# Seminars 2014/15

## Abstracts

### 6th October 2014 - Agelos Georgakopoulos (University of Warwick)

**Group walk random graphs**

I will discuss a new construction of finite random graphs motivated by the study of random walks on infinite groups, and show connections to the Poisson boundary and Sznitman's random interlacements.

### 13th October 2014 - Rob van den Berg (CWI, Amsterdam)

**Disjoint-occurrence inequalities
**Consider a finite sequence of independent 0-1 valued random variables. In the early eighties, Kesten and I proved, for a special class (important in e.g. percolation) of events A and B, that the probability that A and B 'occur disjointly' is at most the product of the probability of A and the probability of B. Our conjecture that the inequality holds for all events was proved by Reimer in the mid-nineties.

A few years ago, in joint work with Jonasson and joint work with Gandolfi, we obtained extensions to certain classes of dependent random variables. In spite of this progress, I think that this is still a relatively small part of a yet to be discovered much more general framework. In this talk I will review the above mentioned notions and results, and discuss some specific open problems and ongoing work.

### 20th October 2014 - Beatrice Acciaio (London School of Economics)

**Arbitrage of the first kind and filtration enlargements**

Let a financial market be given where no arbitrage profits are possible, and let there be an agent with additional information with respect to it. In this framework, I investigate whether the extra information can generate arbitrage profits. I will first justify why the right concept of arbitrage to consider here is the so-called Arbitrage of the First Kind (or, equivalently, Unbounded Profit with Bounded Risk). Then I will illustrate a simple and general condition ensuring that no arbitrage is available to the informed agent either. The preservation of No-Arbitrage under additional information is shown for a general semimartingale model both when this information is disclosed progressively in time and when it is fully added at the initial time. In addition, I will provide a characterization of such a stability in a robust context, that is, for all possible semimartingale models.

This talk is based on a joint work with C. Fontana and K. Kardaras.

### 27th October 2014 - Frank Redig (Delft University of Technology)

**Duality theory via examples**

I will explain our recently developed Lie algebraic approach to duality and how processes such as the SEP (symmetric exclusion process), KMP (Kipnis, Marchioro Presutti model), and SIP (symmetric inclusion process) naturally arrize in the framework of this formalism. By going from classical Lie algebras to their q-deformed counterparts, we obtain new asymmetric processes such as an ASEP with k particles per site, the ASIP and a diffusion process of Wright Fisher type with selection. These processes have by construction dualities and self-dualities which can be exploited e.g. to compute exponential moments of the current.

Based on joint work with G. Carinci, C. Giardina (Modena), T. Sasamoto (Tokyo).

### 3rd November 2014 - Dirk Erhard (University of Warwick)

**The parabolic Anderson model in a dynamic random environment: random conductances
**The parabolic Anderson model is a differential equation, which describes the evolution of a field of particles performing independent nearest neighbor simple random walks with binary branching: particles jump at rate 2dK, K > 0, split into two and die at rates determined by the environment. We denote by u(x,t) the mean number of particles at site x at time t conditioned on the evolution of the environment. My main object of interest is the exponential growth rate of the solution. In this talk I will review some results concerning its qualitative behaviour as a function of K and I will compare this exponential growth rate to the case in which the rate of jump of the particles from x to y is given by 2dK(x,y), for a random field of strictly positive conductances K(x,y) for x,y in Z^d.

This is joint work in progress with Frank den Hollander (Leiden) and Gregory Maillard (Marseille).

### 10th November 2014 - Emmanuel Jacob (University of Bath)

**Convergence of uniform planar maps with n edges to the Brownian map
**Random planar maps are natural models of random discrete geometries. A general aim is to establish convergence results of properly rescaled large maps, in the sense of Gromov-Hausdorff topology, to a random compact continuous metric space called the Brownian map. The convergence has been fully established only recently, independently by Miermont and Le Gall, but only for a limited number of models, including uniform triangulations and 2p-angulations. We will establish such a convergence result for uniform planar maps with n edges, using the recently discovered Ambjorn-Budd bijection. This bijection is in some sense the dual of the Cori-Vauquelin-Schaeffer bijection, and allows to couple directly a uniform planar map with n edges with a uniform quadrangulation with n faces, without changing too much the topology of the map.

