Seminars this semester

Oct 29	Thu	Mathew Joseph (Sheffield)	Probability seminar
15:00		Longest increasing path within the critical strip
		LT 7
	Abstract: Consider the square $[0,n]^2$ with points from a Poisson point process of intensity 1 distributed within it. In a seminal work, Baik, Deift and Johansson proved that the number of points $L_n$ (length) on a maximal increasing path (an increasing path that contains the most number of points), when properly centered and scaled, converges to the Tracy-Widom distribution. Later Johansson showed that all maximal paths lie within the strip of width $n^{\frac{2}{3} +\epsilon}$ around the diagonal with probability tending to 1 as $n \to \infty$. We shall discuss recent work on the Gaussian behaviour of the length $L_n^{(\gamma)}$ of a maximal increasing path restricted to lie within a strip of width $n^{\gamma}, \gamma< \frac{2}{3}$.
Nov 5	Thu	Mohammud Foondun (Loughborough)	Probability seminar
15:00		Approximations of a class of stochastic heat equations
		LT 7
	Abstract: In this talk, we are going to explore some results about approximations of a class of stochastic heat equations. To be more precise, we will indicate how one can approximate stochastic heat equation by infinitely dimensional interacting SDEs. From this, a lot of interesting properties of the heat equation follow
Nov 19	Thu	Christian Andres Fonseca Mora (Sheffield)	Probability seminar
15:00		Stochastic integration with respect to Lévy processes in the dual of a nuclear space
		LT 7
	Abstract: Apart from Banach spaces, nuclear spaces and its strong duals are the most important classes of infinite dimensional spaces used in functional analysis. Among the most important examples are the classical spaces used in the theory of distributions $\mathcal{S}(\mathbb{R}^{d})$, $\mathcal{S}'(\mathbb{R}^{d})$, $\mathcal{D}(\mathbb{R}^{d})$ and $\mathcal{D}'(\mathbb{R}^{d})$. Motivated by the study of stochastic partial differential equations, in this talk we introduce a theory of stochastic integration for operator-valued processes taking values in the dual of a nuclear space with respect to some classes of martingale-valued measures. The stochastic integral is defined using some novel techniques. In particular, this theory allow us to define stochastic integrals with respect to Lévy processes via the Lévy-Itô decomposition. No prior knowledge on stochastic analysis in infinite dimensional spaces is assumed and all the necessary background will be provided within the talk.
Feb 11	Thu	Sarah Penington (Oxford)	Probability seminar
15:00		The front location in Branching Brownian motion with decay of mass
		Hicks Seminar Room J11
	Abstract: We add a competitive interaction between nearby particles in a branching Brownian motion (BBM). Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles' masses decay. In standard BBM, we may define the front displacement 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 O(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. Joint work with Louigi Addario-Berry.
Feb 25	Thu	Markus Riedle (Kings College London)	Probability seminar
15:00		Stochastic integration with respect to cylindrical Lévy processes
		Hicks Seminar Room J11
	Abstract: Cylindrical Lévy processes are a natural generalisation of cylindrical Wiener processes and Gaussian white noise. However, since a cylindrical Lévy process does not enjoy a cylindrical version of the semi-martingale decomposition, one cannot apply one of the standard approaches to define stochastic integrals with respect to cylindrical Lévy processes. In this talk, we will introduce a completely novel approach to stochastic integration. In this approach the integrator is not decomposed into a martingale and a bounded variation process. As a consequence, the sequence of stochastic integrals for simple integrands can only be considered as a sequence in the space $L^0$ of Hilbert space valued random variables. Convergence is established by tightness arguments utilising an approach called decoupled tangent sequences. This talk is based on a joint work with Adam Jakubowski.
