## Friday December 8, 2023. Sparse random boundary conditions on the Ising model on trees

Speaker: Irene Ayuso Ventura, Université Paris-Est Créteil (UPEC).

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-103,

or via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

Motivated by the study of the Ising model on tree-like random graphs with random boundary conditions, we define a random artificial boundary for a supercritical branching process. We show that the phase transition for the root magnetisation on the Ising model with this type of boundary is equivalent to that of the "+"-boundary conditions for a certain inhomogeneous tree model: the Pruned tree. We introduce the model and show criteria for the root magnetisation in terms of the 3-capacity of the Pruned tree.

## Friday December 1, 2023. Normal approximation for the posterior in exponential families

Speaker: Robert Gaunt, University of Manchester.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-103,

or via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

The Bernstein-von Mises (BvM) Theorem is a cornerstone in Bayesian statistics. Loosely put, this theorem reconciles Bayesian and frequentist large sample theory by guaranteeing that, under regularity conditions, suitable scalings of posterior distributions are asymptotically normal. In particular, this implies that the contribution of the prior vanishes in the asymptotic posterior.

In this talk, we demonstrate how the probabilistic technique Stein's method can be used to derive explicit optimal order total variation and Wasserstein distance bounds to quantify this distributional approximation for posterior distributions in exponential family models. We apply our general bounds to some classical conjugate prior models and observe that the resulting bounds have an explicit dependence on the prior distribution and on sufficient statistics of the data from the sample, and thus provide insight into how these factors may affect the quality of the normal approximation.

This is joint work with Adrian Fischer, Gesine Reinert and Yvik Swan

## Friday November 24, 2023. Reconstructing demography and detecting selection from genomic data - extensions of inference methods for non-standard genealogy models.

Speaker: Fabian Freund, University of Leicester.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-103,

or via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

Population genetic inference for past selection processes and demography is often based on estimating parameters of and selecting between coalescent processes that represent different evolutionary and demographic processes. Such quantitative methods make assumptions on the underlying evolutionary processes that recently have been challenged for a variety of species - most notably the assumption that evolution roughly works like a Wright-Fisher or Moran model, or equivalently that genealogies can be represented by Kingman's $n$-coalescent.

I present two ideas for extending such an inference method which originally uses the sequential Markovian coalescent as a hidden Markov model and assumes Kingman's $n$-coalescent as the underlying genealogy model. The extension can either be done purely mathematically by replacing Kingman's $n$-coalescent as the genealogy model by a larger class of genealogy models (Beta $n$-coalescents), or computationally by using a machine learning approach trained on simulations of ancestral recombination graphs based on this larger class of genealogy models.

## Friday November 17, 2023. An Improved Lower Bound on the Largest Common Subtree of Random Leaf-Labeled Binary Trees

Speaker: Ali Khezeli, Inria Paris Centre.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

It is known that the size of the largest commonsubtree (i.e., the maximum agreement subtree) of two independent random binary trees with n given labeled leaves is of order between n^(0.366) and n^(1/2). We improve the lower bound to order n^(0.4464) by constructing a common subtree recursively and by proving a lower bound for its asymptotic growth. The construction is a modification of an algorithm proposed by D. Aldous by splitting the tree at the centroid and by proceeding recursively.

## Friday November 10, 2023. Normal Approximation of Poisson Functional via Malliavin-Stein Method

Speaker: Tara Trauthwein, University of Luxembourg.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

Given a functional of a Poisson measure, it is possible to derive a central limit theorem for it using only information on the variance and the first and the second add-one costs, the differences when adding a point (or two) to the Poisson measure. This derives from a result by Last, Peccati and Schulte from 2016, where moment bounds of the add-one costs of order $4+\epsilon$ were required. In this talk, we will present an extension of this result to minimal $2+\epsilon$ moment requirements, as well as an application to edge-functionals of the Online Nearest Neighbour Graph, thereby closing a conjecture from (Wade 2009; Penrose and Wade 2009).

## Friday October 20, 2023. Diffusions Derived From Branching Diagram

Speaker: Youzhou Zhou, Xi'an Jiaotong-Liverpool University.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract:

In this talk we will discuss a family of branching diagrams. One may also construct up Markov chains and down Markov chains in this branching diagram. Then the diffusions can be derived from a family of Up-down Markov chains in the branching diagrams. The dual processes of these diffusions are continuous time pure-death process on the branching diagrams, and their embedded chains are usually the down Markov chain in the branching diagrams. Making use of these dual processes, the transition density of these diffusion processes can be derived.

## Friday October 6, 2023. Non-Directed Polymers in random environments on Z^d

Speaker: Ran Wei, Xi'an Jiaotong-Liverpool University.

Abstract:

In this talk, we introduce a relatively new random polymer model on $Z^d$, called the non-directed polymer model. The model consists of a simple random walk on $Z^d$ (representing the polymer chains) and a family of i.i.d. random variables $w_x:=(w_x)_{x∈Z^d}$ (representing the random environments). The interaction between the random walk and the random environments is defined via a Gibbs transform, that is, the law of the random walk up to time $N$ is tilted by $\exp(\sum_{x \in R_N} (b w_x -h))$, where $R_N$ is the range of the random walk, $b$ is the inverse temperature, and $h$ is a non-random external field. It can be viewed as a perturbed version of the random walk penalized (rewarded) by its range or a variant of the classic directed polymer model. We assume that the tail probability of the environment w decays polynomially with exponent $a$, and $b = b_N$ and $h = h_N$ grow or decay polynomially over time with exponents $\gamma$ and $\zeta$ respectively. Under the above conditions, we investigate the phase diagram for $a$, $\gamma$, $\zeta$, and we also study the scaling limits for partition functions and the wandering fluctuations of the random walk under the Gibbs measures in different phases.