This is joint work with Jérémie Bettinelli and Grégory Miermont.

### 17th November 2014 - Cyril Labbe (University of Warwick)

**The Eve property for CSBP
**A continuous state branching process (CSBP) can be seen as the evolving size of a continuous population. There is a natural notion of genealogy associated to this process, which allows one to decompose, at any given time, the whole population into families of individuals that share a common ancestor at time 0. We are interested in the asymptotic behaviour of the relative sizes of these families, in particular we will say that the Eve property holds if as time goes to infinity one family becomes overwhelming. Our main result is a complete classification of all the possible behaviours according to the branching mechanism. Joint work with Thomas Duquesne.

### 24th November 2014 - Marc Lelarge (INRIA-ENS Paris)

**Community detection in the stochastic block model via spectral methods
**Community detection consists in identification of groups of similar items within a population. In the context of online social networks, it is a useful primitive for recommending either contacts or news items to users. We will consider a particular generative probabilistic model for the observations, namely the so-called stochastic block model, and generalizations thereof. We will describe spectral transformations and associated clustering schemes for partitioning objects into distinct groups. Exploiting results on the spectrum of random graphs, we will establish consistency of these approaches under suitable assumptions, namely presence of a sufficiently strong signal in the observed data.

This is joint work with Charles Bordenave, Laurent Massoulie and Jiaming Xu.

### 1st December 2014 - Simon Griffiths (University of Oxford)

**Explosion and linear transit times in trees
**Let T be an infinite rooted tree with weights w_e assigned to its edges, and write m_n(T) for the minimum weight of a path from the root to a node of the nth generation. We call T is explosive if

lim_{n\to \infty} m_n(T) < \infty,

and say that T exhibits linear growth if

liminf_{n\to \infty} m_n(T)/n > 0.

Using a result of Kingman, we classify which trees in a class of generalised Poisson-weighted infinite trees exhibit linear growth. This result may be applied to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.

Based on joint work with Omid Amini, Luc Devroye and Neil Olver.

### 8th December 2014 - Alejandro Ramirez (PUC Chile)

**Quenched central limit theorem for random walk in ergodic space-time environment
**We prove a quenched central limit theorem for random walk in space-time ergodic random environment.

This is a joint work with Xiaoqin Guo and Jean-Dominique Deuschel.

### 12th January 2015 - Alain-Sol Sznitman (ETH Zürich)

**Disconnection for random walk and random interlacements
**How costly is it for the simple random walk or for random interlacements to disconnect a large box from the boundary of a larger concentric box, in dimension d ≥ 3 ? In this talk we will recall some facts about random interlacements and their percolative properties, and describe some recent progresses in the understanding of the above mentioned questions.

### 19th January 2015 - Sayan Banerjee (University of Warwick)

**Maximal couplings and geometry**

Maximal couplings are couplings of Markov processes where the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian couplings are coupling strategies where neither process is allowed to look into the future of the other before making the next transition. These are easier to describe and play a fundamental role in many branches of probability and analysis. Hsu and Sturm proved that the reflection coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada proved that to have a MMC for Brownian motions on a Riemannian manifold, the manifold should have a reflection structure, and thus proved the first result connecting this purely probabilistic phenomenon (MMC) to the geometry of the underlying space.

In this work, we investigate general elliptic diffusions on Riemannian manifolds, and show how the geometry (dimension of the isometry group and flows of isometries) plays a fundamental role in classifying the space and the generator of the diffusion for which an MMC exists. We also describe these diffusions in terms of Killing vector fields (generators of rigid motions on manifolds) and dilation vector fields around a point.

This is joint work with W.S. Kendall.

