# Archive 2019

**December 16, 2019**: Franco Flandoli (University of Pisa)

When: Monday, December 16, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*Brownian particles with local interaction: attempts and open problems on the PDE macroscopic limit*

Opposite to the theory of interacting particle systems on Z^d where the macroscopic behavior is usually well understood, the case of Brownian particles moving in R^d and subject to local interaction is less complete. We have been motivated to investigate this direction by the problem of modeling cell adhesion in biology; an overview of models proposed in the literature on this topic will be given but it is clear that the interaction is usually mean field or intermediate between mean field and local, like in the works of Karl Oelschleger. The case of true local interaction has been studied by Varadhan and few other authors and the results are fragmentary and less explicit from the quantitative viewpoint. We have devised heuristic computation and numerical test which produce some agreement and some discrepancy or open cases, and also in the case of agreement a rigorous proof is missing. The purpose of the talk is to illustrate this topic and the relative conjectures.

**December 02, 2019**:Thomas Nagler (Leiden University/TU Munich)

When: Monday, December 02, 16:00

Where:TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*Vine copula regression*

Vine copulas are graphical models for the dependence in a random vector.

In regression problems, we are interested in some aspects of the distribution of a response variable conditional on a set of predictors, e.g., conditional means, probabilities, or quantiles. Vine copulas can be used to model the dependence between response and predictors. There are two main questions: how can we tailor the vine structure to the regression problem? And how to extract the regression function from the joint dependence model? In this talk, I review several variants that were developed in recent years and discuss open problems.

**November 25, 2019**: Extreme TiDE seminar [link: http://evt-seminar.nl/]

When: Monday November 25, 15:00 - 17:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F).

**November 18, 2019**: Michele Salvi (École Polytechnique)

When: Monday, November 18, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*Scale-free percolation in continuous space*

Random graphs are a fundamental tool for the analysis of large

real-world networks (such as social networks, communication networks,

inter-banking systems and so on) which are not directly treatable, often

because of their size. The scale-free percolation random graph features

three properties that are never present at once in classical models, but

that are relevant for applications: (1) Scale-free: the degree of the

nodes follows a power law; (2) Small-world: two nodes are typically at a

very small graph distance; (3) Positive clustering coefficient: two

nodes with a common neighbour have a good chance to be linked.

We study a continuous version of scale-free percolation and try to infer

why it is a suitable model for the cattle trading network in France. Our

final goal is to understand how an epidemic would spread on this kind of

structures.

**October 28, 2019**: Barbara Franci (TU Delft)

When: Monday, October 28, 16:00

Where:TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture room D@ta

*Stochastic Nash Equilibrium problems*

Nash equilibrium problems have been widely studied and number of results are present concerning algorithms and methodologies to find an equilibrium. On the other hand, the analysis of the stochastic case is not fully developed yet. Several problems of interest cannot be modelled without uncertainty as, for instance, transportation systems, electricity markets or gas markets. One possible motivation for this lack of results is the presence of the expected value cost functions that can be hard to compute. The aim of this talk is therefore to describe the stochastic Nash equilibrium problem and a possible approach to find equilibria.

**October 14, 2019**: Maite Wilke Berenguer (University of Bochum)

When: Monday, October 14, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*The seed bank coalescent with spontaneous and simultaneous switching*

Population Genetics is an area of probability theory where mathematical structures arises from biological problems. Such is the case for the geometric seed bank model we introduced to describe a population with an active and a dormant form (picture plants with seeds). It models spontaneous switching, where individuals become active/dormant at a constant rate independently of each other as well as simultaneous switching, i.e. a correlation in their behaviour where positive fractions of the population become active/dorman simultaneously. Its scaling limits going backwards and forwards in time respectively are the seed bank coalescent and the seed bank diffusion (with spontaneous and simultaneous switching) and retain the moment duality.

We will compare the effect of both spontaneous and simultaneous switching through the property of "coming down from infinity" (or not) of the coalescent structures.