## Friday May 5, 2023. Pathwise large deviations for white noise chaos expansions

Speaker: Alexandre Pannier, Université Paris-Cité.

Abstract:

We consider a family of continuous processes, measurable with respect to a white noise measure, and characterised by their Wiener chaos expansion. We provide sufficient conditions for their large deviations principle to hold in path space, thereby refreshing a problem left open by Pérez-Abreu in 1993 in the Brownian motion case. The proof is based on the weak convergence approach to large deviations: it involves demonstrating the convergence in distribution of certain perturbations of the original process, and thus the main difficulties lie in analysing and controlling the perturbed multiple stochastic integrals. Moreover, adopting this representation offers a new perspective on pathwise large deviations and induces a variety of applications thereof.

## Friday April 28, 2023.Gambler’s ruin in a game with many players

Speaker: Denis Denisov, University of Manchester.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-210,

Abstract:

Recently Diaconis and Ethier [1] discussed a game with three players. The players 1, 2 and 3 start to play this game with some respectively positive capitals A,B and C. At each round a pair of players is chosen uniformly at random and then a fair coin is tossed to decide who win in this round. After that one pound is transferred from one of players in this pair to another. Players are eliminated when their capital becomes zero. The game ends when only one player is left. The question of interest is the probability of winning this game for player 1 when A and B are fixed and total capital N=A+B+C increases to infinity. We use the Brownian approximation for random walks in cones to find exact asymptotics for this probability.

The talk is based on a joint work with V. Wachtel [2].

[1] Diaconis and Ethier (2022). Gambler’s ruin and ICM. Statistical Science, 37(3), 289-305, https://doi.org/10.1214/21-STS826.

[2] Denisov and Wachtel (2022). Harmonic measure in a multidimensional gambler’s problem. arXiv: 2212.11526, https://arxiv.org/abs/2212.11526.

## Friday March 17, 2023. Quantitative fluctuation analysis of multiscale dynamical systems

Speaker: Solesne Bourguin, Boston University.

Abstract:

In this talk, we consider multiscale dynamical systems perturbed by a small Brownian noise and study the limiting behavior of the fluctuations around their deterministic limit from a quantitative standpoint. Using a second order Poincare inequality based on Malliavin calculus, we obtain rates of convergence for the central limit theorem satisfied by the slow component in the Wasserstein metric.

This is joint work with K. Spiliopoulos.

## Friday March 03, 2023. Strong laws for growth-fragmentation processes with bounded cell size

Speaker: Alex Watson, University College London.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-210,

Abstract:

A growth-fragmentation is a stochastic process representing cells with continuously growing mass, which experience sudden splitting events. Growth-fragmentations are used to model cell division and protein polymerisation in biophysics. It is interesting to ask whether these processes converge toward an equilibrium, in which the number of cells is growing exponentially and the distribution of cell sizes approaches some fixed asymptotic profile. In this work, we study a process in which the growth and splitting of an individual cell is largely independent of its mass, with the exception that the mass is bounded above, so it cannot exceed a given constant. We give precise conditions to ensure that, almost surely, the process exhibits this equilibrium behaviour, and express the asymptotic profile in terms of an underlying Lévy process. Finally we will compare this with other recent work on the connection with quasi-stationary distributions.

This is joint work with Emma Horton (Inria Bordeaux) and Denis Villemonais (University of Lorraine).

## Friday February 17, 2023. Point process convergence of random walks

Speaker: Jorge Yslas Altamirano, University of Liverpool.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-210,

Abstract:

In this talk, we study point process convergence for sequences of random walks. First, we focus on point process convergence for sequences of i.i.d. random walks, with the objective of deriving asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. In particular, we show convergence of the maximum random walk to the Gumbel distribution under the existence of a (2 + $\delta$)th moment. Then, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size.

## Friday February 03, 2023. A growth-fragmentation-isolation process on random recursive trees and contact tracing

Speaker: Linglong Yuan, University of Liverpool.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-210,

Abstract:

The evolution of random recursive trees (RRT) through growth, fragmentation and isolation is studied to model the outbreak of pandemic. A RRT is to represent a finite set of patients connected by edges if the infector-infectee information is retrievable in the contact tracing records. Growth means a RRT infects a new individual; fragmentation happens when the information of who infected a patient is lost, resulting in a RRT split into two subtrees; isolation occurs if a patient in a RRT is detected so we put every patient in the RRT to self-isolation. By assigning occurrence rates to growth, fragmentation and isolation, we show when the number of (non-isolated) patients goes to infinity or to zero, and the existence of a phase transition, and give the convergence speed upon survival of pandemic. The main tool is a theorem for non-conservative semigroups and the martingale approach applied to a Markov branching process with infinite types.

This is a joint work with Vincent Bansaye (Ecole Polytechnique) and Chenlin Gu (Tsinghua University).

## Friday Dec 9, 2022. A Feynman Kac duality for structured CSBP with competition under cell division

Speaker: Anita Winter, Universität Duisburg-Essen.

