# Utrecht Stochastics Seminar

Unless otherwise stated, the Stochastics Seminar takes place on Wednesday, 15:30-16.15, in room 610 or 611 of the Math building.

## Up next

 September 25, 2017 Room 610 11:00 - 11:45 Pieter Trapman (Stockholm U.) End of an SIR epidemic on a configuration model network

## Recent past

 May 11, 2017 Room 610 1:30 - 2:15 Wioletta Ruszel (TU Delft) Long-range Ising models: phase transitions and interfaces May 3, 2017Room 611 Sandjai Bhulai (VU Amsterdam) Value Function Discovery In Markov Decision Processes April 26, 2017 Room DDW 1.22 Sandór Kolumbán (TU/e) Criticality in the brain, what we can and cannot see April 19, 2017 Matej Stehlik (Université Grenoble Alpes) Edge- and face-width of projective quadrangulations March 29, 2017Room 611 Karma Dajani (Utrecht) Invariant measures, matching and the frequency of 0 for signed binary expansions March 1, 2017Room 611 Derong Kong (Leiden) On the bifurcation set of unique Expansions February 22, 2017Room 610 Reka Szabo (Groningen) From survival to extinction of the contact process by the removal of a single edge February 15, 2017Room 611 Reem Yassawi (Trent University, Canada) Random orders on Bratteli diagrams January 26 - 27, 2017 STAR workshop on Random Graphs 2017 (Utrecht) December 21, 2016 Francesca Collet (Delft) Self-sustained oscillations in dissipative mean field interacting particle systems December 14, 2016 Room 611 Christian Spitoni (postponed) (Utrecht) t.b.a. December 7, 2016 Luca Avena (Leiden) RW kernels, spanning forests and multiscale analysis on graphs. November 30, 2016 Arnoud den Boer (UvA) Decision-based model selection November 23, 2016 Maria Remerova (UvA) Queueing systems with random fluid limits October 26, 2016 Mathew Penrose (Bath, UK) The strong giant in a random digraph October 18, 2016 11:15 - 12:00 Moumanti Podder (NYU) The strange logic of Galton-Watson trees September 28, 2016 Johan van Leeuwaarden (TU/e) Solvable random network model for disordered sphere packings in all dimensions September 5, 2016 13:00 - 13:45 Alessandra Cipriani (WIAS Berlin) The Membrane model June 28, 2016 David Coupier (Lille 1) Navigation on Geometric Random Graphs June 21, 2016 Willem van Zuijlen (Leiden) Mean-field Gibbs-non-Gibbs transitions June 16, 2016 10:00 - 10:45 Matej Stehlik (Université Grenoble Alpes) Topological bounds on the chromatic number June 7, 2016 Rene Conijn (Utrecht) Largest clusters in Percolation and Conformal Measure Ensembles 31 May 2016 Federico Camia (NYU Abu Dhabi) From Brownian Loops to Conformal Fields 24 May 2016 Christoph Hofer-Temmel (NLDA Den Helder) Disagreement percolation for the hard-sphere model Wednesday 18 May 2016 15:00, Room 610 Krerley Oliveira (Universidade Federal de Alagoas, Maceió Brazil) On Uniqueness of Equilibrium States for some Iterated Function Systems 10 May 2016 11:15, Room 610 Shota Gugushvili (Leiden) Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation

## More distant past

Lists of stochastics talks that took place in the seminar before 2016 can be found here, here, here, here and here.

# Abstracts

 Speaker: Shota Gugushvili (Leiden) Title: Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation Abstract: We consider the problem of non-parametric estimation of the deterministic dispersion coefficient of a stochastic differential equation based on discrete time observations on its solution. We take a Bayesian approach to the problem and under suitable regularity assumptions derive the posterior contraction rate. This rate turns out to be the optimal posterior contraction rate. Joint work with Peter Spreij.

