**To access the abstracts and links to the video-recordings of the talks and PDF files with talks’ slides, click the title of the respective seminar below. **

**#1.** **Yamazaki Kazutoshi** (Kansai University, Japan, 18/06/2020): ** Lévy processes observed at Poisson arrival times and their applications in stochastic control**

**#2.** **Linxiao Chen** (University of Helsinki, Finland, 25/06/2020): ** Phase transition in the Ising model on a random 2D lattice**

**#3** **Adrian Röllin** (National University of Singapore, Singapore, 02/07/2020)**: Higher-order fluctuations in dense random graph models**

**#4** **Stephen Muirhead** (The University of Melbourne, Australia, 24/09/2020):** **** The phase transition for planar Gaussian percolation models without positive associations **

**The phase transition for planar Gaussian percolation models without positive associations**

**#5.** **Ross Maller** (Australian National University, Australia, 01/10/2020): ** A Generalized Dickman Distribution and the Number of Species in a Negative Binomial Process Model**

**#6.** **Gesine Reinert** (Oxford University, England, 15/10/2020): ** Stein’s method for multivariate distributions, kernelized goodness of fit statistics, and exponential random graph models**

**#7.** **Wie Qian **(CNRS & Université Paris-Saclay, France, 22/10/2020): ** Generalized disconnection exponents for Brownian loop-soup**

**#8.** **Fima Klebaner **(Monash University, Australia, 29/10/2020): ** A new result on the long term behaviour of a dynamical system perturbed by small noise and started near an unstable fixed point, with an application to the Bare Bones evolution model for an establishment of a mutant**

**#9.** **Elie Aidekon**** **(Laboratoire de Probabilités et Modèles Aléatoires, France, 05/11/2020): **Cluster explorations of the loop soup on a metric graph related to the Gaussian free field**

**#10.** **Takis Konstantopoulos **(The University of Liverpool, England, 12/11/2020): **Longest and heaviest paths in random directed graphs**

**#11.** **Antar Bandyopadhyay**** **(Indian Statistical Institute, India, 19/11/2020): **Random Recursive Tree, Branching Markov Chains and Urn Models**

**Random Recursive Tree, Branching Markov Chains and Urn Models**

**#12.** **Laurent Tournier** (Université Sorbonne Paris Nord, France, 26/11/2020): **The phase transition in three-speed ballistic annihilation**

**#13.** **Xiao Fang** (The Chinese University of Hong Kong, HK, 17/12/2020): **Error bounds in multivariate normal approximations **

**Xiao Fang**(The Chinese University of Hong Kong, HK, 17/12/2020):

**Error bounds in multivariate normal approximations**

**#14.** **Jevgenijs Ivanovs** ** (Aarhus University, Denmark, 04/03/2021): ****Recovering Brownian and jump parts from high-frequency observations of a Lévy process**

**Recovering Brownian and jump parts from high-frequency observations of a Lévy process**

**#15.** **David Croydon** ** (Kyoto University, Japan, 18/03/2021): ****Anomalous scaling regime for one-dimensional Mott variable-range hoppings**

**Anomalous scaling regime for one-dimensional Mott variable-range hoppings**

**#16.** **Erwin Bolthausen** ** (Universität Zürich, Switzerland, 22/04/2021): Approximate message passage and the TAP equation**

## #1. Thursday 18 June 2020

**Yamazaki Kazutoshi** (Kansai University, Japan): **Lévy processes observed at Poisson arrival times and their applications in stochastic control**

__Abstract__: Lévy processes are used in various fields such as finance, insurance and queues. Their recent developments have enabled us to achieve realistic models, generalizing greatly the traditional Brownian motion models. Lévy processes are examples of affine processes and are closely related to other stochastic processes, such as branching and self-similar processes. In this talk, I will review recent developments on Lévy processes observed at Poisson arrival times and discuss their applications in stochastic optimal control.

Video-recording of the talk: https://youtu.be/g0kaQHdElsc

PDF file with slides: PVSeminar_talk_2020_06_18

## #2. Thursday 25 June 2020

**Linxiao Chen** (University of Helsinki, Finland): **Phase transition in the Ising model on a random 2D lattice**

__Abstract__: The Ising model is one of the first statistical mechanics models known to have a non-trivial phase transition in two dimensions. On regular lattices, this phase transition has been extensively studied. In this talk, I will present an annealed Ising model on a random 2D lattice, introduced first in the Physics literature as a model of quantum gravity in 2D. We will see that the partition function of this model is exactly solvable. We show that this model has a phase transition at a unique temperature by examining its free energy, its critical exponents, and the scaling limit some interface lengths. In particular, the result confirms the physical intuition that a random lattice coupled to a non-critical Ising model has a geometry similar to a uniform random lattice with an instance of Bernoulli percolation on it. I will also discuss how the phase transition takes place in the near-critical window. Based on arxiv:1806.06668, arXiv:2003.09343 and a joint work in progress with Joonas Turunen.

