**Random walk on dynamical percolation**

We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate μ, while at the same time a random walker moves on G at rate 1, but only along edges which are open. On the d-dimensional torus with side length n, when the bond parameter is subcritical, the mixing times for both the full system and the random walker were determined by Peres, Stauffer and Steif. I will talk about the supercritical case, which was left open, but can be analysed using evolving sets (joint work with Y. Peres and J. Steif).

### 30th March 2017 - Mathew Penrose (University of Bath - PIMS seminar)

**Note: this seminar is in 6W 1.2
Optimal cuts of random geometric graphs**

Given a `cloud' of n points sampled independently uniformly at random from a Euclidean domain D, one may form a geometric graph by connecting nearby points using a distance parameter r(n). We consider the problem of partitioning the cloud into two pieces to minimise the number of `cut edges' of this graph, subject to a penalty for an unbalanced partition. The optimal score is known as the Cheeger constant of the graph. We discuss convergence of the Cheeger constant (suitably rescaled) for large n with suitably chosen r(n), towards an analogous quantity defined for the original domain D.

### 3rd April 2017 - Neil O'Connell (University of Bristol)

**From Pitman’s 2M-X theorem to random polymers and integrable systems**

Pitman’s (1975) '2M-X' theorem relates one-dimensional Brownian motion to the three-dimensional Bessel process in a surprising way. It has many variations and generalizations which are closely related to representation theory and integrable systems, on one hand, and to random matrices and statistical physics, including the study of random polymers, on the other. This talk will be a survey.

### 27th April 2017 - Sam Johnston (University of Bath - PIMS seminar)

**The coalescent structure of continuous-time Galton-Watson trees**

Take a continuous-time Galton-Watson tree. If the system survives until a large time T, then choose k particles uniformly from those alive. What does the ancestral tree drawn out by these k particles look like? Some special cases are known but we give a more complete answer.

### 4th May 2017 - Pascal Maillard (Paris-Sud University)

**Note: this seminar is on a Thursday at 1.15pm
Branching Brownian motion with absorption at critical and near-critical drift**

I will review results obtained in the last years on (1D-) branching Brownian motion with drift towards and absorption at the origin. There is a minimal drift at which the system dies out almost surely; I will focus on the cases where the drift is at or just below this critical point. I will also present work in progress on the critical case (joint work with Julien Berestycki and Jason Schweinsberg).

### 8th May 2017 - Nikos Zygouras (University of Warwick)

**Scaling limits of disordered systems: disorder relevance and universality**

We consider statistical mechanics models defined on a lattice, in which disorder acts as an external random field (e.g. pinning models, directed polymers, random field Ising model). Such models are called disorder relevant, if arbitrarily weak disorder changes the qualitative properties of the model . Via a Lindeberg principle for multilinear polynomials we show that disorder relevance manifests itself through the existence of a disordered high-temperature limit for the partition function, which is given in terms of Wiener chaos and is model specific. When disorder becomes marginally relevant a fundamentally new structure emerges, which leads to a universal scaling limit for all different (currently directed polymer) models that fall in this class. A notable such representative is the two dimensional SHE with multipicative space-time white noise (which in the SPDE language is characterised as “critical”). In this case certain analogies with Gaussian Multiplicative Chaos and log-correlated Gaussian fields also appear.

Based on joint works with Francesco Caravenna and Rongfeng Sun.

### 15th May 2017 - Sunil Chhita (Durham University)

**The two-periodic Aztec diamond**

Simulations of uniformly random domino tilings of large Aztec diamonds give striking pictures due to the emergence of two macroscopic regions. These regions are often referred to as solid and liquid phases. A limiting curve separates these regions and interesting probabilistic features occur around this curve, which are related to random matrix theory. The two-periodic Aztec diamond features a third phase, often called the gas phase. In this talk, we introduce the model and discuss some of the asymptotic behavior at the liquid-gas boundary. This is based on joint works with Vincent Beffara (Grenoble), Kurt Johansson (Stockholm) and Benjamin Young (Oregon).

### 18th May 2017 - Benjamin Gess (Max Planck Institute Leipzig)

**Note: this seminar is on a Thursday at 1.15pm
Well-posedness by noise for scalar conservation laws**

In certain cases of (linear) partial differential equations random perturbations have been observed to cause regularizing effects, in some cases even producing the uniqueness of solutions. In view of the long-standing open problems of uniqueness of solutions for certain PDE arising in fluid dynamics such results are of particular interest. In this talk we will extend some known results concerning the well-posedness by noise for linear transport equations to the nonlinear case.

### 19th May 2017 - Vladas Sidoravicius (NYU Shanghai) - Landscapes colloquium

**Note: this seminar is on a Friday at 3.15pm
Self-interacting random walks - old questions, new surprises**

Simple random walks, random walks in random environments or random sceneries for nearly a century fascinated and challenged probabilists and physicists. I will focus on a very particular class of walks, namely self interacting walks, walks which remember their past trajectory, and interact with it in the future. It could be self-attracting, when the walker wants to stay close to where it was before, or it could be self-repelling, when the walker tries to move away from its own trajectory. What is the long term behaviour of such processes? It naturally depends on the type and strength of interaction. During the last three decades several deep and beautiful mathematical studies were done on this topic. However many questions are still open and are challenging mathematicians. During my talk I will explain some old and new results, which involve phase transitions, shape theorems, etc.

### 25th May 2017 - Francis Lane (University of Bath - PIMS seminar)

**The largest fragment of a homogeneous fragmentation process**

Fragmentation processes describe objects falling apart randomly as time passes. We study the size of the largest fragment, and identify the polynomial correction to its exponential rate of decay. The proof uses spines, changes of measure, and the close relationship between fragmentation processes and branching random walk.

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