UK Easter Probability Meeting

Speakers

Minicourses

Nathanaël Berestycki (University of Vienna)

Dimers and imaginary geometry Slides

Nathan Ross (University of Melbourne)

Stein's method and applications Slides (1) Slides (2) Slides (3) Slides (4)

Amandine Véber (Univ. Paris Cité/CNRS)

Stochastic models of evolution in a population living in a continuum Slides Notes References

In this course, we shall consider a model for evolution in a spatial continuum, called the spatial Lambda-Fleming-Viot process, which was introduced by Alison Etheridge (Univ. of Oxford) and Nick Barton (IST Vienna) in 2008. We shall discuss its dual "genealogical" process and give examples of interesting behaviour of the local allele frequencies under different parameter regimes and on different space and time scales.

Invited talks

Francesco Caravenna (University of Milano-Bicocca)

The critical 2d Stochastic Heat Flow Slides

We study the 2d Stochastic Heat Equation, that is the heat equation in two space dimensions with a multiplicative random potential (space-time white noise). This equation is ill-defined due to the singularity of the potential and we regularise it by discretising space-time. We prove that, in the limit when discretisation is removed and the noise strength is rescaled in a critical way, the solution has a well-defined and unique limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow. We investigate its features, showing that it cannot be the exponential of a generalised Gaussian field.

(joint work with R. Sun and N. Zygouras)

Hanna Döring (Osnabrück University)

Volume and Coverage Threshold for Poisson Cylinder Sets

Motivated by telecommunication systems, we study functionals such as the volume and the number of isolated cylinders in inhomogeneous Poisson cylinder models. For fixed intensity and radius, we study these functionals in a growing window and prove a central limit theorem applying Stein's method. In addition, we define the coverage threshold to be the minimal cylinder radius it needs to cover a fixed box of volume one. We study the asymptotics of this coverage threshold with respect to the intensity.

Bénédicte Haas (Université Sorbonne Paris Nord)

Tail asymptotics for exponential functionals of subordinators and extinction times of self-similar fragmentations Slides

Exponential functionals of subordinators have been thoroughly investigated, since they play a key role in several facets of modern probabilities, as they correspond to the extinction times of non-increasing self-similar Markov processes. As such, they are involved in the description of various processes ranging from the analysis of algorithms to coagulation or fragmentation processes. In this talk we will provide the exact large-time equivalents of the density and upper tail distribution of the exponential functional of a subordinator in terms of its Laplace exponents. This improves previous results on the logarithmic asymptotic behaviour of this tail.

We will then see how this result can be used to determine the large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity. The extinction time of a typical fragment in such a process is an exponential functional of a subordinator. But the tail of the extinction time of the whole fragmentation process decreases much more slowly in general. We will quantify this difference by determining the asymptotic ratio of the two tails.

Andreas Kyprianou (University of Warwick)

Yaglom limits for general non-local Branching Markov processes Slides

Little has been written about moments higher than 2 for branching processes. In this talk we explore some very general results for non-local spatial branching Markov processes and non-local superprocesses alike which give straightforward limiting results under a natural Perron-Frobenius type assumptions for the first moment semigroup. The method is so robust that we are also able to establish moment growth for the occupation measure of the same classes of spatial branching processes. In the critical setting this can be used to give a Yaglom limit. This is joint work is based on two papers with coauthors: Isaac Gonzalez, Emma Horton, Simon Harris, Minmin Wang.

Giovanni Peccati (University of Luxembourg)

Multidimensional fluctuations of additive functionals of fractional Brownian motion Slides

I will present some recent results on first and second-order multidimensional fluctuations of a class of additive functionals of a fractional Brownian motion. Our findings demonstrate radically different behaviors, according to the value of the Hurst parameter H. When H is greater or equal 1/3 the limit distribution is an independent Brownian motion, time-changed with the local time (with a multiplicative logarithmic correction to the variance in the critical case H=1/3); when H < 1/3 (rough case), the limit is a constant multiple of the derivative of the local time. Some results in discrete time - having a direct interpretation in terms of high-frequency time-series - will be also discussed. The techniques behind the proofs involve Fourier analysis, Itô, and Malliavin calculus. Based on joint works with A. Jaramillo and I. Nourdin, and with A. Jaramillo, I. Nourdin and D. Nualart.

Sarah Penington (University of Bath)

Genealogy of the N-particle branching random walk with polynomial tails Slides

The N-particle branching random walk is a discrete time branching particle system with selection consisting of N particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant.

I will discuss recent results and open conjectures about the long-term behaviour of this particle system when N, the number of particles, is large. In the case where the jump distribution has regularly varying tails, building on earlier work of J. Bérard and P. Maillard, we prove that at a typical large time the genealogy is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Based on joint work with Matt Roberts and Zsófia Talyigás.