 Speaker: Krerley Oliveira (Universidade Federal de Alagoas, Maceió Brazil) Title: On Uniqueness of Equilibrium States for some Iterated Function Systems Abstract: In this talk we discuss two recent results on uniqueness of equilibrium states for some IFS. In the first setting, we prove existence of relative maximal entropy measures for certain random dynamical systems that are skew products of the type $F(x,y)=(\theta(x), f_x(y))$, where $\theta$ is an invertible map preserving an ergodic measure $\mathbb{P}$ on a Polish space and $f_x$ is a local diffeomorphism of a compact Riemannian manifold exhibiting some non-uniform expansion. As a consequence of our proofs, we obtain an integral formula for the relative topological entropy as the integral the of logarithm of the topological degree of $f_x$ with respect to $\mathbb{P}$. When $F$ is topologically exact and the supremum of the topological degree of $f_x$ is finite, the maximizing measure is unique and positive on open sets. The second setting we discuss uniqueness of equilibrium states for Partially Hyperbolic Horseshoes studied in a previous article with R. Leplaideur and I. Rios and in a previous article by Diaz, Horita, Sambarino and Rios. These families of horseshoes have interesting features, as dense sets of segments in its central direction on its non- wandering set. They have heteroclinical cycles and are extensions of the golden shift. From one hand, they have phase transitions for smooth potentials. From the other hand, one expect uniqueness for potentials that are close to zero. We make use of the semiconjugacy with the shift to caracterize the set of Holder potentials with unique equilibrium state.

 Speaker: Rene Conijn (Utrecht) Title: Largest clusters in Percolation and Conformal Measure Ensembles Abstract: Consider an n x n-box in the triangular lattice. The asymptotic behaviour, as n tends to infinity, of the largest percolation clusters in this box was well studied by Borgs, Chayes, Kesten and Spencer in (1999 and 2001). However some questions remained open. If we restrict ourself to critical percolation the size of the largest cluster is of the order n^(91/48). The first natural question is: does there exist a limiting distribution for the size of the largest cluster scaled by its order? In this talk we discuss the existence of the limiting distribution and introduce conformal measure ensembles as a key ingredient. Furthermore we will see an interesting application of these measure ensembles to the FK-Ising model. Based on joint work with Rob van den Berg, Federico Camia and Demeter Kiss.

 Speaker: Christoph Hofer-Temmel (NLDA Den Helder) Title: Disagreement percolation for the hard-sphere model Abstract: We study uniqueness conditions for the Gibbs measure of the hard-sphere model on $R^d$ in the the high temperature case. Classic sufficient conditions for uniqueness in the high-temperature case are cluster expansions and Dobrushin's uniqueness criterion. We generalize disagreement percolation, another sufficient condition, from the lattice case to the point process case. Disagreement percolation, introduced by Maes and van den Berg, compares the competing influence of different boundary conditions for finite volume specifications with a product field and then uses percolation bounds on the product field to derive the uniqueness of the Gibbs measure. Using the same approach together with bounds on the percolation threshold of the Boolean disc model, we derive lower bounds on the extent of the uniqueness region of the infinite volume Gibbs measure. The key tools are a dependent, recursive and simultaneous thinning of a Poisson point process to several dominated point processes and a concept of densities of couplings of simple point processes. For every finite volume and all choices of boundary conditions, we are able to stochastically dominate two instances of the hard-sphere model with these boundary conditions at a given activity simultaneously by a Poisson point process of the same activity. The resulting disagreement percolation bounds are better than the theoretically best possible lower bounds by cluster expansion techniques. I close with a list of possible extensions and an outlook of future work around marked Gibbs point processes.