Mar 10	Thu	Thomas Cass (Imperial College London)	Probability seminar
15:00		Gaussian rough path analysis, a Stratonovich-to-Skorohod conversion formula
		Hicks Seminar Room J11
	Abstract: Rough path theory has made important recent strides in the understanding of nonlinear differential systems driven by non-semimartingale stochastic processes. We recall key aspects of this theory as applied to continuous Gaussian process, and show how the theory can be integrated with existing analytic tools, such as Malliavin's calculus, for the study of functionals on abstract Wiener space. We discuss the so-called CLL tail estimates and, as an extended application, derive a novel Stratonovich-to-Skorohod integral conversion formula for integrands given as path-level solutions to (rough) differential equations driven by Gaussian rough paths. This is based on joint work with Nengli Lim of Imperial College London and the National University of Singapore.
Apr 14	Thu	Andrew Wade (Durham)	Probability seminar
15:00		Convex hulls of planar random walks
		Hicks Seminar Room J11
	Abstract: On each of n unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area? I will describe recent work with Chang Xu (Strathclyde) on the convex hull of planar random walk, concerned in particular with the large-n asymptotics of its perimeter length and area. We assume finite second moments for the steps of the walk. First-order results for the perimeter length include a remarkable expectation formula due to Spitzer and Widom, and a law of large numbers due to Snyder and Steele, who also proved a variance upper bound. We complement these results by variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.
Apr 28	Thu	Ben Hambly (Oxford)	Probability seminar
15:00		A simple probabilistic model for interfaces in a martensitic phase transition
		LT 4
	Abstract: 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.
May 12	Thu	Inaki Esnaola (Sheffield)	Probability seminar
15:00		Information in Electricity Grids
		Hicks Seminar Room J11
	Abstract: The smart grid paradigm is founded on the integration of existing power grids with advanced sensing and communication infrastructures. While the benefits provided by this setting are crucial for the future development of power grids, it also increases the dependency on data acquisition and system monitoring procedures. Central to the control and optimization of power systems is the state estimation problem in electricity grids. In this talk, we first analyze the state estimation problem with imperfect knowledge of the statistical structure of the underlying random process modelling the state variables. Within this setting, we study the theoretical limits of state estimation when partial prior knowledge is available using information theoretic measures and random matrix theory. The second problem addressed in this talk is the characterisation of the secrecy limits of the information generated by the sensing infrastructure in the grid. We provide a closed form expression for the secrecy region as a function of the network characteristics using the replica method. In the last part of the talk, we design decentralized practical attack strategies in a game theoretic setting. The validity of the presented procedures in real settings is studied through simulations in IEEE test systems.
Jun 15	Wed	Robert Gaunt (Oxford)	Probability seminar
14:00		Rates of convergence for multivariate normal approximations by Stein's method
		LT 4
	Abstract: Stein's method is a powerful technique for obtaining distributional approximations in probability theory. We begin by reviewing Stein’s method for normal approximation. We then consider how this approach can be adapted to limits other than the normal. In particular, we see how Stein’s method for normal approximation can be extended relatively easily to the approximation of statistics that are asymptotically distributed as functions of multivariate normal random variables. We obtain some general bounds and a surprising result regarding the rate of convergence. We end with an application to the rate of convergence of Pearson's chi-square statistic. Part of this talk is based on joint work with Alastair Pickett and Gesine Reinert.
Sep 21	Wed	Antal Jarai (Bath)	Probability seminar
14:15		Sum of inverse powers of Poisson distances.
		Hicks LT10
	Abstract: Consider a planar Poisson process of constant intensity, and let S be the sum of the inverse 4-th powers of the distances of the points from the origin, which is a positive stable random variable of index 1/2. We give an approximation scheme, with explicit error bounds, for the conditional density of the point nearest to the origin given S = s. The precision of the approximation is uniform in the conditioning.
Sep 21	Wed	Jan Swart (UTIA, Academy of Sciences of the Czech Republic)	Probability seminar
15:45		Self-organised criticality on the stock market.
