Talks
Talks will take place in building L, Room L003
Eleanor ArcherQuenched critical percolation on Galton-Watson trees [pdf]
We consider critical percolation on a supercritical Galton-Watson tree with mean offspring \(m > 1\). It is well known that the critical percolation probability for this model is \(1/m\) and that the root cluster has the distribution of a critical Galton-Watson tree. For this reason, many properties of the cluster are well understood, such as asymptotic tail decay for long range survival probabilities, the size of the \(n\)-th generation conditioned on survival (the “Yaglom limit”), and convergence of the entire cluster to a branching process/stable tree. All of these results as stated are annealed, that is, we take the expectation with respect to the distribution of the tree and the percolation configuration simultaneously. The goal of this talk is to consider the quenched regime: are the same properties true for almost any realisation of the tree? We will see that this is indeed the case, although some scaling constants will depend on the tree. Based on joint works with Quirin Vogel and Tanguy Lions.
Theo AssiotisInteracting particle systems, conditioned random walks and the Aztec diamond [pdf]
I will talk about a general class of integrable models of interacting particles in inhomogeneous space, containing various types of inhomogeneous pushTASEPs and zero range processes, and how they are connected to determinantal point processes, random walks conditioned to never intersect and random tilings of the Aztec diamond with inhomogeneous weights. The integrability of these models comes from a natural generalisation of Toeplitz matrices which satisfy certain intertwining relations.
Théo BalluShape of a leaky sandpile model via a killed random walk [pdf]
The sanpile model was created by physicists in the 1980s to illustrate the phenomenon of self-organized criticality. It is a cellular automaton that simulates the behavior of a sandpile from a microscopic point of view. One of the main questions about the sandpile model is to describe the shape of the sandpile as the amount of sand goes to infinity. In a joint work with C. Boutillier, S. Mkrtchyan and K. Raschel, we studied an alternative sandpile model, called "leaky", in which a portion of the sand disappears each time sand flows. Using a killed random walk, we proved the convergence of the sandpile to a limit shape and described this limit shape in terms of the Laplace transform of the random walk.
Elisabetta CandelleroBoundary behavior of branching random walks [pdf]
Let \(G\) be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between t-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph \(U\) we study two aspects of branching random walks on \(U\): when the trajectories visit \(U\) infinitely often (survival) and when they stay inside \(U\) forever (persistence). We also provide some examples and counterexamples. (Based on joint works with T. Hutchcroft, D. Bertacchi and F. Zucca.)
Louis ChataignierAsymptotics of the overlap distribution of branching Brownian [pdf]
We are interested in a question about branching Brownian motion, coming from spin glass theory: Given some point \(\alpha\) between \(0\) and \(1\), if we pick two particles at time \(t\) according to the Gibbs measure at inverse temperature \(\beta\), what is the probability that their last common ancestor died after time \(\alpha t\)? We will focus on the subcritical regime where \(\beta^2<2\).
Guillaume Conchon-KerjanA phase transition for the tree-builder random walk [pdf]
The tree-builder random walk (TBRW) is a simple random walk which adds at each step a random (i.i.d.) number of neighbours to its current position. Thus, if starting from a single vertex, it builds recursively the tree on which it evolves. It has been shown that the walk is transient (even ballistic), regardless of the distribution \(D\) of the leaves added. To temper this strength, we add a constant bias \(r\) towards the initial vertex of the tree. We prove a phase transition depending on \(r\) and \(D\): the walk will be ballistic transient for \(r<1+2\mathbb{E}[D]\), and recurrent otherwise. The proofs rely on the branching structure of the local times of the walk at each vertex (after a given number of returns to the initial vertex), which we study via a well-suited urn process. Joint work with Arthur Blanc-Renaudie, Camille Cazaux, Tanguy Lions and Arvind Singh.