 Speaker: Federico Camia (NYU Abu Dhabi) Title: From Brownian Loops to Conformal Fields Abstract: The study of lattice fields is an important topic in statistical mechanics and modern probability theory. In various two dimensional models that undergo a continuous phase transition, when a continuum scaling limit is performed (i.e., when the lattice spacing is sent to zero) at the critical point, such lattice fields are known or believed to converge to continuum fields endowed with a type of symmetry called conformal invariance. The prototypical example is the spin field in the critical, two-dimensional Ising model. In this talk, I will discuss a class of conformal fields constructed out of planar Brownian loops via a cutoff procedure. The goal is to highlight the main features of the continuum scaling limit and of the emergence of conformal fields within a class of models where both conceptual and technical issues are relatively transparent and easy to treat. (The models I will describe are actually motivated by cosmology, and seem to lead to conformal field theories with novel features, but I will not focus on those aspects in this talk. No prior knowledge of conformal field theory will be assumed. The talk is based on joint work with Alberto Gandolfi and Matthew Kleban: Conformal correlation functions in the Brownian loop soup, Nuclear Physics B 902 (2016), 483-507.)

 Speaker: David Coupier (Universite Lille 1) Title: Navigation on Geometric Random Graphs Abstract: In this talk, we consider three models of Geometric Random Graphs which are defined in a very similar way. Their vertex sets are given by a Poisson point process (in the euclidean plane) and their graph structures obey to the same geometric rule. In fact, only their directions differ. The first one is radial and has been introduced by F. Baccelli and C. Bordenave in 2005 to model telecommunications networks. The second one is directed-- according to a given vector --whereas the third one has no favorite directions. We will study the number of topological ends for these three models.

 Speaker: Willem van Zuijlen (Leiden) Title: Mean-field Gibbs-non-Gibbs transitions Abstract: In this talk I discuss dynamical Gibbs-non-Gibbs transitions for mean-field spin systems. Gibbsianness is related to the number of global minimisers of a large deviation rate function. A unique global minimiser implies Gibbsianness while multiple imply non-Gibbsianness. I explain the background of this relation and list possible scenarios when the spins perform independent Brownian motions. This is joint work with Frank den Hollander and Frank Redig.

 Speaker: Matej Stehlik (Université Grenoble Alpes) Title: Topological bounds on the chromatic number Abstract: Topological bounds on the chromatic number of graphs originate from Lovasz’s celebrated proof of Kneser’s conjecture in 1977. The general idea is to associate a simplicial complex to a graph, and then to bound the chromatic number of the graph in terms of certain topological invariants of the associated complex. In this talk, we will explore an alternative approach using what we call higher-dimensional projective quadrangulations. We will illustrate this on some classes of graphs such as Kneser graphs, and also show how one of the “classical" topological bounds can be expressed purely in terms of projective quadrangulations. This is joint work with Tomas Kaiser.

 Speaker: Alessandra Cipriani (WIAS Berlin) Title: The Membrane model Abstract: The membrane or bilaplacian model was first introduced in the physics literature to model random interfaces with constant curvature, and studied mathematically for the first time by Sakagawa and Kurt. It is a centered multivariate Gaussian whose covariance is given by the discrete bilaplacian operator on the lattice. It is tempting to think of it as a kin of the discrete Gaussian free field (GFF), and indeed many results can be deduced with the same methods for both, as for example the fluctuations of the maximum in higher dimensions. Moreover, as the GFF, it also can be seen as a generalised Gaussian variable arising as scaling limit of discrete models, for example in the odometer of the divisible sandpile or height fluctuations in uniform spanning forest. We will discuss some of the interesting features of the model and some open conjectures. The talk is an overview of the area and should be accessible to everybody. Based on joint works with Alberto Chiarini, Rajat Subhra Hazra and Wioletta Ruszel.

 Speaker: Moumanti Podder (NYU) Title: The strange logic of Galton-Watson trees Abstract: This talk will focus on probabilities of first order (FO) and existential monadic second order (EMSO) sentences on the Galton-Watson (GW) tree. I shall limit myself to Poisson offspring distributions, though many of the results can be extended to very general distributions. Fix $k \in \mathbb{N}$. Conditioned on survival of the tree, we show that all FO sentences of quantifier depth $k$ are determined by a local neighbourhood of the root (of radius $\approx 3k+2$). We devise a recursive procedure to compute these probabilities, conditioned on the tree’s survival. The probabilities are very nice functions of $\lambda$ and $p \lambda$, the survival probability. Using Ehrenfeucht games and corresponding equivalence classes, we show that the probability of any FO sentence is analytic in $\lambda$. I shall discuss some of our ongoing work: rogue solutions, i.e. solutions of equations derived from tree automata that do not admit any interpretation; an example of a sentence that is not expressible almost surely as an EMSO, etc. I shall end with speculations and conjectures that hopefully the audience will find fascinating. Parts of this are joint work with Joel Spencer, Alexander Holroyd, Yuval Peres, Tobias Johnson, Fiona Skerman, Avi Levy.