		Hicks LT10
	Abstract: In this talk, I will discuss a number of interacting particles on the real line that have two features in common. First, they are rank-based, in the sense that the only spatial structure that is relevant for the dynamics is the relative order of the particles. Second, they all seem to exhibit a form of behaviour known as self-organized criticality. Special attention will be given to a model for traders placing orders on a stock market, due to Stigler (1964) and Luckock (2003). Although this is a toy model that is known to be rather unrealistic, it gives important insight into some of the mechanisms that are at work on a real market. Moreover, I will show how with just some minor modifications, the model can be made much more realistic. This is joint work with M. Formentin, J. Plackova, and V. Perzina.
Oct 7	Fri	Alex Watson (Manchester)	Probability seminar
15:00		Spines in growth-fragmentations
		LT3
	Abstract: 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 in this talk I will focus on the homogeneous case, where the rate of fragmentation does not depend on the mass of the cell. I will discuss some recent results on spine methods for Bertoin's 'compensated' fragmentation processes, with applications to the growth-fragmentation equation and derivative martingales. Based on joint work with Jean Bertoin and Quan Shi.
Nov 3	Thu	Amanda Turner (Lancaster University)	Probability seminar
14:00		Scaling limits of Laplacian random growth models
		LT E
	Abstract: The idea of using conformal mappings to represent randomly growing clusters has been around for almost 20 years. Examples include the Hastings-Levitov models for planar random growth, which cover physically occurring processes such as diffusion-limited aggregation (DLA), dielectric breakdown and the Eden model for biological cell growth, and more recently Miller and Sheffield's Quantum Loewner Evolution (QLE). In this talk we will discuss ongoing work on a natural variation of the Hastings-Levitov family. For this model, we are able to prove that both singular and absolutely continuous scaling limits can occur. Specifically, we can show that for certain parameter values, under a sufficiently weak regularisation, the resulting cluster can be shown to converge to a randomly oriented one-dimensional slit, whereas under sufficiently strong regularisations, the scaling limit is a deterministically growing disk.
Dec 1	Thu	Chistian Fonseca Mora	Probability seminar
14:00
		Hicks Seminar Room J11
Dec 8	Thu	Balint Toth	Probability seminar
14:00		Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment
		LT E
	Abstract: We prove a CLT under diffusive scaling for the displacement of a random walk on $Z^d$ in stationary and ergodic doubly stochastic random environment, under the $H_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. Based on joint work with Gady Kozma (Weizmann Institute).
Dec 15	Thu	Neils Jacob (University of Swansea)	Probability seminar
14:00		Is there some semi-classical limit theory possible related to Levy processes?
		LT3
	Abstract: A Levy process is generated by a pseudo-differential operator the symbol of which is the negative of a continuos negative definite function. You may add to the generator a potential and look at this new operator as a type of Schroedinger operator, for example so called relativistic Schroedinger operators fall into this class. The problem we want to address, better we want to present first ideas to, is the following : Consider the Hamiltonian function associated with the Schroedinger operator and use this as starting point to develop a “classical mechanics”. Can we consider “classical trajectories” with respect to this “mechanics” as limits of scaled solutions of the Schroedinger operator.
Mar 2	Thu	Mark Walters (Queen Mary)	Probability seminar
14:00
Mar 23	Thu	Weijun Xu (Warwick)	Probability seminar
14:00
Apr 27	Thu	Nick Bingham	Probability seminar
14:00
May 18	Thu	Sean Ledger (Bristol)	Probability seminar
14:00
		Hicks LT E
Oct 19	Thu	Xue-Mei Li (Imperial)	Probability seminar
14:00		Brownian motions, Brownian Bridges and all that…
		LT3
	Abstract: BMs are well understood, Brownian bridges are conditioned Brownian motions and are well understood as such. On an Euclidean space, each induces a Gaussian measures on the space of paths. They are the Wiener measures. These measures can be used to construct Dirichlet forms and Ornstein-Uhlenbeck processes. Brownian bridges play the role of a delta measure and can be used for heat kernel estimates. In this talk we explore Brownian bridges, semi-classical bridges and even `generalized Brownian bridges’ for general elliptic differential operators.