Abstract:

In this talk we introduce a 2-level model which describe cells hosting a virus population. The cells split binary and at a split event divide their virus load to the two daughter cells, while the virus populations within the cells perform independent CSBPs with quadratic competition. We formulate a well-posed martingale problem. An important tool for uniqueness will be a Feynman-Kac duality relation, which can also be used to derive the basic longterm behaviour. We will give criteria for extinction and survival of the virus load within a typical cell sampled at a late time $t$.

Joint work with Luis Osorio

## Friday Nov 25, 2022. Limit theorems for a random walk on oriented percolation

Speaker: Timo Schlüter, University of Basel.

Abstract:

We consider a directed random walk X = (X_{n})_{n} in a random environment on Z^{d}. The environment is given by the backbone of an oriented percolation cluster. We show the existence of a measure Q that is absolutely continuous with respect to the original measure P and invariant with respect to the point of view of the particle. Furthermore, we prove a quenched local limit theorem using the Radon-Nikodym derivative obtained from Q with respect to P as a suitable prefactor.

## Friday Nov 11, 2022. Poisson Hulls

Speaker: Ilya Molchanov, University of Bern.

Abstract:

We introduce a hull operator on Poisson point processes, the easiest example being the convex hull of the support. Assuming that the intensity measure of the process is known on the set generated by the hull operator, we discuss estimation of the expected symmetric statistics built on the Poisson process. The results are based on a spatial Markov property of the underlying Poisson process with respect to the hull. In special cases, our general scheme yields the estimator of the volume of the convex support or the estimator of an integral of a H\"older function. The setting is extended to higher order symmetric statistics, and it is shown that the estimation error is given by the Kabanov--Skorohod integral with respect to the underlying Poisson process. We derive the rate of normal convergence for the estimation error, and illustrate it on an application to estimators of integrals of a H\"older function.

Joint work with Guenter Last (Karlsruhe)

## Friday Oct 28, 2022. Nodal length of random spherical harmonics and Wiener chaos.

Speaker: Maurizia Rossi, University of Milano-Bicocca.

Abstract:

In this talk we study geometric properties of random eigenfunctions on manifolds. In particular, motivated by both Yau's conjecture and Berry's ansatz, we investigate the nodal length of random spherical harmonics (Gaussian Laplace eigenfunctions on the 2-dimensional sphere) for large eigenvalues. The main result is an asymptotic equivalence with the so-called sample trispectrum, allowing one to establish a central limit theorem for the nodal length. The key ingredients are Kac-Rice formulas, chaotic decomposition and fourth moment theorems by D. Nualart, I. Nourdin and G. Peccati.

This talk is mainly based on a joint work with D. Marinucci and I. Wigman.

## Friday Oct 21, 2022. Approximations for random sums with equally correlated summands

Speaker: Fraser Daly, Heriot-Watt University.

Abstract:

Let $Y=X_1+\cdots+X_N$ be a sum of a random number of random variables, where the random variable $N$ is independent of the $X_j$. Such random sums arise in many applications, including in the areas of financial risk, hypothesis testing and physics. Classically, the $X_j$ are assumed to be independent, in which case central limit theorems and other distributional approximation results for $Y$ are well known. However, this assumption of independent $X_j$ may be unrealistic in some applications. We relax this restriction, instead assuming that these random variables come from a generalized multinomial model. In this setting, we prove error bounds in Gaussian, Gamma and Poisson approximations for $Y$ which allow us to investigate the effect of the correlation parameter on the quality of the approximation, while also providing competitive bounds in the special case of independent $X_j$. Proofs make use of Stein's method in conjunction with size-biased and zero-biased couplings.

## Friday Oct 14, 2022. Conformally Invariant Random Fields and Liouville Quantum Gravity on Compact Riemannian Manifolds of Even dimension.

Speaker: Ronan Herry, University of Bonn.

Abstract:

In this talk, I will present a recent construction of the Liouville Quantum Field Theory in dimension $2n$ for $n \geq 1$. Our approach consists in giving a precise mathematical meaning to Polyakov's intuition in dimension 2, that is by quantization of the classical Liouville action.

More precisely, on admissible manifolds, we construct:

- A log-correlated and conformally invariant Gaussian field $h$, whose covariance is given by the Green kernel associated with a perturbation of the poly-Laplacian.
- The Gaussian multiplicative chaos associated with $h$, that is informally the random measure $e^{h(x)} dx$.
- The Polyakov-Liouville measure, that is informally the Gibbs measure associated with the classical Liouville action. We show that this measure is conformally quasi-invariant.

Even though motivated by physics, the talk will stay purely on the mathematical side and no prerequisites in physics are required. Joint work with L. Dello Schiavo, E. Kopfer, K-T. Sturm.

## Friday May 13, 2022. New CLTs and FCLTs for U-statistics with size-dependent kernels

Speaker: Christian Döbler, Heinrich-Heine-Universität Düsseldorf.

