Overview Accommodation and travel Participants Talks Programme

**Branching Structures**

The fourth Bath-Paris meeting

Institut Henri Poincaré, 27-29 June 2016

**Talks**

The following people will give talks:

Vincent Bansaye - **Queueing for an infinite bus line and aging branching process**
We study a queueing system with Poisson arrivals on a bus line indexed by integers. The buses move at constant speed to the right and the time of service per customer getting on the bus is fixed. The customers arriving at station i wait for a bus if this latter is at most at d_i stations before, where d_i is non-decreasing. We determine the asymptotic behavior of a single bus and when two buses eventually merge almost surely by coupling arguments.

The results rely on a connection with aged structured branching processes with immigration and varying environment, for which we need to prove a Kesten-Stigum type theorem, i.e. the a.s. convergence of the successive size of the branching process normalized by its mean. The techniques combine a spine approach for multitype branching processes in varying environment and geometric ergodicity along the spine to control the increments of the normalized process.

Nicolas Broutin - **Scaling limits for inhomogeneous random graphs**

Xinxin Chen - **Range and critical generations of a random walk on Galton-Watson trees**
We consider a randomly biased random walk on a supercritical Galton-Watson tree and focus on the boundary case for the underlying branching potential. We are interested in the (sub)-graph explored by the walker up to time n. On the one hand, Andreoletti and Debs showed that a.s. the first C_1(log n) generations are entirely visited by the walker up to time n. On the other hand, Faraud, Hu and Shi showed that the longest branch that the walker visited is a.s. of length C_2(log n)^3. We want to know more about the intermediate generations, so we begin with the number of visited sites at one such generation. In particular, we count this number at generations of order (log n)^2 which we call critical generations. We show that this number is asymptotically of order n(log n)^3 in probability and also that the range of this walk, mainly from the visited sites of these critical generations, is asymptotically of order n(log n) .

Caroline Colijn - **Understanding disease outbreaks from the shapes of trees**
We use the Crump-Mode-Jagers process as a framework to model outbreaks of an infectious disease. Using a new characteristic function, we relate the basic reproduction number of an outbreak to the number of cherry shapes in binary branching trees. We use this relationship to infer the basic reproduction number, first showing that the approach works on simulated data and then applying it to data on Salmonella typhi and influenza. We describe a method to infer the number of cherries without reconstructing the entire tree.

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

Nina Gantert - **A branching random walk among disasters**
We consider a branching random walk in a random space-time environment of disasters where each particle is killed when meeting a disaster. This is a generalization of the "Random walk in a disastrous random
environment" introduced by Tokuzo Shiga. We give a criterion for positive survival probability. The proofs for the subcritical and the supercritical cases follow standard arguments, which involve branching processes in random environment in the supercritical case. The proof of almost sure extinction in the critical case is more difficult and uses techniques which Olivier Garet and Regine Marchand applied for branching random walk in random environment. We also show that, in the case of survival, the number of particles grows exponentially fast. This is joint work with Stefan Junk.

Benedicte Haas - **Random trees constructed by aggregation**
We study a general procedure that builds random continuous trees by attaching recursively a new branch on a uniform point of the pre-existing tree. This encompasses the famous "line-breaking" construction of the Brownian tree of Aldous. Our aim is to see how the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness, height and Hausdorff dimension. This talk is partly based on a joint work with Nicolas Curien (Université Paris-Sud).

Ben Hambly - **A simple probabilistic model for interfaces in a martensitic phase transition**
A martensitic phase-transformation for a material is a first-order diffusionless transition involving a change of shape of the underlying crystal lattice. In the transition there is a symmetry breaking leading to the formation of different variants with interfaces between them and the original phase. We will consider a simple fragmentation model for the patterns that arise from this phase transition. We can encode the model using a general branching random walk (GBRW) and develop some new results for the GBRW to determine the growth rates for the proportion of interfaces which are of a certain size after a certain time. We calculate explicit descriptions of the interface asymptotics and determine a power law exponent.

Andreas Kyprianou - **Terrorists never congregate in even numbers**
We analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the large-scale limit. Moreover, we discover that, in the limit of small fragmentation rate, these processes exhibit a universal heavy tailed distribution with exponent 3/2. In addition, we observe a strange phenomenon that if coalescence of clusters always involves 3 or more blocks, then the large-scale limit has no even sided blocks. Some complementary results are also presented for exchangeable fragmentation-coalescence processes on partitions of natural numbers. In this case one may work directly with the infinite system and we ask whether the process can come down from infinity. The answer reveals a remarkable dichotomy.

Cyril Labbé - **Weakly asymmetric bridges and the KPZ equation**
Consider the simple exclusion process with N particles on 2N sites, with zero-flux boundary condition. I will present a classification of the static and dynamic behaviour of this model according to the asymmetry imposed to the jump rates. In particular, there is a precise regime of asymmetry for which the fluctuations around the hydrodynamic limit converge to the solution of the KPZ equation on the line, but truncated to a finite interval of time.

Zenghu Li - **Asymptotic results for exponential functionals of Levy processes**
The long time asymptotic behavior of the expectation of some exponential functional of a Levy process is studied. We give not only the exact convergence rate but also explicitly the limiting coefficients. The key of the results is the observation that the asymptotics only depends on the sample paths of the Levy process with local infimum decreasing slowly. This makes it possible for us to determine the limiting coefficients by extending the conditional limit theorems for Levy processes established by Hirano (2001). The constants are represented in terms of some transformations based on the renewal functions. As applications of the results, we give exact evaluation of the decay rate of the survival probability of a continuous-state branching process in random environment with stable branching mechanism.