Oct 26	Thu	Sam Cohen (Oxford)	Probability seminar
14:00		Statistical Uncertainty and nonlinear expectations
		LT 3
	Abstract: In stochastic decision problems, one often wants to estimate the underlying probability measure statistically, and then to use this estimate as a basis for decisions. We shall consider how the uncertainty in this estimation can be explicitly and consistently incorporated in the valuation of decisions, using the theory of nonlinear expectations.
Nov 16	Thu	Sandra Palau (Bath)	Probability seminar
15:30		Extinction properties and asymptotic behaviour of multi-type continuous state branching processes
		LT 7
	Abstract: First, we will discuss how to construct multi-type continuous state branching processes. Under mild conditions, we will see that there exists a lead eigenvalue associated with the first-moment semigroup. The sign of this eigenvalue distinguishes between the cases where there is extinction and exponential growth. Finally, in the supercritical case, we will give the a.s. rate of growth and the convergence of the proportion of each type.
Dec 14	Thu	Ostap Hryniv (Durham)	Probability seminar
14:00		Limiting behaviour of self-avoiding polygons
		LT 3
	Abstract: In the ensemble of two-dimensional self-avoiding polygons enclosing a fixed area (and centred at the origin), consider a probability distribution whose weights decay exponentially in polygon length. We study statistical properties of these polygons in the limit of large values of the enclosed area. Under a natural sub-criticality condition, we show that this probability distribution concentrates on a deterministic Wulff shape, derive a sharp asymptotics of the corresponding partition function, and describe the normal fluctuations of these polygons around the average profile.
Feb 19	Wed	Robin Stephenson (Sheffield)	Probability seminar
15:30
		LT7
Feb 19	Wed	Carina Geldhauser (Sheffield)	Probability seminar
16:15
		LT7
Mar 4	Wed	Dmitri Finkelshtein (Swansea)	Probability seminar
14:00
		LT1
Mar 18	Wed	Marcel Ortgiese (Bath)	Probability seminar
14:00
		LT1
Apr 29	Wed	Ellen Powell (Durham)	Probability seminar
14:00
		LT1
May 6	Wed	Edward Crane (Bristol)	Probability seminar
14:00
May 13	Wed	Tadahiro Oh (Edinburgh)	Probability seminar
14:00
		LT1
May 20	Wed	Alessandra Caraceni (Oxford)	Probability seminar
14:00
		LT1
May 27	Wed	Minmin Wang (Sussex)	Probability seminar
14:00
		LT7
Oct 27	Wed	Alessandra Caraceni (Pisa)	Probability seminar
15:00		Polynomial mixing time for edge flips via growing random planar maps
		Google Meet
	Abstract: A long-standing problem proposed by David Aldous consists in giving a sharp upper bound for the mixing time of the so-called “triangulation walk”, a Markov chain defined on the set of all possible triangulations of the regular n-gon. A single step of the chain consists in performing a random edge flip, i.e. in choosing an (internal) edge of the triangulation uniformly at random and, with probability 1/2, replacing it with the other diagonal of the quadrilateral formed by the two triangles adjacent to the edge in question (with probability 1/2, the triangulation is left unchanged). While it has been shown that the relaxation time for the triangulation walk is polynomial in n and bounded below by a multiple of n^{3/2}, the conjectured sharpness of the lower bound remains firmly out of reach in spite of the apparent simplicity of the chain. For edge flip chains on different models -- such as planar maps, quadrangulations of the sphere, lattice triangulations and other geometric graphs -- even less is known. We shall discuss results concerning the mixing time of random edge flips on rooted quadrangulations of the sphere, partly obtained in joint work with Alexandre Stauffer. A “growth scheme” for quadrangulations which generates a uniform quadrangulation of the sphere by adding faces one at a time at appropriate random locations can be combined with careful combinatorial constructions to build probabilistic canonical paths in a relatively novel way. This method has immediate implications for a range of interesting edge-manipulating Markov chains on so-called Catalan structures, from “leaf translations” on plane trees to “edge rotations” on general planar maps. Moreover, we are able to apply it to flips on 2p-angulations and simple triangulation of the sphere, via newly developed “growth schemes” to appear in an upcoming paper.