William Da SilvaSLE(6) on Liouville quantum gravity as a growth-fragmentation process [pdf]
We study the branching structure induced by a space-filling SLE(6) exploration of the quantum disc with matching parameter. We prove that it can be described as one of the growth-fragmentation processes introduced by Bertoin, Budd, Curien and Kortchemski in the context of planar maps. Importantly, our arguments are elementary, relying only on planar Brownian motion, and requiring no prior knowledge on LQG, once translated through the mating of trees. To this end, we develop new elements of excursion theory for cone excursions of Brownian motion and explore their connections to stable Lévy processes. This set of tools provides new elementary proofs of some of the key properties of the above SLE/LQG coupling. This talk is based on joint work with Ellen Powell (Durham) and Alex Watson (UCL).
Denis DenisovOrdered random walks and the Airy line ensemble [pdf]
I will consider continuous-time ordered random walks. In this model we condition random walks to
stay in the same order for all time using a Doob \(h\)-transform. I will talk
about the regime when the number of random walks grows slower than a certain power of
the number of random walk steps assuming a sufficiently general class of increment
distributions. In this setting I will discuss universality results, such as law of
large numbers, fluctuations of linear statistics and scaling limits of top particles,
where the model of ordered random walks shows behaviour similar to the model of
non-intersecting Brownian motions.
This talk is based on joint work with W. FitzGerald and Vitali Wachtel.
Arxiv preprint can be found at here.
Coexisting of branching populations [pdf]
Consider two one-dimensional branching populations \((Z^1_n, Z_n^2)\) in a joint random environment. Quenched distributions of \(Z^1_n\) and \(Z^2_n\) are assumed independent. Thus, the dependence between populations is caused by the environment only. We are interested in the asymptotic behaviour of coexisting probability \(\mathbb{P}(Z_n^1 > 0, Z_n^2 > 0)\). We are going to show that this problem is deeply connected to a two-dimensional random walk \(\hat{S}_n\) conditioned to stay in a cone. \(\hat{S}_n\) is the Doob \(h\)-transform of a random walk \(S_n\) having i.i.d. increments with zero mean and finite variance and killed at leaving the cone. For the process \(\hat{S}_n\) we estimate the probability of coming close to the boundary of the cone. This will give us upper and lower bounds for the coexistence probability.
Gabriel FlathNumber of particles at sublinear distances from the tip in branching Brownian motion [pdf]
Consider a branching Brownian motion (BBM). It is well known that the rightmost particle is located near \( m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t \). Let \( N(t,x) \) be the number of particles within distance \( x \) from \( m_t \), where \( x = o(t) \) grows with \( t \). We establish a convergence result for \( N(t,x) \) properly renormalized and note that, for \( x \leq t^{1/3} \), the convergence cannot be strengthened to an almost sure result. Moreover, the intermediate steps in our proof provide a path localization for the trajectories of particles in \( N(t,x) \) and their genealogy, allowing us to develop a picture of the BBM front at sublinear distances from \( m_t \). This suggests that the convergence result for \( N(t,x) \) could be strengthened to an almost sure result for \( x\geq\sqrt{t} \).
Clément FoucartConditioning the logistic continuous-state branching process on non-extinction via its total progeny [pdf]
The problem of conditioning a continuous-state branching process with quadratic competition (logistic CB process) on non-extinction is investigated. We first establish that non-extinction is equivalent to the total progeny of the population being infinite. The conditioning we propose is then designed by requiring the total progeny to exceed arbitrarily large exponential random variables. This is related to a Doob's \(h\)-transform with an explicit excessive function \(h\). The \(h\)-transformed process, i.e. the conditioned process, is shown to have a finite lifetime almost surely (it is either killed or it explodes continuously). When starting from positive values, the conditioned process is furthermore characterized, up to its lifetime, as the solution to a certain stochastic equation with jumps. The latter superposes the dynamics of the initial logistic CB process with an additional density-dependent immigration term. Last, it is established that the conditioned process can be starting from zero. Key tools employed are a representation of the logistic CB process through a time-changed generalized Ornstein-Uhlenbeck process, as well as Laplace and Siegmund duality relationships with auxiliary diffusion processes. This is a joint work with Victor Rivero and Anita Winter.
