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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Spring Semester 2015

Date / Time

Speaker

Title

Location

18 February 2015
17:15-19:00

Robin Stephenson
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	Local convergence of large critical multi-type Galton-Watson trees and applications to random maps
Speaker, Affiliation	Robin Stephenson, Universität Zürich
Date, Time	18 February 2015, 17:15-19:00
Location	Y27 H 25
Abstract	We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven when we condition on the number of vertices of one fixed types, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.

Local convergence of large critical multi-type Galton-Watson trees and applications to random mapsread_more

Y27 H 25

25 February 2015
17:15-19:00

Louigi Addario-Berry
MCGill University

Event Details

Seminar on Stochastic Processes

Title	Theta(t^{1/3}) slowdown for branching Brownian motion with decay of mass
Speaker, Affiliation	Louigi Addario-Berry, MCGill University
Date, Time	25 February 2015, 17:15-19:00
Location	Y27 H 25
Abstract	Consider a branching Brownian motion particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles. One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance Theta(t^{1/3}) behind the typical BBM front. Many natural questions about the model remain open.

Theta(t^{1/3}) slowdown for branching Brownian motion with decay of massread_more

Y27 H 25

4 March 2015
17:15-19:00

Massimiliano Gubinelli
Université Paris Dauphine

Event Details

Seminar on Stochastic Processes

Title	Singular Stochastic PDEs and paracontrolled distributions
Speaker, Affiliation	Massimiliano Gubinelli, Université Paris Dauphine
Date, Time	4 March 2015, 17:15-19:00
Location	Y27 H 25
Abstract	Non-linear evolution problems perturbed by singular noise sources arise naturally as scaling limits of certain microscopic evolutions or homogenisation problems. The parabolic anderson model, the Kardar-Parisi-Zhang equation and the stochastic quantisation equation are examples of such systems. Solving (or even giving a meaning to) these equations require a detailed understanding of the propagation of the stochastic perturbations via the non-linear evolution. I will explain how ideas and tools from harmonic analysis can be useful in this analysis and in the related problem of studying the convergence of small scale models to their scaling limits.

Singular Stochastic PDEs and paracontrolled distributionsread_more

Y27 H 25

11 March 2015
17:15-19:00

Jean-François Delmas
CERMICS, Paris

Event Details

Seminar on Stochastic Processes

Title	On the genealogical tree of a stationary (quadratic) branching process
Speaker, Affiliation	Jean-François Delmas, CERMICS, Paris
Date, Time	11 March 2015, 17:15-19:00
Location	Y27 H 25
Abstract	We will first present the genealogy of a population with random size given by a quadratic stationary continuous-state branching processes. We will recall some properties such as the (mild) bottleneck effect at the time of the most recent common ancestor. Then, we will study the total length process for the genealogical tree and give an explicit formula for the one-dimensional marginal. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingman coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model. This work is in collaboration with H. Bi.

On the genealogical tree of a stationary (quadratic) branching processread_more

Y27 H 25

18 March 2015
17:15-18:15

Sasha Sodin
Princeton University

Event Details

Seminar on Stochastic Processes

Title	Wegner estimates for deformed Gaussian ensembles
Speaker, Affiliation	Sasha Sodin, Princeton University
Date, Time	18 March 2015, 17:15-18:15
Location	Y27 H 25
Abstract	The deformed Gaussian ensembles are obtained by adding a deterministic Hermitian matrix to a random matrix drawn from the Gaussian Orthogonal, Unitary (or Symplectic) ensembles. We shall discuss several Wegner-type estimates for these models. As an application, we establish localisation at strong disorder for the Wegner orbital model, with sharp dependence of the localisation threshold on the number of orbitals. Based on joint work with M. Aizenman, R. Peled, J. Schenker, and M. Shamis.

Wegner estimates for deformed Gaussian ensemblesread_more

Y27 H 25

25 March 2015
17:15-18:15

Kevin Schnelli
IST Austria

Event Details

Seminar on Stochastic Processes

Title	Universality for deformed Wigner matrices
Speaker, Affiliation	Kevin Schnelli, IST Austria
Date, Time	25 March 2015, 17:15-18:15
Location	Y27 H 25
Abstract	Bulk universality for random matrices states that the local eigenvalue statistics in the bulk of the spectrum are universal in the sense that they only depend on the symmetry class of the matrix, but are otherwise independent of the details of the entries of the matrix. It was established for a large class of Wigner matrices. In this talk, I explain how it can be derived for deformed Wigner matrices, i.e. matrices of the form H=V+W, where V is a real diagonal matrix and W is a Wigner matrix. I further discuss the edge universality for these models. Based on joint work with J.O. Lee, B. Stetler & H.-T. Yau, and with L. Erdoes.

Universality for deformed Wigner matricesread_more

Y27 H 25

1 April 2015
17:15-18:15

Gaetan Borot
MPI Bonn

Event Details

Seminar on Stochastic Processes

Title	Asymptotics in 1d Coulomb gases and applications
Speaker, Affiliation	Gaetan Borot, MPI Bonn
Date, Time	1 April 2015, 17:15-18:15
Location	Y27 H 25
Abstract	We consider a model of N particles on the real line, confined by an external field and subjected to 2d Coulombian repulsion, and maybe other, regular, many-body interactions. I will summarize a general strategy to estalish the large N asymptotic expansion of the partition function and multilinear statistics, based on the combination of large deviation estimates and the analysis of Schwinger-Dyson equations. Away from criticality, we can show the existence of a 1/N expansion in the one-cut regime, and in general for the multi-cut regime, a quasiperiodic behavior in N at all orders in 1/N. I shall describe two applications (1) the derivation of an all-order asymptotic expansion of skew-orthogonal polynomials away from the bulk ; (2) analytic properties of the perturbative invariants of fiber knots in finite quotients of S^3.