 Speaker: Mathew Penrose (Bath, UK) Title: The strong giant in a random digraph Abstract: Consider a random directed graph on n vertices with out-degrees sampled independently at random from an arbitrary specified probability distribution on the nonnegative integers, and destinations of arcs selected independently and uniformly at random. We say two vertices lie in the same strong component if they intercommunicate. We discuss the emergence of a unique giant component for large n, provided the mean out-degree exceeds 1. Time permitting, we discuss related results for other random graph models.

 Speaker: Johan van Leeuwaarden (TU/e) Title: Solvable random network model for disordered sphere packings in all dimensions Abstract: A notorious open problem in physics is to create a solvable model for random sequential adsorption (RSA) of non-overlapping spheres in continuum spaces for dimensions 2 and higher. In its simplest form, spheres arrive sequentially at uniformly chosen locations in space and are deposited when there is no overlap with a previously deposited sphere. Due to the spatial correlation, characterizing the fraction of accepted spheres or the area covered by the deposited spheres, becomes intractable after a large number of deposition attempts. We investigate such disordered packings of spheres by taking a novel approach that compares continuum RSA with nearest-neighbor blocking on a sequence of clustered random networks with a growing number of vertices. This tractable network model leads to a precise characterization of coverage, including formulas for ensemble averages and fluctuations. We discover laws that describe the packing fraction as a function of density and dimension. By investigating the spatial dimensions two to five over wide density ranges, we show that these laws are universal. The network model advances the theoretical study of higher-dimensional disordered packings and has several potential applications, including the dipole-blockade of ultra-cold Rydberg gas molecules, studied for their potential impact on quantum computing. The predictions from the model also confirm the widely observed sub-Poissonian nature (variance to mean ratio well below one) of disordered packings, and support the conjectured lower bounds on the optimal sphere packings in high dimensions. This is joint work with Souvik Dhara and Debankur Mukherjee. We also discuss recent joint work with Clara Stegehuis and Remco van der Hofstad on hierarchical random graphs and recent works of among other Jaron Sanders and Svante Janson on jamming limits for random graphs.

 Speaker: Maria Remerova (UvA) Title: Queueing systems with random fluid limits Abstract: Fluid limits are law-of-large-numbers type of limits, and as such, they are usually deterministic. In this talk, I will present two queueing models that have random fluid limits: a polling model in overload and a model inspired by synchronization processes in the Bitcoin network.

 Speaker: Arnoud den Boer (UvA) Title: Decision-based model selection Abstract: In data-driven optimization problems, simple mathematical models that discard important factors may sometimes be preferred to more realistic models. This may occur if the parameters of the simple model are easier to estimate than the parameters of the complex model, or if the optimization problem corresponding to the simple model can be solved exactly whereas the optimization problem corresponding to the `realistic model' is intractable. This trade-off between three sources of errors (modeling, estimation, and optimization errors) is encountered in many stochastic optimization problems. The question we address is: how can one determine if it is better to use a simplified model, rather than a more realistic model? In other words: given a particular optimization problem and a data set at hand, how do we know whether the model-misspecification error of a simple model is dominated by estimation and optimization errors of more realistic models?