Abstract:

Since their introduction in a seminal paper by Hoeffding in 1948, U-statistics have played a major role in statistical science. Indeed, many important classical estimators and test statistics are of this form. Moreover, in more recent years, U-statistics have further been in frequent use in other fields like random geometry, big data and telecommunication networks, to name just a few. While in the classical situation, the kernel of the U-statistic remains fixed, modern applications often require that it may depend on the sample size n of the observed data. In this talk I will present several new results about the asymptotic normality of such U-statistics. The conditions for our results are phrased in terms of norms of so-called contraction kernels and are hence of an analytical nature. Moreover, since the arguments leading to our (functional) central limit theorems are based on Stein's method, they are also suitable for providing bounds on the rate of convergence. I will highlight the applicability of our results by discussing an example of subgraph counts in a random geometric graph model, which is also of relevance for goodness of fit testing and for a change point detection problem. If time allows, then I will further address the more general situation of kernels that might even vary with the evaluated data subset. This class of statistics for instance comprises weighted and incomplete U-statistics, which are ubiquitous in many fields of applications.

This talk is partly about joint work with Mikolaj Kasprzak and Giovanni Peccati from the MIT and the University of Luxembourg.

## Friday April 29, 2022. Large Deviations for Randomly Weighted Sums

Speaker: Shui Feng, McMaster University.

Abstract:

Let ${Z_{n}: n \geq 1}$ be a sequence of i.i.d. random probability measures. Independently, for each $n \geq 1$, let (X_{n1},...,X_{nn}) be a random vector of positive random variables that add up to one. The large deviation principles are established for the randomly weighted sum $\sum_{i=1}^{n}X_{ni}Z_{i}$. As applications, we obtain the large deviation principles for a class of randomly weighted means including the Dirichlet mean and the corresponding posterior mean.

## Friday March 25, 2022. Exact simulation of diffusions and their application to inference of natural selection from genetic time series data

Speaker: Paul Jenkins, University of Warwick.

14:00--15:00 (UK time, 45 minutes talk + questions/comments),

Place: Mathematical Sciences Building, Lecture Room MATH-103,

Abstract:

Diffusions arise in many areas of scientific modelling yet often their transition function is intractable, making them difficult to simulate. Furthermore, the fractal nature of a sample path means that it is impossible to store a full realisation using only finite memory. Despite these limitations, in many instances it is nevertheless possible to simulate a diffusion 'exactly', by which we mean to obtain exact realisations from the transition function at a finite collection of times, together with an easy recipe to fill in any further points if desired. The basic idea is to use rejection sampling with Brownian motion as the proposal process. In this talk I will show how these ideas can be extended to simulate exactly from the Wright-Fisher diffusion, a fundamental model of allele frequency evolution arising in population genetics. The chief difficulty here is that, owing to the finite boundaries of the Wright-Fisher diffusion, Brownian motion is unsuitable as a proposal process and we must seek an alternative. I will also show how exact simulation ideas can be embedded within a Markov Chain Monte Carlo algorithm in order to perform 'exact' inference of the parameters of the Wright-Fisher diffusion given noisy, discrete observations from it. This is important because time series of genetic data are becoming more commonly available through, for example, DNA sequencing of ancient skeletal fragments and through long-term tracking of pathogens.

This is joint work with Jere Koskela, Jaromir Sant, and Dario Spanò (University of Warwick).

## Friday March 18, 2022. An Overview of Adversarial Deep Learning

Speaker: George Kesidis, Pennsylvania State University.

Abstract:

Deep learning is a supervised process to heuristically train a model which may have billions of parameters, typically with a very large input-feature dimension, using a commensurately huge labelled training dataset. Though they may achieve state-of-the-art performance for some applications, deep neural networks (DNNs) are vulnerable to attacks on privacy, training data poisoning, and adversarial inputs at test-time. In this talk, we give an overview of such attacks and describe some defenses which are based on statistical anomaly detection in high-dimensional spaces.

## Friday March 04, 2022. Phase transition for the late points of random walk

Speaker: Perla Sousi, University of Cambridge.

Abstract:

Let be a simple random walk in with and let be the expected time it takes for to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set of points that have not been visited by time and prove that it exhibits a phase transition: there exists so that for all and all there exists a coupling between and two i.i.d. Bernoulli sets on the torus with parameters with the property that with probability tending to 1 as . When , we prove that there is no such coupling.

## Friday February 25, 2022. Fragmentation Process derived from $\alpha$-stable Galton-Watson trees

Speaker: Gabriel Hernan Berzunza Ojeda, University of Liverpool.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link

Abstract:

Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on $n$ labelled vertices. In particular, they showed that, after proper rescaling, the above fragmentation process converges as $n \rightarrow \infty$ to the fragmentation process of the Brownian CRT obtained by cutting-down the Brownian CRT along its skeleton in a Poisson manner.

In this talk, we will discuss the fragmentation process obtained by deleting randomly chosen edges from a critical Galton-Watson tree $\mathbf{t}_{n}$ conditioned on having $n$ vertices, whose offspring distribution belongs to the domain of attraction of a stable law of index $\alpha \in (1,2]$. The main result establishes that, after rescaling, the fragmentation process of $\mathbf{t}_{n}$ converges, as $n \rightarrow \infty$, to the fragmentation process obtained by cutting-down proportional to the length on the skeleton of an $\alpha$-stable L\'evy tree. We will also explain how one can construct the latter by considering the partitions of the unit interval induced by the normalized $\alpha$-stable L\'evy excursion with a deterministic drift. In particular, the above extends the result of Bertoin (2000) on the fragmentation process of the Brownian CRT.

The approach uses the well-known Prim's algorithm (or Prim-Jarník algorithm) to define a consistent exploration process that encodes the fragmentation process of $\mathbf{t}_{n}$. We will discuss the key ideas of the proof.