Nov 3	Wed	Ian Letter (Oxford)	Probability seminar
14:00		Hybrid zones and the effect of barriers.
		LT5
	Abstract: Hybrid zones are narrow regions in which two distinct types of individuals reproduce and produce offspring of mixed type. Some mathematical models conclude that hybrid zones of populations with asymmetric selection against heterozygotes evolve, when correctly rescaled, as mean curvature flow plus a constant flow. This conclusion rests on modelling the density of a particular allele as the solution of a partial differential equation in the euclidean space, proving the result in that deterministic setting and finally showing the presence of genetic drift does not disrupt the conclusion. In this talk, I will sketch the main ingredients to adapt this result to capture an effect one would see in a real-life population; the presence of barriers. Barriers refer to environmental obstacles that prevent individuals from invading certain zones. Mathematically this translates into studying the dynamics in a subset of the euclidean space with reflecting conditions on the boundary. We show that in a particular family of domains there is a phase transition; if the domain presents an opening bigger than an explicit constant there is an invasion of the fittest type, but if the opening is smaller than said constant then there is coexistence between the two types of individuals. As a consequence, we get that barriers can provide survival of the less fit homozygote, even if at the initial time the fittest homozygote dominates an unbounded region of the domain. We also mention how the presence of genetic drift (modelled by a Spatial-Lambda-Fleming Viot type process) may change these results. This is work under the supervision of Alison Etheridge.
Dec 8	Wed	Takis Konstantopoulos (Liverpool)	Probability seminar
15:00		Longest and heaviest paths in directed random graph
		Google Meet
	Abstract: In this talk, I will give an overview of results regarding the behaviour of longest paths in Barak-Erdos graphs as well as weighted versions of them, examining their connections to particle systems such as the infinite bin model and to regenerative techniques.
Mar 16	Wed	Andrew Wade (Durham)	Probability seminar
15:00		Deposition, diffusion, and nucleation on an interval
		Hicks LT9
	Abstract: I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studie in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form. This talk is based on joint work with Nicholas Georgiou (Durham): https://arxiv.org/abs/2010.00671
Apr 27	Wed	Wenkai Xu (Oxford)	Probability seminar
15:00		Stein's Method on Testing Goodness-of-fit for Exponential Random Graph Models
		Hicks LT9
	Abstract: In this talk, I will introduce a novel nonparametric goodness-of-fit testing procedure for exchangeable exponential random graph models (ERGMs) when a single network realisation is observed. The test determines how likely it is that the observation is generated from a target unnormalised ERGM density. The test statistics are derived from a kernel Stein discrepancy, a divergence constructed via Stein’s method using functions in a reproducing kernel Hilbert space, combined with a discrete Stein operator for ERGMs. Theoretical properties for the testing procedure for a class of ERGMs will be discussed; simulation studies and real network applications will be presented.
May 4	Wed	Marcel Ortgiese (Bath)	Probability seminar
16:00		Voter models on subcritical inhomogeneous random graphs.
		Hicks LT9
	Abstract: The voter model is a classical interacting particle system modelling how consensus is formed across a network. We analyse the time to consensus for the voter model when the underlying graph is a subcritical scale-free random graph. Moreover, we generalise the model to include a `temperature' parameter. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. Our proofs rely on the well-known duality to coalescing random walks and a detailed understanding of the structure of the random graphs in terms of a thinned Galton-Watson forest.