Ellen Powell (Durham University)

Brownian excursions, conformal loop ensembles and critical Liouville quantum gravity Slides

It was recently shown by Aïdékon and Da Silva how to construct a growth fragmentation process from a planar Brownian excursion. I will explain how this same growth fragmentation process arises in another setting: when one decorates a certain “critical Liouville quantum gravity random surface” with a conformal loop ensemble of parameter 4. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun

Gesine Reinert (University of Oxford)

Stein's Method, Stein's Shrinkage Estimator, and Stein's Unbiased Risk Estimate Slides

Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon his discovery of the famous paradox involving the multivariate Gaussian. A large portion of the subsequent literature only considers the efficiency of shrinkage, and that of an associated procedure known as Stein's Unbiased Risk Estimate, or SURE, in the Gaussian setting of that original work.

We use Stein’s method to extend the domain of validity of shrinkage and SURE away from the Gaussian. As particular tools from Stein’s method, Stein kernels and zero bias couplings are employed.

This is joint work with Max Fathi, Larry Goldstein and Adrien Saumard.

Adrian Roellin (National University of Singapore)

Higher order fluctuations in dense random graph models Slides

Dense graph limit theory is mainly concerned with law-of-large-number type of results. We propose a corresponding central limit theorem — or rather fluctuation theory — based on Janson’s theory of Gaussian Hilbert Spaces and generalised U-statistics from the 1990s. Our approach provides rates and allows for proper statistical inference based on subgraph counts.

Vincent Tassion (ETH Zürich)

Robust noise sensitivity of percolation

Consider critical site Bernoulli percolation on the triangular lattice, where each vertex is colored black or white with probability 1/2, independently of the other vertices. In 1999, Benjamini, Kalai and Schramm proved that crossing probabilities are noise sensitive: resampling a small proportion of the vertices lead to an independent percolation picture. Ten years later, Garban, Pete and Schramm obtained a sharp quantitative version of this result. These works rely on Fourier analysis, and are restricted to Bernoulli percolation (i.e. product measure) and the independent resampling dynamics.

In this talk, we will discuss noise sensitivity for more general percolation models, and more general dynamics. Based on a recent approach to noise sensitivity with Hugo Vanneuville (that relies on geometrical arguments and not on spectral methods), we show noise sensitivity of crossing probabilities for high temperature Ising under Glauber dynamics.

Based on a joint work with Hugo Vanneuville.

Andrew Wade (Durham University)

Deposition, diffusion, and nucleation on an interval Slides

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 popular 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).

Anita Winter (Universität Duisburg-Essen)

Scaling limit of the Aldous-Broder chain on the high-dimensional torus

Contributed talks

Matthew Buckland (University of Oxford)

Branching Interval Partition Diffusions

We construct an interval-partition-valued diffusion from a collection of excursions sampled from the excursion measure of a real-valued diffusion, and we use a spectrally positive Lévy process to order both these excursions and their start times. At any point in time, the interval partition generated is the concatenation of intervals where each excursion alive at that point contributes an interval of size given by its value. Previous work by Forman, Pal, Rizzolo and Winkel considers self-similar interval partition diffusions – and the key aim of this work is to generalise these results by dropping the self-similarity condition. The interval partition can be interpreted as an ordered collection of individuals (intervals) alive that have varying characteristics and generate new intervals during their finite lifetimes, and hence can be viewed as a class of Crump-Mode-Jagers-type processes.

Natalia Cardona (University of Göttingen)

Rates on Yaglom’s Theorem for Galton-Watson processes in varying environment Slides

A Galton–Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. We provide rate of convergence with respect to the Wasserstein metric for the classical theorem of Yaglom for this family of processes. The theorem of Yaglom states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to a standard exponential random variable. This is ongoing work with Arturo Jaramillo (CIMAT, Mexico) and Sandra Palau (UNAM, Mexico).

Chris Dean (University of Bath)

Pólya urns with growing initial compositions Slides

A Pólya urn is a Markov process describing the contents of an urn that contains balls of d colours. At every time step, we draw a ball uniformly for the urn, note its colour, then put it back in the urn along with a set of new balls which depend on the colour drawn. The number of balls of colour j added when colour i is drawn is given by the (i,j)th entry of a predetermined replacement matrix R. The asymptotic behaviour of the urn for most replacement matrices can be inferred from the following two canonical cases. In the case of R the identity, the proportion of each colour in the urn tends to a Dirichlet distributed random variable with parameter given by the urn's initial composition. In the case of R irreducible, this limit is a deterministic vector that only depends on R. Fluctuations around these limits are also known.

Recently, Borovkov showed results on the asymptotic behaviour in the identity case, when the initial number of balls grows together with the number of time steps. In this talk, we show analogous results for the irreducible case. This will include the asymptotic behaviour of the proportion of each colour in the urn and the fluctuations around this limit.