Joint work with Cecilia Holmgren (Uppsala University)

## Friday February 11, 2022. Extremal process of the branching Brownian motion in dimension d

Speaker: Bastien Mallein, Université Paris 13.

14:00--15:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link

Abstract:

The branching Brownian motion is a particle system in which every particles evolves independently of the other. Each particle moves according to a Brownian motion in dimension d, and split into to after an exponential time, independently of its displacement.

We take interest in the asymptotic behaviour of the particles that moved furthest from the origin as time grows. We show that these particles split into groups of particles growing in different directions, sampled according to a random measure Z(d\theta) on the sphere, with norms centered around the atoms of a Poisson point process with exponential intensity.

## Friday December 3, 2021. Minimal observable clade sizes in coalescent processes - properties and an application.

Speaker: Fabian Freund, University of Hohenheim.

Abstract:

We consider $n$-coalescent processes, in particular $\Lambda$-$n$-coalescents. A $\Lambda$-$n$-coalescent process represents a rooted random tree with $n$ leaves, an underlying Markovian structure and exchangeability when permutating leaves. This tree can be used as a model for the genealogy of a sample of $n$ DNA sequences. To then model such sequences, we define a Poisson process on the branches whose points model the differences (representing mutations) between the sequences.

The minimal observable clade of a leaf $i$ is the set of leaves of the smallest subtree containing $i$ whose connecting edge to the rest of the tree bears at least one Poisson point. If there is no such subtree, the clade is $\{1,\ldots,n\}$. In the application context from above, it can be interpreted as the smallest family of sequences containing $i$ that can be distinguished from sequence data.

For the size $O_n(i)$ of this clade for individual $i$, moments can be derived recursively for fixed $n$. Asymptotically, $O_n(i)$ is of order $n$ and moments of it limit can be derived. For the Bolthausen-Sznitman $n$-coalescent, the limiting distribution is a Beta distribution.

Using the minimal observable clade sizes $(O_n(1),\ldots,O_n(n))$ as an addition to other statistics of genetic diversity leads to decreased errors when performing model selection between different genealogy models based on DNA sequence data.

Joint work with Arno Siri-Jégousse (UNAM Mexico City, Mexico)

## Friday November 19, 2021. Iterating Bernstein operators, a fully stochastic approach.

Speaker: Takis Konstantopoulos, University of Liverpool.

Abstract:

A Bernstein operator takes a function and produces a polynomial that approximates the function when its degree tends to infinity, if the function is continuous. This is the classical result of Bernstein (1913) who gave a constructive approach to Weierstrass' approximation theorem (1885). The rate of convergence is of interest to function approximation theorists and is related to composing Bernstein operators ad libitum. We look at this composition as a an functional of an underlying Markov chain and provide a way to understand its limiting behaviour by examining the chain as well as its diffusion approximation. Interestingly, these objects are known in the probability literature as Wright-Fisher chain and diffusion and are arguably the most fundamental models of mathematical biology.

Joint work with Linglong Yuan and Michael Zazanis.

## Friday November 12, 2021. Discrete Self-similar and some ergodic Markov Chains.

Speaker: Rohan Sarkar, Cornell University.

Abstract:

A continuous time Markov chain X on Z_+ issued from n is called discrete self-similar if for any \alpha \in [0, 1] it satisfies the following identity for any t \geq 0, Binom(X(t, n), \alpha) = X(t, Binom(n, \alpha)), in distribution, where Binom is the binomial random variable. In this work we identify a large class of discrete self-similar Markov chains which also includes classical birth-death chains. We show that the semigroups associated with these Markov chains satisfy a "gateway" relation with the semigroups of self-similar Markov processes on R_+. As a consequence of this fact, we prove convergence of these chains to self-similar Markov processes on R_+ after scaling appropriately. By a linear perturbation of the generator of the discrete self-similar Markov chains, we obtain a class of ergodic Markov chains that are non-reversible. In this talk I will explain how we obtain the spectral properties and entropy convergence of these Markov chains to their equillibrium distributions using the concept of intertwining and interweaving relationship.

Joint work with Laurent Miclo and Pierre Patie.

## Friday November 5, 2021. Stopping sets and variance estimates on the Poisson space.

Speaker: Giovanni Peccati, Department of Mathematics, Université du Luxembourg.

Abstract:

I will present several new estimates, allowing one to assess variances of random variables depending on Poisson random point configurations - based on the two notions of "restricted hypercontractivity" and "stopping set". In particular, stopping sets are the crucial elements featured in new intrinsic forms of the "OSSS" and "Schramm-Steif" inequalities - that one can use in order to establish sharp phase transitions and noise sensitivity in continuum percolation models.

Based on joint works with G. Last & D. Yogeshwaran (Karlsruhe and Bangalore) and I. Nourdin & X. Yang (Luxembourg and Brunel University, London).

## Friday October 22, 2021. On semi-Markov processes and evolutions and their interplay with non-local equations.

Speaker: Bruno Toaldo, Department of Mathematics "Giuseppe Peano", University of Turin.

Abstract:

In this talk, we aim to give an introduction to the interplay between semi-Markov processes, semi-Markov evolutions and non-local in time equations. This interplay generalizes the well-known theory of Markov processes, evolutions and Cauchy problems. In the case of semi-Markov processes the effect of the semi-Markov memory is represented in the equations by non-local operators acting in the time variable. In the case of semi-Markov evolutions the memory induces non-local operators acting both in space and time. Possible applications the theory of anomalous diffusion will be also discussed.