### 22nd January 2015 - Remco van der Hofstad (TU Eindhoven)

**Scale-free percolation - Note that this seminar is on a Thursday!
**We propose and study a random graph model on the hypercubic lattice that interpolates between models of scale-free random graphs and long-range percolation. In our model, each vertex x has a weight W_x, where the weights of different vertices are i.i.d. random variables. Given the weights, the edge between x and y is, independently of all other edges, occupied with probability 1-e^{-\lambda W_xW_y/|x-y|^{\alpha}}, where

(a) \lambda is the percolation parameter,

(b) |x-y| is the Euclidean distance between x and y, and

(c) \alpha is a long-range parameter.

The most interesting behavior can be observed when the random weights have a power-law distribution, i.e., when P(W_x>w) is regularly varying with exponent 1-\tau for some \tau>1. In this case, we see that the degrees are infinite a.s. when \gamma =\alpha(\tau-1)/d <1, while the degrees have a power-law distribution with exponent \gamma when \gamma>1.

Our main results describe phase transitions in the positivity of the critical value and in the graph distances in the percolation cluster as \gamma varies. Let \lambda_c denote the critical value of the model. Then, we show that \lambda_c=0 when \gamma<2, while \lambda_c>0 when \gamma>2.

Further, conditionally on 0 and x being connected, the graph distance between 0 and x is of order \log\log|x| when \gamma<2 and at least of order \log|x| when \gamma>2. These results are similar to the ones in inhomogeneous random graphs, where a wealth of further results is known. We also discuss many open problems, inspired both by recent work on long-range percolation (i.e., W_x=1 for every x), and on inhomogeneous random graphs (i.e., the model on the complete graph of size n and where |x-y|=n for every x\neq y).

[This is joint work with Mia Deijfen and Gerard Hooghiemstra.]

### 2nd January 2015 - Nina Gantert (TU Munich)

**A branching random walk among disasters
**Inspired by the "Random walk in a disastrous random environment" model which was introduced by Tokuzo Shiga in 1997, we consider a branching random walk among disasters. We discuss survival and extinction. In particular, the branching random walk dies out in the critical cases. We then show how to deduce new results for the single particle case from the statements about the branching random walk.

The talk is based on joint work (in progress) with Stefan Junk.

### 9th February 2015 - Louigi Addario-Berry (McGill) - *Leverhulme lecture*

**Theta(t^{1/3}) slowdown for branching Brownian motion with decay of mass
**Consider a branching Brownian motion particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles.

One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance Theta(t^{1/3}) behind the typical BBM front.

Many natural questions about the model remain open.

### 16th February 2015 - Ed Crane (University of Bristol)

**Random trees, forest fires and explosions**

The mean field forest fire model of Rath and Toth is a modification of the Erdos-Renyi random graph process. Starting with n isolated vertices, each possible edge appears independently at rate 1/n. In addition, each vertex is independently struck by lightning at a rate that depends on n. When a vertex is struck by lightning, all the edges in its cluster are removed. In the limiting regime where the lightning rate per vertex tends to zero but the overall lightning rate tends to infinity, Rath and Toth showed that the system has a global limit that remains critical after its gelation time. In joint work with Freeman and Toth (arXiv:1405.5044) we identify the local limit of this model, describing the limit of the cluster of a tagged vertex as a process. I will describe this work and also explain how the Brownian continuum random tree arises as the scaling limit of the large clusters in a stationary state of the limit process.

### 23rd March 2015 - Victor Rivero (CIMAT)

**Local limit theorems for first passage times of subordinators
**A subordinator is a process with independent and stationary increments with non-decreasing paths, they appear naturally as the inverse of local times. In this talk we will discuss some recent results about the asymptotic behaviour of the distribution of the first passage time or local time of a subordinator in the domain of attraction of a stable distribution, either at zero or infinity. We will provide estimates for the probability that a subordinator passes above a level x in the time interval (t,t+h] and explain how the estimates depend on the event where the level is crossed continuously or by a jump. The estimates obtained are uniform in h and in x in some frameworks.

This is based in a joint work with Ron Doney.