Thomas Finn (Durham University)

Non-equilibrium multi-scale analysis and coexistence in competing first passage percolation

Consider a two-type growth model on Z^d with the following dynamics. Type 1 occupies the origin at time 0 while type 2 is dormant in seeds that are distributed on sites as a product of Bernoulli measures of parameter p. Type 1 spreads through the edges of Z^d at rate 1 and when a seed is attempted to be occupied by either process that seed is occupied by type 2 and type 2 spreads from that seed at rate \lambda. Once a site is occupied by either type 1 or type 2 it remains occupied by that type henceforth. This model is called first passage percolation in a hostile environment (FPPHE) and was introduced by Sidoravicius and Stauffer to analyse multi-particle diffusion limited aggregation. A major challenge in studying FPPHE is the lack of monotonicity. Counterintuitively, increasing p or \lambda could benefit type 1 by delaying the activation of other seeds. We will discuss known results for FPPHE and demonstrate how both types can concurrently occupy infinite connected components with positive probability in a regime of coexistence. Our proof relies on a novel multi-scale construction to handle non-equilibrium dynamics and non-monotonicity called multi-scale analysis with non-equilibrium feedback that we believe could be applicable to other challenging models. Based on joint work with Alexandre Stauffer.

Moritz Otto (Aarhus University)

Limit theorems for Gibbs functionals: Stein’s method meets disagreement percolation Slides

I will begin with a brief introduction to Gibbs processes and define a disagreement coupling for finite Gibbs processes with different boundary conditions. We will then see how disagreement couplings can be used to couple a functional of a Gibbs process with its reduced Palm version. Using Stein’s method, this will lead to a Poisson approximation result if we assume that the underlying Gibbs process is dominated by a Poisson process with subcritical Boolean model. The convergence rate depends on second-order quantities of the Gibbs functional and one-arm probabilities from continuum percolation. In the last part of the talk, I will explain how a similar approach can be used to obtain normal approximation for Gibbs functionals based on a recent result from Chen, Röllin and Xia (2021). This talk is based on a joint article with Günter Last (Karlsruhe) and on an ongoing project with Christian Hirsch (Aarhus) and Anne Marie Svane (Aalborg).

Léonie Papon (Durham University)

Thick points of the planar GFF are totally disconnected for all γ ≠ 0 Slides

I will talk about a result regarding thick points of the Gaussian free field obtained in collaboration with Juhan Aru and Ellen Powell. We prove that the set of $\gamma$-thick points of a planar continuum Gaussian free field (GFF) with Dirichlet boundary conditions is almost surely totally disconnected for all $\gamma \neq 0$. The proof relies on the coupling between the GFF and the nested CLE_4. In particular, I will show that the thick points of the GFF are the same as those of the weighted CLE_4 nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE_{\kappa}, $\kappa \in (8/3,4]$. As a corollary, I will prove that the set of singular points for supercritical LQG metrics is almost surely totally disconnected.

Location

The meeting will take place in the Department of Mathematics at the University of Manchester, in the Frank Adams room on the first floor of Alan Turing Building.

Alan Turing Building is building 46 on the Campus Map. For more details on reaching the venue, please see Maps and Travel.

Accessibility information can be found here.

Information for speakers

Speakers (invited and contributed) are booked at the Pendulum Hotel, Sackville St, Manchester, M1 3BB. The hotel is around 10 minutes walk from Manchester Piccadilly train station.

Conference dinner

The conference dinner is at Giorgio, 131 Portland St, Manchester, M1 4PY, starting at 6pm. The restaurant is around 20 minutes walk from the meeting venue, or 15 minutes from the Pendulum Hotel.

Places to eat

The following locations are close to the meeting venue; walking time estimates are from Google Maps.

Restaurants

• Tai Wu (Chinese, 2 mins)
• Umami (Japanese, 8 mins)
• Zouk (Indian, veg[an] options, 12 mins)
• Samsi (Japanese, 14 mins)

Pubs that serve dinner

• Brewdog (Mainly burgers, veg[an] options, 4 mins)
• Kro Bar (British and Danish pub food, veg[an] options, 6 mins)
• Sandbar (Neapolitan style pizza, veg[an] options, 7 mins)
• The Flour and Flagon (British pub food, veg[an] options, 7 mins)

Schedule

The meeting schedule appears as an embedded Google Calendar below. You may need to set your adblocker to allow iframes to see it. Alternatively, you can access it at this link: UK Easter Probability Meeting 2023 schedule.

Registration

Registration is now closed.

There is a conference dinner on Wednesday at no additional charge.

Funding is still available to support the attendance of UK research students, with priority for EPSRC-funded students, and to pay for caring costs of participants. Please contact the organising committee if you wish to request this.

Contact

The organising committee is:

• Denis Denisov (University of Manchester)
• Robert Gaunt (University of Manchester)
• Xiong Jin (University of Manchester)
• Alex Watson (UCL)

If you have any questions, please email the committee.

This meeting is funded by grants from the London Mathematical Society, EPSRC and the Applied Probability Trust.