## Thursday October 14, 2021. Collatz orbits of sparser numbers (organised jointly with Dynamics seminar)

Speaker: Professor Mary Rees, University of Liverpool

13:00--14:00 (UK time)

Place: Room MATH-106

Zoom link: https://liverpool-ac-uk.zoom.us/j/94522522974?pwd=ejBHNSthSEd0ZHFiZzUvL1dLdEZDdz09

Meeting ID: 945 2252 2974 Passcode: Fatou_1926

Abstract:

The Collatz map sends an integer n to n/2 if n is even and 3n+1 if n is odd. So the orbit of 1 is 1,4,2,1,4,2,1..

The orbit of 3 is 3,10,5,16,8,4,2,1.. The famously intractable Collatz conjecture, which is thought to have been first circulated by word of mouth in 1950, states that the orbit of every strictly positive integer ends in the cycle 1,4,2..

It is not even known if this problem is decidable. John Conway proved in the 1950's that a more general problem is not decidable. What positive information we have about the Collatz map is largely, although not entirely, of a probabilistic nature. It was shown in the 1970's that, for most positive integers n, in the sense of positive density, the Collatz orbit of n passes below n. Terence Tao's 2019 paper ``Almost all Collatz orbits attain almost bounded values'' is a powerful strengthening of this basic result.

Most work on the Collatz conjecture is focussed on ``typical'' integers, which essentially means those n such that the orbit of n starts by passing below n in a reasonable amount of time. Tao's method shows that the orbits such numbers typically continue to decrease for some time.

This talk will concentrate on numbers which are not necessarily typical -- by looking not at the standard measures of uniform density on intervals of integers, but at other measures on integers: the so called geometric measures, and integers which are typical for such a measure. These are numbers whose Collatz orbits start by increasing, depending on the mean of the geometric measure. A chain of conjectures will be stated which, if true, show that the Collatz orbits of these sparser numbers do later decrease for some time. A first weaker version of the final conjecture in the chain has been proved: not enough to be the basis of an induction, but a step in the right direction.

## Friday October 8, 2021. On the profile of trees with a given degree sequence.

Speaker: Osvaldo Angtuncio-Hernández, Universität Duisburg-Essen.

Abstract:

A degree sequence is a sequence $\bf S=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\p_{{\bf S}}$ on rooted plane trees whose degree sequence equals ${\bf S}$, giving conditions for the convergence of the profile (sequence of generation sizes) as the size of the tree goes to infinity. This provides a more general formulation and a probabilistic proof of a conjecture due to Aldous (1991). Our formulation contains and extends results in this direction obtained previously by Drmota and Gittenberger (1997) and Kersting (2011). A technical result is needed to ensure that trees with law $\p_{{\bf S}}$ have enough individuals in the first generations, and this is handled through novel path transformations and fluctuation theory of exchangeable increment processes. As a consequence, we obtain a boundedness criterion for the inhomogeneous continuum random tree introduced by Aldous, Miermont and Pitman (2004).

This is a joint work with Dr. Gerónimo Uribe Bravo

## Friday Jun 11, 2021. Strong stationary times for finite Heisenberg walks.

Speaker: Professor Laurent Miclo, Institut de Mathématiques de Toulouse, Toulouse School of Economics CNRS and University of Toulouse.

Abstract:

A strong stationary time associated to an ergodic Markov chain $(X_n)_n$ is a finite stopping time $\tau$ such that $\tau$ and $X_\tau$ are independent and $X_\tau$ is distributed according to the invariant probability. Such a time enables to estimate the speed of convergence to equilibrium of $(X_n)_n$ in a probabilistic way. After presenting classical examples and general principles for the construction of strong stationary times, we will introduce a first strong stationary time for finite Heisenberg walks.

## Friday May 14, 2021. Creeping of Lévy processes through curves

Speaker: Professor Loïc Chaumont, University d'Angers.

Abstract: *A real Lévy process X is said to creep upward if for some (and hence for all) x>0, with positive probability, the process X hits x at its first passage time above x. Necessary and sufficient conditions for a Lévy process to creep upward are well known as well as the probability to creep through each level x. In this work we study the creeping property of a Lévy process X issued from 0 through the graph of a continuous, non increasing function f such that f(0)>0. We give an expression of the probability that X creeps through f in terms of the drift coefficients of the upward ladder processes and the density of the entrance law of the excursions of X reflected at its infimum. **This is a joint work with my PhD student Thomas Pellas.*

## Friday Apr 30, 2021. Convergence of ancestral processes from non-neutral Cannings models

Speaker: Dr. Jere Koskela, University of Warwick.

Abstract: Cannings models are a wide class of stochastic models for the evolution of genetic traits in finite populations of fixed size. The population reproduces clonally and in discrete generations, with the next generation being formed from the current one by sampling a random vector specifying the family size for each individual, and with offspring inheriting the trait of their parent. When the model is neutral, i.e. the family size of an individual is independent of their trait, the ancestral tree embedded into a Cannings model by tracking parent-offspring relations backward in time is known to converge weakly to the Kingman coalescent under a natural condition on the third moments of family sizes, and a suitable rescaling of time. I will show that the same convergence takes place under analogous conditions in the non-neutral case, albeit with a less tractable time rescaling, and present some applications for sequential Monte Carlo algorithms, which can be thought of as examples of non-neutral Cannings models.