 Speaker: Luca Avena (Leiden) Title: RW kernels, spanning forests and multiscale analysis on graphs. Abstract: We use ideas from large-scale stochastic dynamics to build a multiresolution scheme to analyse arbitrary functions on graphs. These types of problems emerge naturally in the context of signal processing. The goal is to obtain successive approximations at different scales of arbitrary functions on graphs which are used for signal classification, reconstruction and data compression. When the signal is defined on a graph having enough regularity structures, several methods (such as wavelets) are available in the literature and used in practice. When the regularity structure of the graph is lacking, very few methods are known. Our work aims at addressing this issue by using random spanning forests, loop-erased walks, determinantal structures, random walk kernels and intertwining of Markov chains. Joint work with Castel, Gaudilliere and Melot.

 Speaker: Francesca Collet (Delft) Title: Self-sustained oscillations in dissipative mean field interacting particle systems Abstract: An important issue in Complex Systems is to understand how many interacting units organize to produce “coherent” behavior at macroscopic level. Basic examples include polarization (e.g. spin alignments in Ising-like models) and synchronization (e.g. phase locking in interacting rotators). In this talk we deal with a different, and less understood, phenomenon of self-organization: the emergence of periodic behavior in systems whose units have no tendency to evolve periodically. We will show how the interplay between dissipation and noise can be responsible for self-sustained oscillations in some mean field interacting particle systems. This talk is based on joint work with Paolo Dai Pra and Marco Formentin.

 Speaker: Cristian Spitoni (Utrecht) Title: t.b.a. Abstract: t.b.a.

 Speaker: Reem Yassawi (Trent University, Canada) Title: Random orders on Bratteli diagrams Abstract: The Fisher-Wright model describes gene survival in a variable size population, giving conditions for gene heterogeneity as a function of population growth. We recreate this model in the context of Bratteli-Vershik dynamical systems} $(X,T)$. Bratteli diagrams were first studied in operator algebras, and provide partial information about the dynamical system under consideration. Complete information is given by the order imposed on the Bratteli diagram. In our setting the Fisher-Wright model describes the nature of the resulting dynamical system, and gives us the dichotomy: a random order on a slowly growing Bratteli diagram admits a homeomorphism, while a random order on a quickly growing Bratteli diagram does not. This is joint work with Jeannette Janssen and Anthony Quas. The talk is aimed at an audience with only elementary knowledge of probability theory!

 Speaker: Reka Szabo (Groningen) Title: From survival to extinction of the contact process by the removal of a single edge Abstract: The contact process is one of the most well-studied stochastic models of the spread of epidemics on a graph, and can also be seen as a continuous-time version of oriented percolation. Its behaviour can be affected by local modifications in the underlying graph. In this talk I will give a construction of a tree in which the contact process with any positive infection rate survives but, if a certain privileged edge is removed, one obtains two subtrees in which the contact process with infection rate smaller than 1/4 dies out.

 Speaker: Karma Dajani (Utrecht) Title: Invariant measures, matching and the frequency of 0 for signed binary expansions Abstract: We introduce a parametrized family of maps $S_{\alpha}$, the so called symmetric doubling maps, defined on $[-1,1]$ by $S_{\alpha}(x)=2x-d\alpha$, where $d\in \{-1,0,1 \}$ and $\alpha\in [1,2]$. Each map $S_\alpha$ generates binary expansions with digits -1,0 and 1. The transformations $S_{\alpha}$ have a natural invariant measure $\mu_{\alpha}$ that is absolutely continuous with respect to Lebesgue measure. We show that for a set of parameters of full measure, the invariant measure of the symmetric doubling map is piecewise smooth. We also study the frequency of the digit 0 in typical expansions, as a function of the parameter $\alpha$. In particular, we investigate the self similarity displayed by the function $\alpha\to \mu_{\alpha}([-1/2,1/2]$, where $\mu_{\alpha}([-1/2,1/2]$ denotes the measure of the cylinder where digit zero occurs. This is joint work with Charlene Kalle.

 Speaker: Derong Kong (Leiden) Title: On the bifurcation set of unique Expansions Abstract: Expanding a real number in a non-integer base is an interesting topic in mathematics. Most notably it has close connection with dynamical systems, number theory and fractal geometry. Following the works of Paul Erdos and Vilmos Komornik in the 1990s we investigated the geometrical properties of unique expansions in a non-integer base as well as the set of bases for which 1 has a unique expansion. This is a joint work with Kalle, Li and Lu.