**September 23, 2019**: Philipp Sibbertsen (University of Hannover)

When: Monday, September 23, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*Robust Multivariate Local Whittle Estimation and Spurious Fractional Cointegration*

This paper derives a multivariate local Whittle estimator for the memory parameter of a possibly long memory process and the fractional cointegration vector robust to low frequency contaminations. This estimator as many other local Whittle based procedures requires a priori knowledge of the cointegration rank. It is shown that low frequency contaminations bias inference on the cointegration rank. We, therefore, also provide a robust estimator of the cointegration rank. Both estimators are obtained by trimming the periodogram. As all of our procedures are periodogram based we further derive some insights in the behaviour of the periodogram of a process under very general types of low frequency contaminations which may be of some interest on its own. An extensive Monte Carlo exercise shows the usefulness of our estimators in small samples. Our procedures are applied to realized betas of two American energy companies discovering that the series are fractionally cointegrated. As the series exhibit low frequency contaminations, standard procedures were unable to detect this relation.

**September 20, 2019: **Debleena Thacker, Uppsala University

When: Friday, September 20, 11:00

Where: TU Delft, Building 28, van Mourik Broekmanweg 6, Hilbert room, west second floor.

*Embedding balanced infinite color urn models into trees.*

Based on joint works with Antar Bandyopadhyay and Svante Janson. In this work the authors introduce the embedding into random recursive trees to study classical and generalized balanced urn models with non-negative balanced replacement matrices, for both finite and infinitely many colors. We provide a coupling of the balanced urn model with branching Markov chain on a random recursive tree, and use the properties of the later to deduce results for the former. We use this embedding to calculate the covariance between the proportions of any two colors when the replacement matrix is irreducible, aperiodic, positive recurrent and uniformly ergodic. This proves the strong law of large numbers for the proportion of colors. This method is especially useful for infinitely many colors, since the use of operator theory leads to technical difficulties for infinitely many colors.

**September 19, 2019**: Bernardo N.B. de Lima (extra talk at the Optimization seminar)

When: Thursday, September 19, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, Room Chip

*The Constrained-degree percolation model*

In the Constrained-degree percolation model on a graph (V,E) there are a sequence, (Ue)e∈E, of i.i.d. random variables with distribution U[0,1] and a positive integer k. Each bond e tries to open at time Ue, it succeeds if both its end-vertices would have degrees at most k−1. We prove a phase transition theorem for this model on the square lattice L2, as well on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase. Joint work with R. Sanchis, D. dos Santos, V. Sidoravicius and R. Teodoro.

**September 9, 2019: **Mini-workshop "*Critical behaviour of spin systems: phase transition, metastability and ergodicity*"

When: September 9, 10:00

Where: * Please mind the new location! *TU Delft Building 26; Van der Burghweg 1 A0.360

Program**:**

10.00 – 10.45 *Pierre-Yves Louis, U Poitiers*

11.00 – 11.45 *Christof Kϋlske, U Bochum*

12.00 – 13.30 Lunch break

13.30 – 14.15 *Aernout van Enter, U Groningen*

14.30 –15.15 *Bruno Kimura, TU Delft*

15.30 –16.00 Coffee break

16.00 –16.45 *Evgeny Verbitskiy, U Leiden*

**Titles and abstracts:**

Pierre-Yves Louis

*Systems of reinforced processes through mean-field interaction*

Abstract: Reinforced processes are used to study urns (Polya, Friedman rules), stochastic algorithms and in many applications... We consider systems of stochastic processes where the interaction holds through the reinforcement. Each component (urn) is updated in a parallel way at discrete time steps. We consider a mean field type interaction. We will present a class of such systems introduced these last years. Issues we will address are : long time behaviour, existence of an a.s. limit shared by the whole system (synchronization), nature of this limit : random or deterministic. Fluctuations are studied through central limit theorems. This talk is based on joint works with I. Crimaldi, P. Dai Pra, I. Minelli (hal-01277974, hal-01287461) and M. Mirebrahimi ⟨hal-01856584v2⟩.

Christof Kϋlske

*Metastates and measurable extremal decomposition in random spin systems *

Metastates are measures on the infinite-volume states of a random spin system (introduced by Newman and Stein) which depend measurably on the realization of the random environment.

They are useful in the presence of phase transitions to describe the large-volume asymptotics, also when chaotic volume-dependence may occur. We show that, for any metastate (possibly supported on non-extremal states) there is an associated decomposition metastate which has the same barycenter, and which is fully supported on the extremal states.