Asymptotics in 1d Coulomb gases and applicationsread_more

Y27 H 25

15 April 2015
17:15-18:15

Emmanuel Jacob
CNRS, Lyon

Event Details

Seminar on Stochastic Processes

Title	Spatial Preferential Attachment Networks
Speaker, Affiliation	Emmanuel Jacob, CNRS, Lyon
Date, Time	15 April 2015, 17:15-18:15
Location	Y27 H 25
Abstract	We study random networks constructed sequentially by adding vertices, then connecting them prefrably to high degree vertices nearby. The model can roughly be seen as an interpolation between preferential attachment networks and random geometric graphs. Its preferential attachment feature yields a power law degree distribution, while its spatial structure yields clustering. Both properties are desirable in the sake of modeling real networks, and rarely both naturally present in a random graph model.We will discuss in particular the robustness of the network, exhibiting a non-trivial (and non-complete) phase transition depending on both the power law exponent and the clustering features. Techniques are partly inspired by percolation theory and include a suprising application of the BK-inequality.

Spatial Preferential Attachment Networksread_more

Y27 H 25

22 April 2015
17:15-18:15

Simone Warzel
TU München

Event Details

Seminar on Stochastic Processes

Title	Resonant delocalization
Speaker, Affiliation	Simone Warzel, TU München
Date, Time	22 April 2015, 17:15-18:15
Location	Y27 H 25
Abstract	Mathematical analysis of the delocalised phase of random operators has so far turned to be more elusive than the proofs of localization. There is therefore the need to develop insight into mechanisms by which extended states may form in the presence of extensive, though possibly weak, frozen disorder. One such mechanism is resonant delocalisation. In this talk, I will explain the main ideas behind this proposal. The analysis rests on an improvement of the Simon-Wolff criterion which allows to characterise the regime of delocalisation through the divergence of $\ell^2$-sums of the Green function with only positive probability. (Joint work with M. Aizenman.)

Resonant delocalizationread_more

Y27 H 25

29 April 2015
17:15-18:15

Roman Vershynin
University of Michigan

Event Details

Seminar on Stochastic Processes

Title	Concentration of random graphs
Speaker, Affiliation	Roman Vershynin, University of Michigan
Date, Time	29 April 2015, 17:15-18:15
Location	Y27 H 25
Abstract	Do random graphs concentrate near their ``expectations''? This question can be asked rigorously using one of the matrices classically associated with graphs, namely the adjacency and Laplacian matrices. The answer depends on sparsity. Relatively dense random graphs concentrate well. Sparse graphs do not, but they can be forced to concentrate. Theoretical analysis of concentration brings together tools from random matrix theory, combinatorics, and Grothendieck's inequalities. Practical motivation comes from analysis of networks, and in particular community detection problems.

Concentration of random graphsread_more

Y27 H 25

6 May 2015
17:15-18:15

Jan Maas
Universität Bonn

Event Details

Seminar on Stochastic Processes

Title	Entropic gradient flows in Markov chains and chemical reaction networks
Speaker, Affiliation	Jan Maas, Universität Bonn
Date, Time	6 May 2015, 17:15-18:15
Location	Y27 H 25
Abstract	A seminal result by Jordan-Kinderlehrer-Otto (1998) asserts that the heat equation can be viewed as the gradient flow of the Boltzmann entropy in a geometric structure on the space of probability measures induced by an optimal transport problem. This result has been the starting point for many developments at the interface of analysis, probability theory, and geometry. In this talk I will show how these ideas can be adapted to the context of discrete Markov chains and discuss applications to chemical reaction networks. This is based on joint works with Matthias Erbar and Alexander Mielke.

Entropic gradient flows in Markov chains and chemical reaction networksread_more

Y27 H 25

13 May 2015
17:15-18:15

Mark Rudelson
University of Michigan

Event Details

Seminar on Stochastic Processes

Title	Approximation complexity of convex bodies
Speaker, Affiliation	Mark Rudelson, University of Michigan
Date, Time	13 May 2015, 17:15-18:15
Location	Y27 H 25
Abstract	Consider the approximation of an n-dimensional convex body by a projection of a section of an N-dimensional simplex, and call the minimal N for which such approximation exists the approximation complexity of the body. The reason for such strange definition lies in computer science. A projection of a section of a simplex is the feasible set of a linear programming problem, and so it can be efficiently generated. So far, there are no deterministic examples of convex bodies whose approximation complexity is higher than linear. We will show how probabilistic methods allow to construct convex bodies of exponential complexity.

Approximation complexity of convex bodiesread_more

Y27 H 25

20 May 2015
17:15-18:15

Roberto Fernandez
Massachusetts Institute of Technology

Event Details

Seminar on Stochastic Processes

Title	Stability of phase transitions under discretization
Speaker, Affiliation	Roberto Fernandez, Massachusetts Institute of Technology
Date, Time	20 May 2015, 17:15-18:15
Location	Y27 H 25
Abstract	When simulating a continuous model it is assumed that increasingly refined discretizations lead to more faithful lattice versions. Mathematically, this fact is not automatically guaranteed. I will discuss a number of issues related to what is understood by discretization, and will present some general approach that covers point processes and many important models in statistical mechanics. The approach includes a novel criterion for the stability of phase diagrams that may lead to a purely probabilistic alternative to Pirogov-Sinai theory.

Stability of phase transitions under discretizationread_more

Y27 H 25

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

Archive: SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 SS 21 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11 SS 11 AS 10 SS 10 AS 09