### 2nd March 2015 - Marcelo Hilario (UFMG, Brazil)

**Some percolation processes with infinite-range dependencies
**Consider the hyper-cubic lattice and remove the lines parallel to the coordinate axis independently at random. What are the properties of the set of remaining vertices? Does this model undergoes a sharp phase transition as the probability of removing the lines vary? How many connected components are there? What if we remove cylinders from the Euclidian space in a isometry invariant way?

In this talk we discuss some of these questions. We also discuss for Bernoulli percolation processes in the square lattice, how enhancing the parameter in a set of vertical lines chosen uniformly at random changes the critical point.

### 9th March 2015 - Jason Schweinsberg (UC San Diego)

**Rigorous results for a population model with selection
**We consider a model of a population of fixed size N in which each individual acquires beneficial mutations at rate \mu. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. We obtain rigorous results for the rate at which mutations accumulate in the population, the distribution of the fitnesses of individuals in the population at a given time, and the genealogy of the population. Our results confirm predictions of Desai and Fisher (2007), Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).

### 23rd March 2015 - Omer Angel (UBC, Canada)

**Unimodular planar maps
**We study random hyperbolic planar graphs by using their circle packing embedding to connect their geometry to that of the hyperbolic plane. This leads to several results: Identification of the Poisson and geometric boundaries, a connection between hyperbolicity and a form of non-amenability, and a new proof of the Benjamini-Schramm recurrence result. Based on works with subsets of Martin Barlow, Ori Gurel-Gurevich, Tom Hutchcroft, Asaf Nachmias and Gourab Ray.

### 13th April 2015 - Noah Forman (University of Oxford)

**The quantile rearrangement of random walk and Brownian motion increments
**From a simple random walk one may obtain a random permutation of indices [1,n] via the lexicographic ordering first on the value of the walk at a given time, and second on the time itself. We demonstrate that by rearranging the increments of a random walk bridge according to this quantile permutation, we obtain a Dyck path. Passing to a Brownian limit gives a novel proof and a generalization of a theorem of Jeulin (1985) describing Brownian local times as a time-changed Brownian excursion.

### 27th April 2015 - Firas Rassoul-Agha (University of Utah)

**The growth model: Busemann functions, shape, geodesics, and other stories
**We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppalainen.

### 11th May 2015 - Ioan Manolescu (Geneva)

**Scaling limits and influence of the seed graph in preferential attachment trees
**We investigate two aspects of large random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees. Starting with a given tree (called the seed), a random sequence of trees is built by adding vertices one by one, connecting them to one of the existing vertices chosen randomly with probability proportional to its degree.

Bubeck, Mossel and Racz conjectured that the law of the trees obtained after adding a large number of vertices still carries information about the seed from which the process started. We confirm this conjecture using an observable based on the number of ways of embedding a given (small) tree in a large tree obtained by preferential attachment.

Next we study scaling limits of such trees. Since the degrees of vertices of a large preferential attachment tree are much higher than its diameter, a simple scaling limit would lead to a non locally compact space that fails to capture the structure of the object. Yet, for a planar version of the model, a much more convenient limit may be defined via its loop tree. The limit is a new object called the Brownian tree, obtained from the CRT by a series of quotients.

Based on join work with Nicolas Curien, Thomas Duquesne and Igor Kortchemski.

### 18th May 2015 - Krzysztof Burdzy (University of Washington)

**Twin peaks
**I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to estimate the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks.

Joint work with Sara Billey, Soumik Pal, Lerna Pehlivan and Bruce Sagan.

### 8th June 2015 - Takashi Kumagai (Kyoto University)

**Heat kernel estimates and local CLT for random walk among random conductances with a power-law tail near zero
**We study on-diagonal heat kernel estimates and exit time estimates for continuous time random walks (CTRWs) among i.i.d. random conductances with a power-law tail near zero. For two types of natural CTRWs, we give optimal exponents of the tail such that the behaviors are ‘standard’ (i.e. similar to the random walk on the Euclidean space) above the exponents. We then establish the local CLT for the CTRWs. We will also compare our results to the recent results by Andres-Deuschel-Slowik.