(Joint with Codina Cotar and Benedikt Jahnel, ECP Volume 23 (2018), paper no. 95)

Aernout van Enter

*Dyson models with random boundary conditions*

I discuss the behaviour of Dyson (long-range Ising) models with random boundary conditions. At low temperature, there is chaotic size-dependence, non-convergence of the Gibbs measure. The metastate , the distributional limit, is shown to be dispersed, and qualitatively a difference is shown to occur between decay power faster than 3/2 where the metastate is concentrated on mixed states and decay power slower than 3/2 when the metastate is concentrated on extremal Gibbs measures

Bruno Kimura

*Nucleation for 1D long range Ising models*

The rigorous study of metastability in the setting of stochastic dynamics is a relatively recent topic.

One of the most interesting problems that have been investigated is the study of the dependence on the dynamics of metastable behavior and nucleation toward the stable phase . Such class of problems appears in the literature considering several dynamic regimes, however, in most of them the microscopic interactions are assumed to be of shot range.

Therefore, the following questions naturally arise: Does indeed a long range interaction change substantially the nucleation process? Are we able to define in this framework a critical configuration triggering the crossover towards the stable phase?

In this talk I will show that under very general assumptions, the 1D long range Ising mode with a weak uniform field (without loss can be assumed to be positive) evolving according to the Metropolis dynamics, the state **-1** is a metastable configuration that nucleates toward the stable phase **+1**. It is possible to determine the tunneling time and the critical configurations that triggers the nucleation. Some model-dependent examples and generalizations are also discussed.

Evgeny Verbitskiy

*On the relation between one-sided and two-sided Gibbs measure*

We will discuss the relation between Gibbs measures on the lattices Z_+ and Z. Joint work with S. Berghout, A. van Enter, and R. Fernandez.

June 3, 2019: Alberto Chiarini (ETH Zürich)

When: Monday, June 3rd, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*Entropic repulsion for the occupation-time field of random interlacements by disconnection.*

The model of random interlacements was introduced in 2007 by A.-S. Sznitman, motivated by questions about the disconnection of discrete cylinders or tori by the trace of simple random walk. Since then, it has gained popularity among probabilists due to its percolative properties and also because of its connections to the free field. Random interlacements on transient graphs can be constructed as a Poisson point process of doubly infinite trajectories. After reviewing this model, we will focus on the rare event that these trajectories disconnect a macroscopic body from infinity, in the strongly percolative regime. We will ask the following question: What is the most efficient way for random interlacements to enforce such disconnection? In other words, how do the trajectories of random interlacements look like conditionally on disconnection?

This talk is based on joint work with M. Nitzschner.

**May 20, 2019:** Nestor Parolya (TU Delft)

When: Monday, May 20th, 16:00

Where: TU Delft, Faculty EWI, Mekelweg 4, EWI-Lecture hall F

*Large dimensional random matrices and their applications*

The random matrix theory (RMT) is originated from the multivariate statistics, nuclear physics and quantum mechanics under the strong impetus of Dyson, Gaudin, Mehta, Wigner, Wishart and others in the 1960's and 1970's. In particular, in 1967 two Ukrainian mathematicians Marchenko and Pastur derive the celebrated equation for the limiting spectral measure for the large dimensional sample covariance matrix. RMT has emerged as an extremely powerful tool for a variety of applications, especially in statistical signal processing, wireless communications, statistical finance and econometrics. Estimation of covariance/precision matrices is particularly important in portfolio allocation and risk assessment in finance, classification and large scale hypothesis testing in statistics or forecasting of time series in macroeconomics. In this talk we will give a short introduction to the theory of large random matrices and discuss our recent results on applications in high-dimensional statistics and finance.

**April 15, 2019**: Richard Kraaij (TU Delft)

When: Monday, April 15th

Where: TU Delft, Faculty 3mE, Mekelweg 2, 3mE-CZ F (Simon Stevin)

*How close is the critical Kac-Ising of ferromagnetism to solutions of the Allen-Cahn equation?*

The Kac-Ising model for ferromagnetism is used in statistical physics to study phase transitions in lattice systems. If we study the dynamic Kac-Ising model close to its critical temperature, it is known that field of local magnetizations converges to a solution of the Allen-Cahn equation as lattice spacing is sent to 0. The Allen-Cahn equation is a PDE that is used for the study of phase-separation phenomena.