## Friday Apr 23, 2021. The functional Breuer-Major theorem

Speaker: Professor Ivan Nourdin, University of Luxemburg.

Abstract: Let $X$ be a zero-mean stationary Gaussian sequence of random variables with covariance function $p$ satisfying $p(0)=1$. Let $f$ be a function of Hermite rank $d$ bigger than 1. The celebrated Breuer–Major theorem asserts that, if $p^d$ is summable then the finite dimensional distributions of the normalized sum of $f(X_i)$ converge to those of a standard Brownian motion (up to some multiplicative constant). Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the space of càdlàg functions endowed with the Skorohod topology was still missing. We will explain in this talk how to fill this gap. This is joint work with David Nualart, based on a recent paper published in PTRF.

## Friday Apr 16, 2021. Voronoi cells in random split trees

Speaker: Dr. Cécile Mailler, University of Bath.

Abstract: Take k nodes uniformly at random in a n-node graph, and let k competing epidemics spread along edges at constant speed from these initial nodes, infecting nodes when they reach them. A node can only get infected by one of the epidemics, the first one that reaches it. We are interested in the sizes of the final territories of the k epidemics, which we call the Voronoi cells of the k initial nodes. In this joint work with Markus Heydenreich (Munich) and Alex Drewitz (Cologne), we prove limiting theorems for the sizes of the Voronoi cells of k nodes taken uniformly at random in an n-node random split tree.

## Friday Mar 19, 2021. The Abelian sandpile model and its stochastic variants

Speaker: Dr. Thomas Selig, Xi'an Jiaotong-Liverpool University

13:00--14:00 (UK time, 45 minutes talk + questions/comments), via the permanent Zoom link https://liverpool-ac-uk.zoom.us/j/92609198321?pwd=QktiRlVzOGhPUHo4R1doTWJRTDIzUT09

Abstract: The sandpile model is a dynamic process on a graph G. At each unit of time, a grain of sand is added to a randomly selected vertex of G. When this causes the number of grains at that vertex to exceed a certain threshold (usually its degree), that vertex is said to be unstable, and topples, sending grains to (some of) its neighbours in G. This may cause other vertices to become unstable, and they topple in turn. A special vertex called the sink absorbs excess grains from the system, and so the process eventually stabilises. In the standard Abelian sandpile model (ASM), topplings are deterministic: one grain is sent to each neighbour of an unstable vertex. Stochastic variants of the ASM makes these topplings random: an unstable vertex sends grains to a random subset of its neighbours. There are a number of such variants in the literature, depending on how this random subset is chosen.

In this talk, we will expose some of the important results and recent developments of the ASM and introduce some of its more popular stochastic variants. We will then analyse in more detail one such variant, known as the Bernoulli stochastic sandpile model, in which unstable vertices flip a coin for each neighbour to decide whether to send a grain to it or not.

## Friday Mar 12, 2021. k-CUT MODEL FOR THE BROWNIAN CONTINUUM RANDOM TREE

Speaker: Dr. Minmin Wang, University of Sussex.

Abstract: To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the k-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the k-cut model to isolate the root of a Galton–Watson tree with a finite-variance offspring law and conditioned to have n nodes, when divided by n^{1-1/2k}, converges in distribution to some random variable defined on the Brownian Continuum Random Tree. In this talk, we present a direct construction of the limit random variable, relying upon the Aldous–Pitman fragmentation process and a deterministic time change.

## Friday Feb 26, 2021. Interval-partition evolutions as limits of random walks on compositions, and their applications

Speaker: Dr. Quan Shi, Universität Mannheim.

Abstract: A *composition *of a positive integer *n *is a sequence of positive integers that sum to *n.* In this talk, we* *will introduce a family of interval-partition-valued diffusions as limits of random walks on integer compositions. These infinite-dimensional diffusions have Poisson--Dirichlet (pseudo-)stationary distributions and are closely related to the limits of random walks on integers *partitions* studied by Borodin--Olshanski and Petrov. I will also talk about some applications of our model in population genetics and continuum-tree-valued dynamics.

This talk is based on joint works with Noah Forman (McMaster), Douglas Rizzolo (Delaware), and Matthias Winkel (Oxford).

## Friday Feb 19, 2021. Quantitative two-scale stabilisation on the Poisson space

Speaker: Dr. Xiaochuan Yang, University of Bath.

Abstract: The stabilisation theory was put forward in 2001 by Penrose and Yukich as a kind of high level abstraction of the famous CLT for minimal spanning trees of Kesten and Lee. Since its birth, this theory constitutes one of the most fundamental ideas for proving Gaussian approximation of stochastic geometric models, e.g. coverage processes, random tessellations, spatial networks etc. In a recent joint work with G. Peccati and R. Lachieze-Rey, we develop a quantitative stabilization theory which gives rates of multivariate Gaussian approximation for general stabilizing Poisson functionals, extending some estimates from a recent paper of Chatterjee and Sen on the rate of normal convergence of minimal spanning trees. Several examples are worked out to illustrate our results, including the online nearest neighbor graphs, edge statistics of Euclidean minimal spanning trees, and excursion of heavy tailed shot noise random fields. Based on the preprint https://arxiv.org/abs/2010.13362.