Félix Foutel-RodierSelf-similar superprocesses
In this presentation, I will introduce a class of superprocesses verifying a self-similarity condition. They correspond to the scaling limits of near-critical branching systems where particles perform independent self-similar motions. This class of objects includes a model for the introgression of genetic material under infinitesimal selection, which was our original motivation. These superprocesses display a surprisingly rich behaviour with two phase transitions and I will describe some of their asymptotic properties (survival probability, distribution of particle locations, genealogies). This is partly based on an ongoing work with Alison Etheridge (Oxford) and Nick Barton (ISTA).
Max HelmerPersistence for Spherical Fractional Brownian Motion [pdf]
We consider spherical fractional Brownian motion \((S_H(\eta))_{\eta\in\mathbb{S}_{d-1}}\), which is obtained by taking fractional Brownian motion indexed by the (multi-dimensional) sphere \(\mathbb{S}_{d-1}\), and calculate its persistence exponent. Persistence in this context is the study of the decay of the probability \( \mathbb{P}\left( \sup_{\eta \in \mathbb{S}_{d-1}} S_H(\eta) \leq \varepsilon \right) \) when the barrier \(\varepsilon \searrow 0\) becomes more and more restrictive. Our main result shows that the persistence probability of spherical fractional Brownian motion has the same order of polynomial decay as its Euclidean counterpart.
Nathalie KrellBranching processes and bacterial growth [pdf]
Through the modeling of bacterial growth, we explore the construction of a multi-type
branching process. We consider the growth of a cell population using a piecewise
deterministic Markov branching tree. In this model, each cell divides into two offspring
at a division rate \(B(x)\), which depends on its size \(x\). The size of each cell grows
exponentially over time, with an individual-specific growth rate.Building upon the model
studied in (M. Doumic, M. Hoffmann, N. Krell and L. Robert. Statistical estimation of a
growth-fragmentation model observed on a genealogical tree. Bernoulli, 21(3):1760–1799, 2015.),
we extend the framework to account for two types of bacteria: those with a young pole and
those with an old pole. We will show that the proposed branching process is rigorously
defined and satisfies a many-to-one formula. Additionally, we prove that the mean empirical
measure of this process satisfies a growth-fragmentation equation, where size, growth rate,
and type are used as state variables.
I will conclude by presenting ongoing work in collaboration with Benoîte de Saporta, Bertrand Cloez, and Tristan Roget.
Explosion speed of continuous state branching processes indexed by the Esscher transform [pdf]
A branching process \(Z\) is said to be non conservative if it hits \(\infty\) in a finite time with positive probability. It is well known that this happens if and only if the branching mechanism \(\varphi\) of \(Z\) satisfies \(\int_{0+}d\lambda/|\varphi(\lambda)|<\infty\). We construct on the same probability space a family of conservative continuous state branching processes \(Z^{(\varepsilon)}\), \(\varepsilon\ge0\), each process \(Z^{(\varepsilon)}\) having \(\varphi^{(\varepsilon)}(\lambda)=\varphi(\lambda+\varepsilon)-\varphi(\varepsilon)\) as branching mechanism, and such that the family \(Z^{(\varepsilon)}\), \(\varepsilon\ge0\) converges a.s. to \(Z\) as \(\varepsilon\rightarrow0\). Then we study the speed of convergence of \(Z^{(\varepsilon)}\), when \(\varepsilon\rightarrow0\), referred to here as the explosion speed. More specifically, we characterize the functions \(f\) with \(\lim_{\varepsilon\rightarrow0} f(\varepsilon)=\infty\) and such that the first passage times \(\sigma_\varepsilon=\inf\{t:Z^{(\varepsilon)}_t\ge f(\varepsilon)\}\) converge toward the explosion time of \(Z\). Necessary and sufficient conditions are obtained for the weak convergence and convergence in \(L^1\). Then we give a sufficient condition for the almost sure convergence. Work in collaboration with Loïc Chaumont.
Bastien MalleinA fragmentation toy-model for the perimeter of the convex hull of a Brownian motion in a disk [pdf]
Consider a Brownian motion in the unit disk, reflected orthogonally at its boundary.