I will present work in progress in which I use the probabilistic technique of large deviations to study how close the dynamic Kac-Ising model is to the solution of the Allen-Cahn equation.

**April 1, 2019****: **Timothy Budd (Radboud University)

When: Monday, April 1st

Where: TU Delft, EWI-Lecture hall F

*Geometry of random planar maps with high degrees*

For many types of random planar maps, i.e. planar graphs embedded in the

sphere, it is known that their geometry possesses a scaling limit

described by a universal random continuous metric space known as the

Brownian sphere. One way to escape this universality class is to

consider random planar maps that harbor vertices of very high degree.

In this talk I will describe a peeling exploration that allows us to

study distances in such maps. Based on the results we conjecture the

existence of a new one-parameter family of random continuous metric

spaces, referred to tentatively as the stable spheres.

**March 11, 2019**: Botond Szabó (Leiden Univeristy)

When: Monday, March 11

Where: EWI-Lecture hall F

*Bayesian nonparametric approach to log-concave density estimation*

In the beginning of the talk I will give a (somewhat) lengthier introduction to Bayesian nonparametric methods and their theoretical analysis. Then I will focus on estimating log-concave densities, which is a canonical problem in the area of shape-constrained nonparametric inference. I will present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the corresponding posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. I demonstrate the applicability of the proposed method for estimating the underlying log-concave density and its mode in a simulation study and compare our Bayesian method with the classical MLE. Finally, I will briefly talk about potential application in cluster analysis.

It is a joint work with Ester Mariucci and Kolyan Ray.

**February 25, 2019**: Ayan Bhattacharya (CWI Amsterdam)

When: Monday, February 25

Where: EWI-Lecture hall F

*Large deviation for extremes of branching random walk with regularly varying tails.*

We consider discrete time branching random walk on real line where the

displacements have regularly varying tail. Using the one large jump

asymptotics, we derive large deviation for the extremal processes

associated to the suitably scaled positions of particles in the nth

generation where the genealogical tree satisfies Kesten-Stigum

condition. The large deviation limiting measure in this case is

identified in terms of the cluster Poisson point process obtained in

the underlying weak limit of the point processes. As a consequence of

this, we derive large deviation for the rightmost particle in the

nth generation giving the heavy-tailed analogue of recent work by

Gantert and Höfelsauer(2018).

Reference: Large deviation for extremes of branching random walk with regularly varying displacements (https://arxiv.org/abs/1802.05938v1).

**February 11, 2019**: Jaron Sanders (TU Delft)

When: Monday, February 11

Where: EWI-Lecture hall F

*Clustering in Block Markov Chains*

In this talk, I will discuss our recent paper that considers cluster detection in Block Markov Chains (BMCs). These Markov chains are characterized by a block structure in their transition matrix. More precisely, the n possible states are divided into a finite number of K groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. In this paper we devise a clustering procedure that accurately, efficiently, and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the BMC, and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense.

This is joint work with Alexandre Proutière and Se-Young Yun.

**January 28, 2019**: Pasquale Cirillo (Tu Delft)

When: Monday, January 28th

Where: TU Delft, EWI-Lecture hall F

*The arithmetic of finance*

Take finance, remove the trendy words, remove the acronyms and the obscure jargon, admit you will not be rich, but be happy because you will not lose money either. What do you get? Interestingly you discover that all those changes of measure, those coherent risk measures, even the pay-off of a European option, not to mention utility theory in its many declinations, are nothing but the result of sums and products. So, let’s come back to basics.

**January 14, 2019**: Lixue Pang (TU Delft)

When: Monday, January 14th

Where: TU Delft, EWI-Lecture hall F

*Bayesian estimation of a decreasing density*

Consider the problem of estimating a decreasing density function, with special interest in zero. It is well known that the maximum likelihood estimator is inconsistent at zero. This has led several authors to propose alternative estimators which are consistent. As any decreasing density can be represented as a scale mixture of uniform densities, a Bayesian estimator is obtained by endowing the mixture distribution with the Dirichlet process prior. Assuming this prior, we derive contraction rates of the posterior density at zero. Several choices of base measure are numerically evaluated and compared. In a simulation various frequentist methods and a Bayesian estimator are compared. Finally, the Bayesian procedure is applied to current durations data.