## Friday Dec 11, 2020. The scaling limit of a critical random directed graph.

Speaker: Dr. Robin Stephenson, University of Sheffield

Abstract: We consider the random directed graph D(n,p) with vertex set {1,2,...,n} in which each of the n(n − 1) possible directed edges is present independently with probability p. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at p = 1/n, with critical window p = 1/n + λn⁻⁴/³ for λ ∈ ℝ . We show that, within this critical window, the strongly connected components of D(n,p), ranked in decreasing order of size and rescaled by n⁻¹/³ , converge in distribution to a sequence (C₁,C₂, ...) of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an L^1 sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős–Rényi random graph G(n,p), whose scaling limit is well understood. We show that the limiting sequence (C₁,C₂, ...) contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.

Joint work with Christina Goldschmidt.

## Friday Dec 04, 2020. Around the almost sure central limit theorem of Salem-Zygmund.

Speaker: Dr. Guillaume Poly, University of Rennes 1

Abstract: We will review some recent results around this limit theorem and provide a simple and self-contained proof. Then, we will discuss applications to the roots of random trigonometric polynomials.

## Friday Nov 20, 2020. Attraction to and repulsion from patches on the hypersphere and hyperplane for isotropic d-dimensional α-stable processes with index in α∈(0,1) and d≥2.

Speaker: Professor Andreas Kyprianou, University of Bath

Abstract: Consider a d-dimensional α-stable processes with index in α∈(0,1) and d≥2. Suppose that Ω is a region of the unit sphere S^{d−1} = {x ∈ R^d : |x| = 1}. We construct the aforesaid stable Lévy process conditioned to approach Ω continuously, either from inside S^{d−1}, from outside S^{d−1} or in an oscillatory way; all of which have zero probability. Our approach also extends to the setting of hitting bounded domains of (d-1)-dimensional hyperplanes. We appeal to a mixture of methods, appealing to the modern theory of self-similar Markov process as well as the classical potential analytic view.

Joint work with Tsogzolmaa Saizmaa (National University of Mongolia), Sandra Palau (UNAM, Mexico) and Mateusz Kwasniki (Technical University of Wroclaw).

## Thursday Nov 19, 2020. The van Dantzig problem and the Riemann hypothesis (organised jointly with Dynamics seminar)

Speaker: Professor Pierre Patie, University of Liverpool

Abstract: In this talk, we start by introducing the intriguing van Dantzig problem which consists in characterizing the subset of Fourier transforms of probability measures on the real line that remain invariant under the composition of the reciprocal map with a complex rotation.

We first focus on the so-called Lukacs class of solutions that is the ones that belong to the set of Laguerre-Pόlya functions which are entire functions with only real zeros. In particular, we show that the Riemann hypothesis is equivalent to the membership to the Lukacs class of the Riemann ξ function.

We state several closure properties of this class including adaptation of known results of Pόlya, de Bruijn and Newman but also some new ones. We proceed by presenting a new class of entire functions, which is in bijection with a set of continuous negative definite functions, that are solutions to the van Dantzig problem and discuss the possibility of the Riemann ξ function to belong to this class.

## Friday Nov 13, 2020. Non-standard limits for a family of autoregressive stochastic sequences

Speaker: Professor Sergey Foss, Heriot-Watt University

Abstract:

We consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We obtain a non-standard limit theorem where the limiting distribution is a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case where there is no the restart mechanism.

Joint work with Matthias Schulte.

## Friday 25th October 2019 - Fatou's Lemmas for Varying Probabilities and their Applications to Sequential Decision Making

Speaker: Distinguished Professor Eugene A. Feinberg, Stony Brook University (USA)

16:00 MATH-211

Abstract: The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma.

We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time.

This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Nina Zadoianchuk, and Michael Zgurovsky.

## Wednesday 12th June 2019 - Comparison Theorem of Stochastic Differential Equations

Speaker: Chenggui Yuan (Swansea University)

16:00 MATH-117

Abstract: In this talk, the existence and uniqueness of strong solutions to distribution dependent neutral SFDEs are proved. We give the conditions such that the order preservation of these equations holds. Moreover, we show these conditions are also necessary when the coefficients are continuous. Under sufficient conditions, the result extends the one in the distribution independent case, and the necessity of these conditions is new even in distribution independent case.

## Friday 1st March 2019 - Stochastic Porous Media Equations: An Entropy Approach

Speaker: Konstantinos Dareiotis (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

12:00 MATH-105

Abstract: Nonlinear diffusion equations describe macroscopic phenomena such as flow of gas or fluid in a porous medium, heat propagation with temperature-dependent conductivity, and the evolution of crowd-avoiding populations. Often, a source term is present that forces/removes mass into/from the system. This source term can be random. As a particular example, generalised stochastic porous media equations (SPME) appear as scaling limits of the empirical measure of interacting branching diffusion particle systems. The interaction leads to a non-linear, degenerate second order operator in the drift, while the randomness of the branching mechanism leads to a non-linear noisy source term. We will discuss the main difficulties towards well posedness of these equations and introduce the concept of entropy solutions. In the class of entropy solutions we obtain well-posedness, L 1 -contraction, and stability estimates. Our results cover the full range of powers of the porous medium operator and allow for nonlinearities in the noise that are 1/2-Hölder continuous. This is a joint work with Mát´e Gerencs´er and Benjamin Gess.