Seminar on Stochastic Processes

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Spring Semester 2014

Date / Time Speaker Title Location
19 February 2014
17:15-19:00
Alex Watson
Universität Zürich
Event Details

Seminar on Stochastic Processes

Title The hitting time of zero for stable processes, symmetric and asymmetric
Speaker, Affiliation Alex Watson, Universität Zürich
Date, Time 19 February 2014, 17:15-19:00
Location Y27 H 25
Abstract We will discuss a method to characterise explicitly the law of the first time at which a stable process reaches the point zero, using theories of self-similar Markov processes. When the process is symmetric, the Lamperti representation characterises the law of the hitting time of zero as equal to that of the exponential functional of a Lévy process. When the stable process is asymmetric, things are not quite so simple. However, using the newly-developed theory of real self-similar Markov processes, we demonstrate that the hitting time of zero is equal to the exponential functional of a Markov additive process. These laws have not been investigated very thoroughly in the past, but remarkably, we are able to set up and solve a two-dimensional functional equation for a vector-valued Mellin transform. Moreover, we can even write down the density of the hitting time. Finally, we will discuss an application to the stable process conditioned to avoid zero. This is joint work with Alexey Kuznetsov (York, Canada), Juan-Carlos Pardo (CIMAT) and Andreas Kyprianou (Bath, UK).
The hitting time of zero for stable processes, symmetric and asymmetricread_more
Y27 H 25
26 February 2014
17:15-19:00
Bénédicte Haas
Paris Dauphine & ENS
Event Details

Seminar on Stochastic Processes

Title Behavior near the extinction time in self-similar fragmentations
Speaker, Affiliation Bénédicte Haas, Paris Dauphine & ENS
Date, Time 26 February 2014, 17:15-19:00
Location Y27 H 25
Abstract We consider a Markovian model for the random fragmentation of an object. At each time, the state consists of a collection of blocks. Each block waits an exponential amount of time with parameter given by its size to some power alpha<0, independently of the other blocks. Every block then splits randomly, but with the same distribution. In such a model, small blocks split intensively and it turns out that the whole state is reduced to ``dust'' in a finite time, almost surely (we call this the extinction time). In this talk we will see how the fragmentation process behaves as it approaches its extinction time. In particular, we will prove a scaling limit for the block sizes which, as a direct consequence, gives us an expression for an invariant measure for the fragmentation process. This is based on a joint work with Christina Goldschmidt (Oxford).
Behavior near the extinction time in self-similar fragmentationsread_more
Y27 H 25
5 March 2014
17:15-19:00
Lorenzo Zambotti
Paris UPMC
Event Details

Seminar on Stochastic Processes

Title The generalized KPZ equation
Speaker, Affiliation Lorenzo Zambotti, Paris UPMC
Date, Time 5 March 2014, 17:15-19:00
Location Y27 H 25
Abstract The generalized KPZ equation is a SPDE driven by a multiplicative space-time white noise with a quadratic nonlinearity in the space-derivative multiplied by a generic smooth non-linear function of the solution. We discuss how one can treat this equation using the recent theory by Martin Hairer on the renormalization of SPDEs.
The generalized KPZ equationread_more
Y27 H 25
12 March 2014
17:15-19:00
Federico Camia
VU Amsterdam & NYU Abu Dhabi
Event Details

Seminar on Stochastic Processes

Title Brownian Loops, Cosmic Bubbles and Conformal Field Theory
Speaker, Affiliation Federico Camia, VU Amsterdam & NYU Abu Dhabi
Date, Time 12 March 2014, 17:15-19:00
Location Y27 H 25
Abstract Poissonian ensembles of Brownian loops have attracted considerable attention in recent years, particularly because of their conformal invariance and connections to the Schramm-Loewner Evolution. In this talk I plan to present some recent results involving such ensembles. I will first discuss how they appear as scaling limits of Poissonian ensembles of random walk loops, then I will use them to define a one-parameter family of conformal field theories whose original motivation comes from cosmology. (This is partly joint work with Tim van de Brug and Marcin Lis, and with Alberto Gandolfi and Matthew Kleban. No prior knowledge of conformal field theory or cosmology is required.)
Brownian Loops, Cosmic Bubbles and Conformal Field Theoryread_more
Y27 H 25
19 March 2014
17:15-18:15
Titus Lupu
Université Paris-Sud
Event Details

