Departement Mathematik

Seminar on stochastic processes

Members of the probability group are involved in co-organizing remote specialized seminars that take place on Tuesdays and Thursdays:

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Herbstsemester 2018

Datum / Zeit Referent:in Titel Ort
26. September 2018
17:00-18:00 		Jiří Černý
Universität Basel
Details

Seminar on Stochastic Processes

Titel The maximal particle of branching random walk in random environment
Referent:in, Affiliation Jiří Černý, Universität Basel
Datum, Zeit 26. September 2018, 17:00-18:00
Ort HG G 43
Abstract The behaviour of the maximal particle of branching random walk have been subject to intensive research recently. It is natural to ask how these properties change when a spatially dependent random branching rates are introduced to the process. In my presentation, I will describe the first results in this direction, in particular a CLT for the position of the maximal particle, and explain their consequences for other models of interest: the randomized Fisher-KPP equation, and the parabolic Anderson model.
The maximal particle of branching random walk in random environmentread_more
HG G 43
3. Oktober 2018
17:00-18:00 		Nina Holden
ETH Zürich
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Seminar on Stochastic Processes

Titel Cardy embedding of uniform triangulations
Referent:in, Affiliation Nina Holden, ETH Zürich
Datum, Zeit 3. Oktober 2018, 17:00-18:00
Ort HG G 43
Abstract A uniformly sampled triangulation is a canonical model for a discrete random surface. The Cardy embedding is a discrete conformal embedding of triangulations which is based on percolation observables. We present a series of works in progress where we prove convergence of uniform triangulations to the continuum random surface known as Liouville quantum gravity under the Cardy embedding. The project is a collaboration with Xin Sun, and is also based on our joint works with Bernardi, Garban, Gwynne, Lawler, Li, and Sepulveda.
Cardy embedding of uniform triangulationsread_more
HG G 43
10. Oktober 2018
17:15-18:15 		Alejandro Rivera
Université de Grenoble
Details

Seminar on Stochastic Processes

Titel Decorrelation of topological events for smooth Gaussian fields
Referent:in, Affiliation Alejandro Rivera, Université de Grenoble
Datum, Zeit 10. Oktober 2018, 17:15-18:15
Ort HG G 43
Abstract Consider f a planar smooth centered stationary Gaussian field with covariance K(x)=E[f(x)f(0)]. Under mild non-degeneracy assumptions on K, the zero set Z of f is a.s. a random locally finite collection of disjoint smooth loops on the plane and perhaps some smooth curves going to infinity. Assuming that K(x) goes to zero as |x| goes to infinity, we would like to prove that the topologies of the level lines on distant regions become independent from each other. This type of information is useful to derive central limit theorems or to prove percolation estimates for level sets of smooth Gaussian fields. However, if the field is, say, analytic, conditioning on the field inside an open subset determines the field on the whole plane. I will discuss different ways of surpassing this obstacle, including a recent result obtained in collaboration with Hugo Vanneuville, and some applications to component counting and percolation.
Decorrelation of topological events for smooth Gaussian fieldsread_more
HG G 43
17. Oktober 2018
17:15-18:15 		Cécile Mailler
University of Bath
Details

Seminar on Stochastic Processes

Titel The Monkey walk: a Markov process with random reinforced relocations.
Referent:in, Affiliation Cécile Mailler, University of Bath
Datum, Zeit 17. Oktober 2018, 17:15-18:15
Ort HG G 43
Abstract In this joint work with Gerónimo Uribe-Bravo, we prove and extend results from the physics literature about a random walk with random reinforced relocations. The "walker" evolves in $\mathbb Z^d$ or $\mathbb R^d$ according to a Markov process, except at some random jump-times, where it chooses a time uniformly at random in its past, and instatnly jumps to the position it was at that random time. This walk is by definition non-Markovian, since the walker needs to remember all it’s past.
We prove that, under moment conditions on the inter-jump-times, and provided that the underlying Markov process verifies a distributional limit theorem, we show a distributional limit theorem for the position of the walker at large time. The proof relies on exploiting the branching structure of this random walk with random relocations; we are able to extend the model further by allowing the memory of the walker to decay with time.
The Monkey walk: a Markov process with random reinforced relocations.read_more
HG G 43
24. Oktober 2018
17:15-18:15 		Sebastien Martineau
Université Orsay
Details

Seminar on Stochastic Processes

Titel Strict monotonicity of percolation thresholds under covering maps
Referent:in, Affiliation Sebastien Martineau, Université Orsay
Datum, Zeit 24. Oktober 2018, 17:15-18:15
Ort HG G 43
Abstract How does the critical parameter of percolation depend on the graph under consideration? Several theorems and conjectures attempt to shed qualitative light on this question, and I will present a recent result fitting this paradigm. If some transitive graph H is a strict quotient of a transitive graph G, then pc(G)<1 => pc(H)>pc(G). We also obtain the same result for the uniqueness parameter pu instead of pc, under the additional assumption that the quotient map has bounded fibres. This is joint work with Franco Severo.
Strict monotonicity of percolation thresholds under covering mapsread_more
HG G 43
31. Oktober 2018
17:00-18:00 		Aran Raoufi
IHES, France
Details