Video-recording of the talk: https://youtu.be/37L83dfz9KA

PDF file with slides: PVSeminar_talk_2020_06_25

## #3. Thursday 2 July 2020

**Adrian Röllin** (National University of Singapore, Singapore)**: Higher-order fluctuations in dense random graph models**

__Abstract__: Dense graph limit theory is essentially a first-order limit theory analogous to the classical Law of Large Numbers. Is there a corresponding central limit theorem? We believe so. Using the language of Gaussian Hilbert Spaces and the comprehensive theory of generalised U-statistics developed by Svante Janson in the 90s, we identify a collection of Gaussian measures (aka white noise processes) that describes the fluctuations of all orders of magnitude for a broad family of random graphs. We complement the theory with error bounds using a new variant of Stein’s method for multivariate normal approximation, which allows us to also generalise Janson’s theory in some important aspects. This is joint work with Gursharn Kaur.

Video-recording of the talk: https://youtu.be/h69yggVJzAM

PDF file with slides: PVSeminar_talk_2020_07_02

## #4. Thursday 24 September 2020

**Stephen Muirhead ** (The University of Melbourne, Australia)**: The phase transition for planar Gaussian percolation models without positive associations **

__Abstract__: Given a smooth stationary centred Gaussian field *f* on the plane and a level ℓ in ℝ, we study the connectivity properties of the set {*f* < ℓ}. We prove that the critical level is ℓ_{c} = 0 under only symmetry and (very mild) correlation-decay assumptions, which includes the important example of the *random plane wave.* Since these models are not necessarily positively associated (i.e. they do not enjoy the “Fortuin-Kasteleyn-Ginibre (FKG) inequality”), many classical arguments from percolation/statistical mechanics do not apply, and so these are a rare example of non-FKG models whose critical point can be rigorously computed.

Although many arguments are specific to the Gaussian setting we hope that our techniques may be adapted to analyse other non-FKG models, of which there are many important examples (e.g. anti-ferromagnetic Ising models, certain regimes of the FK model and *O*(*n*) loop models, random current models, Boolean models on non-Poisson point processes etc). This is joint work with Hugo Vanneuville and Alejandro Rivera and will appear on arXiv very soon.

Video-recording of the talk: https://youtu.be/2nvKDKnxVeU

PDF file with slides: pvseminar_talk_2020_09_24

## #5. Thursday 1 October 2020

**Ross Maller** (Australian National University, Australia): **A Generalized Dickman Distribution and the Number of Species in a Negative Binomial Process Model.**

__Abstract__: In Ipsen, Maller, Shemehsavar (*J. Theoret. Prob*., 2019) we defined a new class of distributions related to Kingman’s PD_{α} distribution, which we called PD_{α}^{(r)}. It has two parameters, α∈(0,1) and* r* > 0.

A version of Ewens’ sampling formula was derived for it, and we obtained a formula for the probability mass function of *K _{n}*, the number of allele types/species observed in a sample of size

*n*from PD

_{α}

^{(r)}.

The aim of this seminar is to sketch a derivation of the large-sample distribution of *K _{n}* as

*n*→ ∞.

We cite a set of genetics data on the near-threatened marsupial quoll as a nice motivation. [Joint work with Yuguang Ipsen and Soudabeh Shemehsavar.]

Video-recording of the talk: https://youtu.be/tt0nn7xnOII

PDF file with slides: PVSeminar_talk_2020_10_01.pdf

## #6. Thursday 15 October 2020

**Gesine Reinert **(Oxford University, England): **Stein’s method for multivariate distributions, kernelized goodness of fit statistics, and exponential random graph models**

**Abstract**: Assessing the goodness of fit of models with continuous distributions for which the likelihood cannot be evaluated directly can be tackled using kernels which are based on Stein’s method for continuous multivariate distributions. Often independent replicas are assumed for this method. When the data are given in the form of a network, usually there is only one network available. If the data are hypothesised to come from an exponential random graph model, the likelihood cannot be calculated explicitly. Using a Stein operator for these models we introduce a kernelized goodness of fit test and illustrate its performance.