This can be thought of as the trajectory of an animal exploring its territory, foraging for food.
To estimate the domain covered by this animal during its search for t units of time, one can
estimate the area, or the perimeter, of the convex hull of its positions. This question was
studied by De Bruyne et al, who conjectured a convergence rate \(\Theta(t^{1/4} e^{-2 t^{1/2}})\)
for these quantities, towards the area and perimeter of the unit disk respectively.
We introduce in this talk an inhomogeneous fragmentation process in which particles of mass
\(m \in (0, 1)\) split at a rate proportional to \(\frac{1}{|\log m|}\), as a toy-model
for the study of this speed of convergence. This process does not belong to the well-studied
family of self-similar fragmentation processes. Our main results characterize the Laplace
transform of the typical fragment of such a process, at any time, and its large time behavior.
Based on a joint work with Bénédicte Haas.
Critical branching processes with immigration: scaling limits of local extinction sets [pdf]
We establish the joint scaling limit of a Bienaymé-Galton-Watson process with immigration (BGWI) and its (counting) local time at zero to the corresponding self-similar continuous-state branching process with immigration (CBI) and its (Markovian) local time at zero. We do this by analysing the structure of excursions from zero and from positive levels. The convergence of the hitting times of the BGWI to those of the CBI plays a key role in our proof. This is joint work with G. Uribe Bravo and B. Povar.
Da Cam PhamA local limit theorem for lattice oscillating random walks [pdf]
In this paper, we obtain a local limit theorem for the Kemperman's model of oscillating random walk on \(\mathbb{Z}\); it extends the existing results for ordinary random walks on \(\mathbb Z\) or reflected random walks on \(\mathbb N_0\). The key technical point here is controlling the long-term behavior of the embedding subprocess that characterizes the oscillations of the original random walk between \(\mathbb Z^-\) and \(\mathbb Z^+\) in both recurrent and transient cases.
Benjamin PovarContinuous Branching process with Immigration conditioned to stay positive [pdf]
Consider a (discrete time) Bienaymé-Galton-Watson process with Immigration (BGWI) such that it has a non-trivial scaling limit. It is known that the limit must be a self-similar Continuous Branching process with Immigration (CBI). We study the behaviour of the BGWI conditioned on non-extinction up to some large time, and derive the so-called Yaglom limit. We are able to extend it to the whole trajectory using the results of Durrett, hence constructing a Continuous Branching process with Immigration (CBI) conditioned to stay positive for a given amount of time. As a by-product of this, letting the amount of time in the conditioning go to infinity, we produce a self-similar CBI conditioned to stay positive (forever). We show that the conditioned CBI is a Doob \(h\)-transform of the original CBI and give the function \(h\) explicitly. Finally, through studying the Lévy process governing the Lamperti transform of the self-similar CBI, we verify that \(h\) applied to the killed CBI is a martingale. This allows us to extract the behaviour of tails of excursions, as well as to contruct the scale function of the CBI.
Stjepan ŠebekIterated-logarithm laws for convex hulls of random walks with drift
We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our starting point is a version of Strassen’s functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest.
Aleksandr TarasovBerry-Esseen inequality and reflection principle
We consider a one-dimensional random walk conditioned to stay positive. It is well known that, under the assumption of zero-mean and finite-variance increments, the distribution converges to the Rayleigh distribution. In this talk, we discuss the key ideas, including a generalized reflection principle, that allow us to derive a Berry-Esseen type estimate with rate of convergence of order \(n^{-1/2}\).
Vitali WachtelLower deviations for critical branching processes with immigration
Let \(\{Y_{n}\), \(n \geq 1\}\) be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation probabilities for \(Y_{n}\). More precisely, assuming that \(k,n \to \infty\) such that \(k=o\left(n \right)\), we investigate the asymptotics of \(\mathbf{P}\left(Y_{n} \leq k \right)\) and \(\mathbf{P}\left(Y_{n} = k \right)\).
