The Department of Mathematics at the University of Manchester will host the UK Easter Probability Meeting, a week-long workshop on probability and its applications, with three minicourses, 12 invited talks and opportunities for contributed talks from early-career researchers.

- Minicourse Speakers
- Nathanaël Berestycki (University of Vienna)
- Dimers and imaginary geometry
- Alison Etheridge (University of Oxford)
- Mathematical models from population genetics
**Abstract:**For over a century, mathematical models have played a fundamental role in theoretical population genetics. Indeed, genetics provided some of the earliest applications of the Ito calculus. In turn, population genetics provides a wealth of mathematical challenges.

In these lectures, we shall focus on the models which arise when we try to model the interplay between the forces of evolution (mutation, selection, random genetic drift etc) acting on a population. We shall begin with some classical models for so-called panmictic populations, before turning to the complications that arise when we try to incorporate spatial structure.The challenge is to provide consistent forwards in time models for the way in which the frequencies of different genetic types evolve in the population, and backwards in time models for the ways in which individuals sampled from the population are related to one another.

- Nathan Ross (University of Melbourne)
- Stein's method and applications
- Invited Speakers
- Francesco Caravenna (University of Milano-Bicocca)
**Title: On the two-dimensional KPZ and Stochastic Heat Equation****Abstract:**We consider the Kardar-Parisi-Zhang equation (KPZ) and the multiplicative Stochastic Heat Equation (SHE) in two space dimensions, driven by with space-time white noise. These singular PDEs are "critical" and lack a solution theory, so it is standard to consider regularized versions of these equations - e.g. convolving the noise with a smooth mollifier - and to investigate the behavior of the regularized solutions when the regularization is removed.Based on joint works with Rongfeng Sun and Nikos Zygouras, we show that these regularized solutions undergo a phase transition as the noise strength is varied on a logarithmic scale, with an explicit critical point. In the sub-critical regime, the regularized solutions of both KPZ and SHE exhibit so-called Edwards-Wilkinson fluctuations, i.e. they converge to the solution of the *additive* Stochastic Heat Equation (after centering and rescaling), with a non-trivial constant on the noise. We finally discuss the critical regime, where many questions are open.

- Hanna Döring (Universitat Osnabruck)
**Title: Limit theorems in random dynamic networks****Abstract:**TBC - Bénédicte Haas (Université Paris 13)
**Title: Scaling limits of multi-type Markov Branching trees****Abstract:**TBC - Andreas Kyprianou (University of Bath)
- Giovanni Peccati (Luxembourg University)
- Sarah Penington (University of Bath)
**Title: Genealogies in pushed waves****Abstract:**Consider a population in which each individual carries two copies of each gene, and suppose that a particular gene occurs in two different types, a and A. Suppose that individuals carrying AA have a higher evolutionary fitness than aa individuals, and that aA individuals have a lower evolutionary fitness. We can model this situation using a stepping stone model on the integers, and show that (under certain conditions) as the number of individuals at each site goes to infinity, the genealogy of a sample of type A genes from the population (under a suitable time scaling) converges to a Kingman coalescent. Joint work with Alison Etheridge. - Ellen Powell (Durham University)
- Gesine Reinert (University of Oxford)
**Title: Covariance Inequalities via Stein’s Method****Abstract:**Covariance inequalities are a cornerstone in probability. In this talk we shall give covariance inequalities which are based on probabilistic representations for solutions to Stein equations. These representations lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit and easily computable in most cases. Applications of the Cauchy-Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincare’ inequalities. Classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries.This is joint work with Marie Ernst and Yvik Swan.

- Guillaume Remy (Columbia University)
**Title: A probabilistic construction of conformal blocks for Liouville CFT****Abstract:**Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by A. Polyakov in the context of string theory. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will present a probabilistic construction of the conformal blocks of Liouville CFT on the torus. These are the fundamental objects that allow to understand the integrable structure of CFT using the conformal bootstrap equation. We will also mention the connection with the AGT correspondence. Based on joint work with Promit Ghosal, Xin Sun, and Yi Sun. - Adrian Roellin (National University of Singapore)
**Title: Higher-order fluctuations in dense graph limit theory****Abstract:**Dense graph limit theory is now a well-established approach to describe the structure of large dense networks. While this theory is mainly concerned with the first-order behaviour of graphs in the same way as the Law of Large Numbers describes the first order behaviour of sums of random variables in classical limit theory, a theory to describe second or even higher-order fluctuations around those first order limits has not yet been developed. However, we argue that the tools to understand these fluctuations have already been around since the 90s, although in the more abstract form of generalised U-statistics. We discuss how this theory can be used to describe fluctuations of large graphs, and we complements these results with rates of convergence using Stein’s method and with generalisations that are relevant to dense graph limit theory, but which go beyond the usual framework of generalised U-statistics. - Vincent Tassion (ETH Zurich)
**Title: Supercritical percolation on graphs of polynomial growth****Abstract:**We consider Bernoulli percolation on transitive graphs of polynomial growth. In the subcritical regime (p < pc), it is well known that the connection probabilities decay exponentially fast. In this talk we will discuss the supercritical phase (p>pc) and prove the exponential decay of the truncated connection probabilities (probabilities that two points are connected by an open path, but not to infinity). This sharpness result was established by [CCN87] on Zd using the difficult slab result of Grimmett and Marstrand. However the techniques used there are very specific to hypercubic lattices and do not extend to more general geometries. We will present new robust techniques based on the recent progress in the theory of sharp thresholds and the sprinkling method of Benjamini and Tassion. On Zd, the methods give a completely new proof of the slab results of Grimmett and Marstrand. We also study Schramm’s locality conjecture, which states that the percolation critical value pc should only depend on the local structure of the underlying graph. We prove that the locality conjecture holds, when restricted to graphs of polynomial growth. The proof involves new percolation results and geometric group theory.Based on a joint work with Daniel Contreras and Sebastien Martineau.