Pascal Maillard - **Choices and Intervals : about a certain interval fragmentation with dependence between the intervals**
Consider the interval fragmentation where at each time step, two intervals are chosen randomly with probability proportional to their size and only the largest (or, smallest) is randomly split into two. With Elliot Paquette, we proved (for a very general version of this process) that the empirical distribution of the rescaled interval sizes converges almost surely to a deterministic limit. I will present this result together with a glimpse of a work in progress where we show moreover that the empirical distribution of the splitting points converges almost surely to the uniform distribution on the unit interval.

Cecile Mailler - **Non-extensive condensation in reinforced branching processes**
In this joint work with Steffen Dereich and Peter Mörters, we introduce and study a branching process with both a reinforcement and a mutation dynamics. Our model incompasses as a particular case the
Bianconi and Barabási's preferential attachment graph with fitnesses but also a selection and mutation population branching process. We prove that this model exhibits, under some conditions on the parameters, a condensation phenomenon, and more precisely that "the winner does not take it all", disproving a claim made in the physics literature about the Bianconi and Barabási's model.

Bastien Mallein - **Branching random walk in random environment**
A branching random walk in random environment on the real line evolves as follows. At each generation, a new point process law is chosen at random. Every individual currently alive dies, giving birth to children that are positioned around their parent according to i.i.d. version of this point process. We study the asymptotic behavior of the position of the rightmost individual at time n in this process.

Sarah Penington - **The front location in Branching Brownian motion with decay of mass**
We add a competitive interaction between nearby particles in a branching Brownian motion (BBM). Each particle has a mass, which decays at rate proportional to the local mass density at its location. The total mass increases through branching events. In standard BBM, we may define the front location at time t as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from θ(1) to o(1). We can show that in a weak sense this front is ~ c t^{1/3} behind the front for standard BBM.

This is joint work with Louigi Addario-Berry.

Ellen Powell - **Critical behaviour of branching diffusions in bounded domains**
We study branching diffusions in bounded domains in which particles are killed upon hitting the boundary. It was proved by Sevast'yanov and Watanabe that any such process undergoes a phase transition when the branching rate exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality, and prove an asymptotic for the probability of survival up to large time t. This is of the form c(x)/t where c(x) is a constant depending only on the starting position, and is proportional to the first eigenfunction of the generator. Using this asymptotic, we study the system conditioned on survival for a long time. In particular, we obtain a Yaglom type limit theorem for the positions of particles given survival. The result holds under only a C1 assumption on the domain, and is valid for arbitrary branching mechanisms and diffusions satisfying mild assumptions.

Loïc de Raphélis - **Scaling limit of the range of a random walk in random environment**
We consider a nearest-neighbour random walk on a Galton-Watson tree in random environment. We are interested in the range of this walk, that is the set of vertices visited by the walk. As time goes on, this set grows up, and we will see that when correctly re-normalised it converges towards a real tree. Moreover, we will see that depending on the reproduction law of the underlying Galton-Watson tree, different regimes may occur. Partly joint work with Elie Aïdékon.

Yanxia Ren - **L log L criterion for a class of multitype superdiffusions with nonlocal branching mechanism**
Suppose that \{Z_n, n\ge 1\} is a Galton-Watson branching process
with each particle having probability p_n of giving birth to n
children. Let L stand for a random variable with this offspring
distribution. Let m:=\sum^{\infty}_{n=1}n p_n be the mean number of
children per particle. Then Z_n/m^n is a non-negative martingale.
Let W be the limit of Z_n/m^n as n\to\infty. Kesten and Stigum
proved that if 1

This result is usually referred to the Kesten-Stigum L\log L
theorem.

In 1995, Lyons, Pemantle and Peres developed a martingale change of
measure method to give a new proof for the
Kesten-Stigum L\log L theorem for single type branching processes.
Later this method was extended to prove the L\log L theorem for
multiple and general multiple type branching processes.

My former works with Liu and Song extended this method to a class of branching Hunt processes and
superdiffusions and establish a L\log L criterion for Branching Hunt processes and
superdiffusions. In this talk, I will give the L\log L theorem for multitype superdiffusions with nonlocal branching mechanism. The proof of the theorem is based on a spine decomposition of the superdiffusion with nonlocal branching mechanism under a martingale change of measure.

The talk is based on a working paper with Zhen-Qing Chen and Renming Song.

Zhan Shi - **Branching, selection and deviations**
I am going to make some elementary discussions on deviation properties of one-dimensional branching Brownian motion with competition. Joint work with Bernard Derrida.

Alex Watson - **Fragmentation with growth**
In models of fragmentation with growth, one has a number of independent cells, each of which grows continuously in time until a fragmentation event occurs, at which point the cell splits into two or more child cells of a smaller mass. Each of the children is independent and behaves in the same way as its parent. The rate of fragmentation may be infinite, and fragmentation may be homogeneous (where the rate does not depend on the mass of the cell) or self-similar (where the rate is a power of the mass). We will discuss some recent results, with a focus on solutions of the growth-fragmentation equation from a probabilistic perspective.