Dec 7	Wed	Guillaume Conchon-Kerjan (Bath)	Probability seminar
15:00		Scaling limit of a branching process in a varying environment
		F41
	Abstract: A branching process in varying environment is a Galton-Watson tree whose offspring distribution can change at each generation. The evolution of the size of successive generations has drawn a lot of attention in recent years, both from the discrete and continuum points of view (as the scaling limit is a modified Continuous State Branching Process). We focus on the limiting genealogical structure, which is much more delicate to study. In the critical case (all distributions have average offspring 1), we show that under mild second moment assumptions on the sequence of offspring distributions, a BPVE conditioned to be large converges to the Brownian Continuum Random Tree, as in the standard Galton-Watson setting. The varying environment adds asymmetry and dependencies in many places. This requires numerous changes to the usual arguments. In particular we employ a (to our knowledge) new connexion between the Łukasiewicz path and the height process. This is a joint work with Daniel Kious and Cécile Mailler.
May 3	Wed	Debleena Thacker (Durham)	Probability seminar
15:00		Continuous-time digital search tree and a border aggregation model
	Abstract: We consider the continuous-time version of the random digital search tree, and construct a coupling with a border aggregation model as studied in Thacker and Volkov (2018), showing a relation between the height of the tree and the time required for aggregation. This relation carries over to the corresponding discrete-time models. As a consequence we find a very precise asymptotic result for the time to aggregation, using recent results by Drmota et al. (2020) for the digital search tree. This is joint work with Svante Janson.
Oct 4	Wed	Ilay Hoshen (Tel Aviv)	Probability seminar
15:00		Simonovits's theorem in random graphs
		Hicks Seminar Room J11
	Abstract: Let $H$ be a graph with chromatic number $\chi(H) = r+1$. Simonovits's theorem states that the unique largest $H$-free subgraph of $K_n$ is its largest $r$-partite subgraph if and only if $H$ is edge-critical. We show that the same holds with $K_n$ replaced by $G_{n,p}$ whenever $H$ is also strictly 2-balanced and \begin{align*} p \geq C n^{-1/m_2(H)} \log(n)^{1/(e(H)-1)}, \end{align*} for some constant $C > 0$. This is best possible up to the choice of the constant $C$. This (partially) resolves a conjecture of DeMarco and Kahn, who proved the result in the case where $H$ is a complete graph. Moreover, we prove the result with explicit constant $C = C(H)$ that we believe to be optimal. Joint work with Wojciech Samotij.
Nov 15	Wed	Lukas Lüchtrath (Weierstrass Institude)	Probability seminar
16:00
		Hicks Seminar Room J11
Nov 22	Wed	Jan Swart (Czech Academy of Sciences)	Probability seminar
15:00
		Hicks Seminar Room J11
Feb 14	Wed	Robin Stephenson (Sheffield)	Probability seminar
16:00		Where do trees grow leaves?
		Hicks Seminar Room J11
	Abstract: We study a model of random binary trees grown ``by the leaves" in the style of Luczak and Winkler (2004). If $\tau_n$ is a uniform plane binary tree of size $n$, Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure $\nu_{\tau_n}$ such that the tree obtained by adding a cherry on a leaf sampled according to $\nu_{\tau_n}$ is still uniformly distributed on the set of all plane binary trees with size $n+1$. It turns out that the measure $\nu_{\tau_n}$, which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree $\tau_n$. In fact we prove that as $n \to \infty$, with high probability it is almost entirely supported by a subset of only $n^{3 ( 2 - \sqrt{3})+o(1)} \approx n^{0.8038...}$ leaves. In the continuous setting, we construct the scaling limit of this measure, which is a probability measure on the Brownian Continuum Random Tree supported by a fractal set of dimension $ 6 (2 - \sqrt{3})$. We also compute the full (discrete) multifractal spectrum. This work is a first step towards understanding the diffusion limit of the discrete leaf-growth procedure.