Seminar on Stochastic Processes

Titel Percolation Phase Transition via the Gaussian Free Field
Referent:in, Affiliation Aran Raoufi, IHES, France
Datum, Zeit 31. Oktober 2018, 17:00-18:00
Ort HG G 19.1
Abstract Let $G$ be a bounded-degree infinite graph, and $p_c$ be the critical parameter of bond percolation on $G$. That is $p_c$ is the infimum of values of $p$ that you have an infinite cluster almost surely.
In this talk, we prove that if the isoperimetric dimension of $G$ is higher than 4, then $p_c(G)<1$. The theorem settles affirmatively two conjectures of Benjamini and Schramm. Notably, if $G$ is a transitive graph with super-linear growth, then $p_c(G) <1$. In particular, it implies that if $G$ is a Cayley graph of a finitely generated group without a finite index cyclic subgroup, then $p_c(G)<1$.
The proof of the theorem starts with the existence of an infinite cluster for percolation in a certain in-homogeneous random environment governed by the Gaussian free field. Then, by the help of a multiscale decomposition of GFF, we relate the existence of an infinite cluster in percolation in the random environment to that of percolation with a fix parameter $p<1$.
This talk is based on a joint work with H. Duminil-Copin, S. Goswami, F. Severo, and A. Yadin.
Percolation Phase Transition via the Gaussian Free Fieldread_more
HG G 19.1
7. November 2018
17:15-18:15 		Peter Gracar
Universität Köln
Details

Seminar on Stochastic Processes

Titel Spread of infection by random walks - Multi-scale percolation along a Lipschitz surface
Referent:in, Affiliation Peter Gracar, Universität Köln
Datum, Zeit 7. November 2018, 17:15-18:15
Ort HG G 43
Abstract A conductance graph on $\mathbb{Z}^d$ is a nearest-neighbor graph where all of the edges have positive weights assigned to them. We first consider a point process of particles on the nearest neighbour graph $(\mathbb{Z}^d,E)$ and show some known results about the spread of infection between particles performing continuous time simple random walks. Next, we extend consider the case of uniformly elliptic random graphs on $\mathbb{Z}^d$ and show that the infection spreads with positive speed also in this more general case. We show this by developing a general multi-scale percolation argument using a two-sided Lipschitz surface that can also be used to answer other questions of this nature. Joint work with Alexandre Stauffer.
Spread of infection by random walks - Multi-scale percolation along a Lipschitz surfaceread_more
HG G 43
14. November 2018
17:15-18:15 		Sebastien Ott
Université de Genève
Details

Seminar on Stochastic Processes

Titel Ornstein-Zernike theory for lattice spin models, an overview of the construction
Referent:in, Affiliation Sebastien Ott, Université de Genève
Datum, Zeit 14. November 2018, 17:15-18:15
Ort HG G 43
Abstract I will briefly present the original idea of Ornstein and Zernike for obtaining sharp control over two point functions. Then, I will describe a general strategy to implement this idea in lattice spin models, highlighting the robust parts of the analysis and the ones that require model-specific arguments. I will finish with a quick survey of existing results in the field and open questions.
Ornstein-Zernike theory for lattice spin models, an overview of the constructionread_more
HG G 43
28. November 2018
17:15-18:15 		Pierre-François Rodriguez
Université Paris-Saclay
Details

Seminar on Stochastic Processes

Titel Sign cluster geometry of the Gaussian free field
Referent:in, Affiliation Pierre-François Rodriguez , Université Paris-Saclay
Datum, Zeit 28. November 2018, 17:15-18:15
Ort HG G 43
Abstract We consider the Gaussian free field on a class of transient weighted graphs G, and show that its sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with very large probability. Examples of graphs G belonging to this class include cases in which the random walk on G exhibits anomalous diffusive behavior. Our findings also imply the existence of a nontrivial percolating regime for the vacant set of random interlacements on G. Based on joint work with A. Prévost and A. Drewitz.
Sign cluster geometry of the Gaussian free fieldread_more
HG G 43
5. Dezember 2018
17:15-18:00 		Gaultier Lambert
Universität Zürich
Details

Seminar on Stochastic Processes

Titel How much can the eigenvalue of a random matrix fluctuates?
Referent:in, Affiliation Gaultier Lambert, Universität Zürich
Datum, Zeit 5. Dezember 2018, 17:15-18:00
Ort HG G 43
Abstract The goal of this talk is to explain how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations. These are known as "rigidity estimates" in the literature and they play a crucial role in the proof of universality. I will review some of the current results on eigenvalues' fluctuations and present a new approach which relies on the theory of Gaussian Multiplicative Chaos and leads to optimal rigidity estimates for the Gaussian Unitary Ensemble. I will also mention how to deduce a central limit theorem from our proof. This is joint work with Tom Claeys, Benjamin Fahs and Christian Webb.
How much can the eigenvalue of a random matrix fluctuates?read_more
HG G 43
* 12. Dezember 2018
15:15-16:00 		Joseph Najnudel
University of Bristol
Details

Seminar on Stochastic Processes

Titel On the maximum of two log-correlated fields: the logarithms of the characteristic polynomial of the Circular Beta Ensemble and the Riemann zeta function
Referent:in, Affiliation Joseph Najnudel, University of Bristol
Datum, Zeit 12. Dezember 2018, 15:15-16:00
Ort HG G 19.1
Abstract For different random fields whose correlation is logarithmic with respect to the distance between the points, we observe similar behavior for their extreme values, either proven or conjectured, depending on the model. In this talk, we present two different examples of such fields: the logarithm of the characteristic polynomial of the Circular Beta Ensemble (an ensemble of random unitary matrices generalizing the Circular Unitary Ensemble), and the logarithm of the Riemann zeta function, on a random interval of the critical line.
On the maximum of two log-correlated fields: the logarithms of the characteristic polynomial of the Circular Beta Ensemble and the Riemann zeta functionread_more
HG G 19.1

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