Seminar on Stochastic Processes

Title From loop clusters of parameter 1/2 to the Gaussian free field
Speaker, Affiliation Titus Lupu, Université Paris-Sud
Date, Time 19 March 2014, 17:15-18:15
Location Y27 H 25
Abstract To a transient symmetric Markov jump process on a network is naturally associated an infinite measure on loops and the Poisson point process of loops of intensity proportional to this measure are sometimes called “loop soups”. We focus on the “loop soup” of parameter 1/2 and construct a coupling between the Poisson ensemble of loops and the Gaussian free field on the network satisfying two constraints. First of all half the square of the free field must be the occupation field of the loops. Besides that the sign of the free field must be constant on clusters of loops. This is an improvement over the relation between the Poisson ensemble of loops and the Gaussian free field obtained by Le Jan, which did not take in account the sign of of the free field. As a consequence of our coupling we deduce that loop clusters at parameter 1/2 do not percolate on periodic lattices.
From loop clusters of parameter 1/2 to the Gaussian free fieldread_more
Y27 H 25
26 March 2014
17:15-18:15
Florent Benaych-Georges
Université Paris Descartes
Event Details

Seminar on Stochastic Processes

Title Outliers in the Single Ring Theorem
Speaker, Affiliation Florent Benaych-Georges, Université Paris Descartes
Date, Time 26 March 2014, 17:15-18:15
Location Y27 H 25
Abstract This talk well be about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type $A+P$, where the rank of $P$ stays bounded as the dimension goes to infinity and where the matrix $A$ is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if $P$ has some eigenvalues out of the maximal circle of the single ring, then $A+P$ has some eigenvalues (called outliers) in the neighborhood of those of $P$, which is not the case for the eigenvalues of $P$ in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of $A$ around the eigenvalues of $P$ and prove they are distributed as eigenvalues of some finite dimensional random matrices. This fact had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of $P$) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but with non Gaussian models, whereas spiked Ginibre matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new. This is a joint work with Jean Rochet (Université Paris Descartes).
Outliers in the Single Ring Theoremread_more
Y27 H 25
2 April 2014
17:15-18:15
Jean-Dominique Deuschel
TU Berlin
Event Details

Seminar on Stochastic Processes

Title Quenched local limit theorem for ergodic random conductance model
Speaker, Affiliation Jean-Dominique Deuschel, TU Berlin
Date, Time 2 April 2014, 17:15-18:15
Location Y27 H 25
Abstract We prove an almost sure local limit theorem for a symmetric random walk in an ergodic random environment. The proof is based on the parabolic Harnack inequality, which holds under some moment conditions on the conductances and the quenched invariance principle. This is a joint work with S. Andres and M. Slowik.
Quenched local limit theorem for ergodic random conductance modelread_more
Y27 H 25
9 April 2014
17:15-18:15
Dmitry Chelkak
Steklov Institute at St. Petersburg
Event Details

Seminar on Stochastic Processes

Title Surgery of discrete domains and uniform bounds on crossing probabilities of topological quads for the critical FK-Ising model in 2D
Speaker, Affiliation Dmitry Chelkak, Steklov Institute at St. Petersburg
Date, Time 9 April 2014, 17:15-18:15
Location Y27 H 25
Abstract For the critical FK-Ising model in 2D, we obtain uniform bounds on crossing probabilities of topological quadrilaterals in terms of the effective resistance (aka the discrete extremal length) of a quad. This generalizes earlier results available for (a) lattice rectangles with arbitrary boundary conditions, (b) general quads with self-dual (wired/free/wired/free) boundary conditions. The proof relies on new discrete complex analysis techniques (`toolbox') that allows one to construct `nice' (from harmonic measure point of view) cross-cuts of a given discrete quad: the product of partition functions of random walks in the two pieces is of the same order as the partition function of random walks in the original domain. Applications include quasi-multiplicativity properties of arm probabilities and the computation of so-called universal exponents for this model. Based on a joint project with Hugo Duminil-Copin (Geneve) and Clement Hongler (New York/Lausanne).
Surgery of discrete domains and uniform bounds on crossing probabilities of topological quads for the critical FK-Ising model in 2Dread_more
Y27 H 25
16 April 2014
17:15-18:15
James Martin
Oxford University
Event Details