This talk is based on joint work with Guillaume Mijoule, Nathan Ross, Yvik Swan and Wenkai Xu.

Video-recording of the talk: https://youtu.be/2nvKDKnxVeU

PDF file with slides: PVSeminar_talk_2020_10_15

## #7. Thursday 22 October 2020

**Wei Qian **(CNRS & Université Paris-Saclay, France): **Generalized disconnection exponents for Brownian loop-soups.**

**Abstract**: We study the question of whether there exist double points on the boundaries of clusters in Brownian loop-soups. This question is closely related to our earlier works (with Werner) on the decomposition of Brownian loop-soup clusters.

More concretely, we introduce a notion of disconnection exponents which generalizes the Brownian disconnection exponents derived by Lawler, Schramm and Werner in 2001 (conjectured by Duplantier and Kwon in 1988). By computing the generalized disconnection exponents, we make the first prediction of the dimension of multiple points on the cluster boundaries in loop-soups. For the critical intensity of loop-soup, the dimension of double points on the cluster boundaries appears to be zero, making it more difficult to determine whether such points exist for the critical loop-soup. (We plan to rigorously compute these dimensions in an upcoming work.)

Our definition of the exponents is based on a certain general version of radial restriction measures that we construct and study. As an important tool, we introduce a new family of radial SLEs depending on κ and two additional parameters μ, ν, that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial SLEκ(ρ)s.

Video-recording of the talk: https://youtu.be/T8-g-8sfpRU

PDF file with slides: PVSeminar_talk_2020_10_22

## #8. Thursday 29 October 2020

**Fima Klebaner **(Monash University, Australia): **A new result on the long term behaviour of a dynamical system perturbed by small noise and started near an unstable fixed point, with an application to the Bare Bones evolution model for an establishment of a mutant.**

**Abstract**: In the talk the speaker will explain the title.

Video-recording of the talk: https://youtu.be/SK73LdmLg4U

PDF file with slides: PVSeminar_talk_2020_10_29

## #9. Thursday 05November 2020

**Elie Aidekon **(Laboratoire de Probabilités et Modèles Aléatoires, France): **Cluster explorations of the loop soup on a metric graph related to the Gaussian free field****.**

**Abstract**: We give a Markov property for the loop soup on a metric graph which mimics that of the Gaussian free field, and describe the law of the loop soup conditionally on its occupation field.

Video-recording of the talk: https://youtu.be/gS9V9tL7gAk

No PDF file with slides is available as the talk was given “live”: the speaker was writing on his tablet device.

## #10. Thursday 12 November 2020

**Takis Konstantopoulos** (The University of Liverpool, England): **Longest and heaviest paths in random directed graphs.**

**Abstract**: In this talk we give an overview of research in the area of random directed graphs with possibly random edge weights. We are interested in the longest paths between two vertices (or heaviest paths if there are weights). Typically, the longest path satisfies a law of large numbers and a central limit theorem (which gives a non-normal distribution in the limit if the vertex set is not one-dimensional). The constant in the law of large numbers as a function of the graph parameters and weight distributions cannot be computed explicitly except, perhaps, in very simple cases. A lot of work has been done in obtaining bounds and in studying its behaviour. For example, is it a smooth function of the connectivity parameter p? These kinds of graphs appear in several areas: in computer science, in statistical physics, in performance evaluation of computer systems and in mathematical ecology. They originated in a paper by Barak and Erdos but have also been studied independently, in connection with the applications above. The questions asked are related to the so-called last passage percolation problems because we can interpret “longest” in a time sense (what’s the worst case road that will take us from a point to another point?). As such, it is not surprising that in some cases, the limiting behaviour is related to limits of large random matrices. However, the complete picture is not understood and so open problems will also be presented.

Video-recording of the talk: https://youtu.be/cnyrRcaMJ80

PDF file with slides: PVSeminar_talk_2020_11_11

## #11. Thursday 19 November 2020

**Antar Bandyopadhyay** (Indian Statistical Institute, India): **Random Recursive Tree, Branching Markov Chains and Urn Models****.**