- Anita Winter (Universität Duisburg-Essen)
- Contributed Talks
- Eleanor Archer (The University of Warwick)
**Title: Random walks on decorated Galton-Watson trees****Abstract:**We consider a model obtained by taking a Galton-Watson tree, replacing each vertex with an independent copy of a random metric space, and gluing these metric spaces along the tree structure. These kinds of models arise naturally in the context of statistical mechanics models on random planar maps. With these applications in mind, we will focus on the case where the Galton-Watson tree has critical offspring distribution in the domain of attraction of a stable law, and the laws of the inserted metric spaces have some dependence on the degree of the associated vertex in the tree. We will see that the random walk and volume growth exponents in this decorated space undergo a phase transition, depending on how the individual exponents for the inserted metric spaces and the underlying Galton-Watson tree balance out. - Peter Braunsteins (University of Amsterdam)
**Title: Local limit theorems for occupancy models****Abstract:**Local central limit theorems for general sums of independent integer valued random variables are well understood; however, for sums of dependent random variables much less is known. In this talk we present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with $d$ neighbours in a germ--grain model, and the number of degree-$d$ vertices in an Erd\H{o}s--R\'enyi random graph. In both cases, the error rate is optimal, up to logarithmic factors. It is joint work with Nathan Ross and Andrew Barbour. - Benedetta Cavalli (University of Zurich)
**Title: On a growth-fragmentation equation with bounded fragmentation rate and its solution via the Feynman-Kac formula****Abstract:**The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. This question has traditionally been addressed using analytic techniques such as entropy methods or splitting of operators.Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question in the case in which the growth rate is sublinear and the mass is conserved at fragmentation events. This assumption on the growth ensures that microscopic particles remain microscopic.

In this talk, we present a recent work of the speaker, in which we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. Moreover, we drop the hypothesis of conservation of mass when a fragmentation occurs. With the Feynman-Kac approach, we establish necessary and sufficient conditions on the coefficients of the equation that ensure the so-called Malthusian behaviour with exponential speed of convergence to an asymptotic profile. Furthermore, we provide an explicit expression of the latter.

- Aleksander Klimek (Max Planck Institute for Mathematics in the Sciences)
**Title: TBC****Abstract:**It is well known that the dynamics of the rare subpopulation subject to Wright-Fisher diffusion follows Feller diffusion.We discuss a spatial analogue of this result in the presence of selection in random environment by investigating the scaling limits of Spatial Lambda-Fleming-Viot models. We consider two regimes. In the first one, the environment fluctuates in time and space. In the second one, only spatial fluctuations are present. The subpopulation of rare individuals follows the superBrownian motion in random environment and rough superBrownian motion, respectively.

The long time behaviour of the limiting process differs significantly for both cases. While both the standard superBrownian motion and the superBrownian motion in random environment in full space suffers from local weak extinction in dimension $d=1,2$, the rough superBrownian motion persists, even on a torus.

This provides weak circumstantial evidence for Wright's claim that the variation in spatial conditions contributes positively to genetic variety in the populations.

- George Liddle (Lancaster University)
**Title: Scaling limits and fluctuations for random growth under capacity rescaling****Abstract:**Random growth occurs in many real world settings, for example, we see it exhibited in the growth of tumours, bacterial growth and lightning patterns. As such we would like to be able to model such processes to determine their behaviour in their scaling limits. Well studied models to describe these different processes include the Eden model and DLA. In a 1998 paper Hastings and Levitov introduced a one parameter family of conformal maps HL(alpha) which can be used to model Laplacian growth processes and allows us to vary between the previous models by varying the parameter alpha. We will consider a regularised version of this model and analyse its scaling limits and fluctuations for different values of alpha under capacity rescaling. - Mo Dick Wong (University of Oxford)
**Title: Tail universality of critical Gaussian multiplicative chaos****Abstract:**Gaussian multiplicative chaos (GMC) is a one-parameter family of random measures formally defined as the exponentiation of log-correlated Gaussian fields, and it has attracted a lot of attention in recent years due to its appearance in e.g. Liouville conformal field theory and random matrix theory. Motivated by the study of extremal process of log-correlated fields in the discrete literature, we consider GMC measures at criticality and establish an asymptotic power law profile for the (right) tail probability under mild assumptions. The leading order coefficient of the tail asymptotics is fully explicit and does not depend on any local variation of the underlying field, demonstrating a new universality phenomenon.

(M) Minicourse (D) Discussion (T) Talk

The meeting will take place in the Department of Mathematics of the University of Manchester.

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

The room has stepless access.

Registration is opening now. Please use the following link to register:

There will be a conference dinner on Wednesday at no extra charge. The dinner is sponsored by the Applied Probability Trust.

There is an opportunity for early-career researchers and research students to present a short talk (there will be six 20 minute contributed talks) or a poster. Space for the short talks is limited, and participants will be selected based on their proposed abstracts. The deadline for abstract submission is Friday 17 January, and decisions will be communicated by Friday 24 January. Participants selected to present a short talk will receive financial support for their attendance and have the conference fee waived.

We have awarded all our funding for supporting UK research students. Currently there is no funding available.

The organising committee is:

- Denis Denisov
- Robert Gaunt
- Xiong Jin
- Alex Watson
- Tusheng Zhang

If you have any questions, please email the committee.

This meeting is funded by EPSRC and LMS.