Seminar on Stochastic Processes

Title Multi-type interacting particle systems and queueing constructions
Speaker, Affiliation James Martin, Oxford University
Date, Time 16 April 2014, 17:15-18:15
Location Y27 H 25
Abstract I'll survey a collection of one-dimensional interacting particle systems - exclusion processes and variants. Multi-type versions of such models have been studied from various perspectives recently, with motivations coming from combinatorics and physics. There is a natural connection between these particle systems and models of random growth such as first- or last-passage percolation; paths of second-class particles in an exclusion process correspond to interfaces between competing populations in spatial growth models. A variety of Markovian queueing systems, which are very natural from an applied probability perspective, turn out to be important in the analysis of these multi-type systems.
Multi-type interacting particle systems and queueing constructionsread_more
Y27 H 25
30 April 2014
17:15-18:15
Sandrine Péché
Université Paris Diderot
Event Details

Seminar on Stochastic Processes

Title Deformed ensembles of random matrices
Speaker, Affiliation Sandrine Péché, Université Paris Diderot
Date, Time 30 April 2014, 17:15-18:15
Location Y27 H 25
Abstract We review some more or less recent results on deformed random matrices. In particular we discuss some recent results (joint work with M. Capitaine) on the full rank deformations of random matrices
Deformed ensembles of random matricesread_more
Y27 H 25
7 May 2014
17:15-18:15
Prof. Dr. David Wilson
Microsoft Research Redmond (USA)
Event Details

Seminar on Stochastic Processes

Title Local statistics of the abelian sandpile model
Speaker, Affiliation Prof. Dr. David Wilson, Microsoft Research Redmond (USA)
Date, Time 7 May 2014, 17:15-18:15
Location Y27 H 25
Local statistics of the abelian sandpile model
Y27 H 25
14 May 2014
17:00-17:45
Ivan Corwin
University of Colombia
Event Details

Seminar on Stochastic Processes

Title Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
Speaker, Affiliation Ivan Corwin, University of Colombia
Date, Time 14 May 2014, 17:00-17:45
Location Y27 H 25
Abstract Integrable probability has emerged as an active area of research at the interface of probability/mathematical physics/statistical mechanics on the one hand, and representation theory/integrable systems on the other. Informally, integrable probabilistic systems have two properties: 1) It is possible to write down concise and exact formulas for expectations of a variety of interesting observables (or functions) of the system. 2) Asymptotics of the system and associated exact formulas provide access to exact descriptions of the properties and statistics of large universality classes and universal scaling limits for disordered systems. We focus here on examples of integrable probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality class and explain how their integrability stems from connections with symmetric function theory and quantum integrable systems.
Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality classread_more
Y27 H 25
* 14 May 2014
18:15-19:00
Dr. Fabio Toninelli
Université Claude Bernard Lyion 1
Event Details

Seminar on Stochastic Processes

Title Height functions and interacting dimers
Speaker, Affiliation Dr. Fabio Toninelli, Université Claude Bernard Lyion 1
Date, Time 14 May 2014, 18:15-19:00
Location Y27 H 25
Abstract Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance (see e.g. Kenyon, Okounkov, Sheffield '06). As soon as dimers mutually interact, the model is not solvable any more. However, tools from constructive field theory allow to prove that, as long as the interaction is small, the height field still behaves like a gaussian log-correlated field. Work in collaboration with A. Giuliani and V. Mastropietro.
Height functions and interacting dimersread_more
Y27 H 25
21 May 2014
17:15-18:15
Vincent Bansaye
Ecole Polytechnique Paris
Event Details

Seminar on Stochastic Processes

Title Ancestral lineage and limit theorems for Branching Markov chains
Speaker, Affiliation Vincent Bansaye, Ecole Polytechnique Paris
Date, Time 21 May 2014, 17:15-18:15
Location Y27 H 25
Abstract We consider a branching model in discrete time where each individual is characterized by a trait in some general state space. Both the reproduction law and the trait inherited by the offsprings may depend on the trait of the mother and the environment. We study the long time behavior of the population and the ancestral lineage of typical individuals under general assumptions, which we specify for applications to some models motivated by biology. Our results focus on the growth rate, the trait distribution among the population for large time, and we tackle the local densities and the position of extremal individuals. The approach consists in comparing the branching Markov chain to well chosen (possibly non-homogeneous) Markov chains.
Ancestral lineage and limit theorems for Branching Markov chainsread_more
Y27 H 25
28 May 2014
17:15-18:15
Fredrik Viklund
University of Uppsala
Event Details

Seminar on Stochastic Processes

Title Title T.B.A.
Speaker, Affiliation Fredrik Viklund, University of Uppsala
Date, Time 28 May 2014, 17:15-18:15
Location Y27 H 25
Title T.B.A.
Y27 H 25

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Organizers: Jean Bertoin, Erwin Bolthausen, Ashkan Nikeghbali, Pierre Nolin, Martin Schweizer, Alain-Sol Sznitman

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