**Abstract**: In this talk, we will show that a *branching Markov chain* defined on the *random recursive tree* is nothing but a *balanced urn model*. This is a novel connection between these two apparently unrelated probabilistic models. Exploring the connection further we will derive fairly general scaling limits for balanced urn models with colors indexed by any Polish Space. Thus generalizing the classical urn models defined on finitely many colors to a measure-valued process on a general Polish Space. We will use the connection to show that several exiting results on classical finite color urn schemes, as well as, recently introduced infinite color urn schemes, can be derived easily from the general asymptotic. In particular, we will prove that the well-known classical result of the finitely many color balanced urn, when the replacement matrix is irreducible will also hold for countably infinite many colors (under certain technical assumptions). We will also discuss the special case, of null recurrent replacement matrices, which can only arise in the context of infinitely many colors. We will further show that the connection can also be used to derive exact asymptotic of the sizes of the connected components of a random recursive forest, which is obtained by removing the root of a random recursive tree. [Parts are joint work with Debleena Thacker and Svante Janson]

Video-recording of the talk: https://youtu.be/QBMkjFXVw5c

PDF file with slides: PVSeminar_talk_2020_11_19

## #12. Thursday 26 November 2020

**Laurent Tournier** (Université Sorbonne Paris Nord, France): **The phase transition in three-speed ballistic annihilation**

**Abstract**: In the ballistic annihilation model, particles are emitted from a Poisson point process on the line, move at constant speed (chosen i.i.d. at initial time) and mutually annihilate when they collide. This model was introduced in the 1990’s in physics, however its asymptotic behavior remains very poorly understood as soon as the speeds may take at least three values. We will focus on this minimal case when speeds may equal -1, 0 or 1, with symmetric probabilities, and show in particular that a phase transition takes place when 0-speed particles have probability 1/4, and discuss remarkable combinatorial properties of this model. This is based on joint works with J.Haslegrave and V.Sidoravicius.

Video-recording of the talk: https://youtu.be/ojDPxiftMVw

PDF file with slides: PVSeminar_talk_2020_11_26

## #13. Thursday 17 December 2020

**Xiao Fang** (The Chinese University of Hong Kong, HK): Error bounds in multivariate normal approximations

**Abstract**: We extend Stein’s celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. We discuss applications to Wishart matrices and the fourth-moment phenomenon. We also mention some recent progress in large-dimensional normal approximation of sums of independent random vectors and some open problems. This talk is based on joint works with Yuta Koike.

Video-recording of the talk: https://youtu.be/BgKWiJPV5sA

PDF file with slides: PVSeminar_talk_2020_12_17

## #14. Thursday 04 March 2021

**Jevgenijs Ivanovs** (Aarhus University, Denmark): **Recovering Brownian and jump parts from high-frequency observations of a Lévy process**

**Abstract**: We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a carefully chosen threshold, in which case both methods yield the same rate of convergence. This rate depends on the small-jump activity and is given in terms of the Blumenthal-Getoor index. Finally, we discuss possible extensions, including the multidimensional case, and provide numerical illustrations. [Joint work with Jorge González Cázares.]

Video-recording of the talk: https://youtu.be/iMW7JS4_uNU

PDF file with slides: PVSeminar_talk_2021_03_04

## #15. Thursday 18 March 2021

**David Croydon **(Kyoto University, Japan): **Anomalous scaling regime for one-dimensional Mott variable-range hopping
**

**Abstract**: I will present an anomalous, sub-diffusive scaling limit for a one-dimensional version of the Mott random walk, which was originally introduced as a model for electron transport in a disordered medium. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. A main goal of the talk is to explain the proof strategy, which relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces, which has also proved useful for other examples of random walks in random environments. [This talk is based on joint work with Ryoki Fukushima and Stefan Junk (both at the University of Tsukuba.]

Video-recording of the talk: https://youtu.be/pMT_TpsTAws

PDF file with slides: PVSeminar_talk_2021_03_18

## #16. Thursday 22 April 2021

** Erwin Bolthausen **(Universität Zürich, Switzerland):

**Approximate message passage and the TAP equation**

**Abstract**: Message passage is a method to approximately compute marginal distributions of probability distributions having an underlying graphical structure. It was reinvented several times. One of its first appearances was in decoding algorithms for error correcting codes. More recently, versions of it where used as an alternative to LASSO type methods in compressed sensing. Depending on the underlying graphical structure, it is possible to reduce the algorithmic complexity by an approximation step which results in what is called “Approximate Message Passage”. The high efficiency of the corresponding algorithms in compressed sensing was first observed numerically by Donoho and Montanari. A theoretical understanding was first obtained in spin glass theory where it is closely connected with the Thouless-Anderson-Palmer equation.

The talk aims at giving an overview. If the time allows, we will present some recent developments of the AMP-TAP approach to compute free energies.

Video-recording of the talk: https://youtu.be/37enYln53Uk

PDF file with slides: PVSeminar_talk_2021_04_22