 Speaker: Sandjai Bhulai (VU Amsterdam) Title: Value Function Discovery In Markov Decision Processes Abstract: In this talk, we introduce a novel method for discovery of value functions for Markov Decision Processes (MDPs). This method is based on ideas from the evolutionary algorithm field. Its key feature is that it discovers descriptions of value functions that are algebraic in nature. This feature is unique, because the descriptions include the model parameters of the MDP. The algebraic expression can be used in several scenarios, e.g., conversion to a policy, control of systems with time-varying parameters. We illustrate its application on an example MDP.

 Speaker: Matej Stehlik (Université Grenoble Alpes) Title: Edge- and face-width of projective quadrangulations Abstract: Graphs which embed in the projective plane so that all faces are bounded by four edges are called projective quadrangulations. They are a fascinating class of graphs with some surprising properties. In this talk, we will give upper bounds on the edge-width and the face-width of non-bipartite projective quadrangulations. The former is sharp, while the latter is a constant away from the optimal. We will also discuss some related questions. Joint work with Louis Esperet

 Speaker: Sandór Kolumbán (TU/e) Title: Criticality in the brain, what we can and cannot see Abstract: Classical dynamical models (like the Ising model) have very different behavior in their sub-, super- and critical phase and this is universal to some extent. A large amount of empirical evidence shows that brain functionality resembles this universal behavior around criticality. During this presentation we will briefly overview some of the empirical evidence supporting that the brain is a system around criticality. We will also try to go beyond directly measurable properties of the brain based on this insight of criticality. A version of the Ising model is introduced that captures particular aspects of the brain behavior. Examining this model around criticality boils down to classical results in some cases but its behavior is suspected to be different in other cases. We will discuss some open questions related to this model. This is ongoing joint work with Remco van der Hofstad and Julia Komjáthy.

 Speaker: Wioletta Ruszel (TU Delft) Title: Long-range Ising models: phase transitions and interfaces Abstract: Long-range models are very special since they allow a phase transition to occur even in 1d lattice models which is in general not possible for ordinary spin models. Even more surprising, this phenomenon persists in the presence of fields which usually “stear” a spin model towards a certain preference in phase. In this talk we will first review some recent results on phase transition phenomena of 1d long-range Ising models with external fields, announced in [1]. In the second part we will discuss some work in progress concerning Dobrushin interfaces and its fluctuations. A Dobrushin interface is a separation line which appears when the lattice is conditioned to be in one phase in one half of the lattice and in the other phase on the other half. For short-range models this has been done for example using the relation between Ising models and percolation. There is also a corresponding Fortuin-Kasteleyn representation for long-range models and we are trying currently to extend the analysis for the long-range case. This is joint work with: Rodrigo Bissacot (Sao Paolo), Eric Endo (Sao Paolo), Aernout van Enter (Groningen), Bruno Kimura (Delft), Arnaud Le Ny (Paris), [1] Dyson models under renormalization and in weak fields R. Bissacot, E.O. Endo, A.C.D. van Enter, B. Kimura, A. Le Ny, W.M. Ruszel arXiv:1702.02887, 2017

 Speaker: Pieter Trapman (Stockholm U.) Title: End of an SIR epidemic on a configuration model network Abstract: We consider an SIR (Susceptible-Infective-Recovered) epidemic on a static (configuration model) random graph. We describe the final stages of the epidemic and analyse the time until extinction of the epidemic. Furthermore, we consider the effect of control measures, such as vaccination on the duration of the epidemic. The analysis heavily relies on theory on branching processes. We show that the epidemic is always exponentially declining in the final stages, even if the degree distribution of the network has infinite variance. We also show that control measures, although decreasing the size of the epidemic, might well increase the duration of the epidemic, which might have serious public health and economic implications. Joint work with Ana Serafimovic and Abid Ali Lashari