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 2012

Date / Time Speaker Title Location
22 February 2012
17:15-18:15
Martin Hairer
University of Warwick, UK
Event Details

Seminar on Stochastic Processes

Title Solving the KPZ equation
Speaker, Affiliation Martin Hairer, University of Warwick, UK
Date, Time 22 February 2012, 17:15-18:15
Location Y27 H 25
Abstract The solution to the KPZ equation is believed to be a ``universal'' object describing the crossover between the Gaussian universality class and the KPZ universality class. The mathematical proof of its universality however is still an open problem, in particular because of the lack of a good approximation theory for the equation. We will present a new notion of solution to the KPZ equation that bypasses the use of the Cole-Hopf transform. It allows to factorise the solution map into a ``universal'' (independent of initial condition) measurable map, composed with a solution map with good continuity properties. This lays the foundations for a robust approximation theory to the KPZ equation, which is needed to prove its universality. As a byproduct of the construction, we obtain very detailed regularity estimates on the solutions, as well as new homogenisation results.
Solving the KPZ equationread_more
Y27 H 25
29 February 2012
17:15-18:15
Gaëtan Borot
Université de Genève
Event Details

Seminar on Stochastic Processes

Title All order finite size corrections in Coulomb gases
Speaker, Affiliation Gaëtan Borot, Université de Genève
Date, Time 29 February 2012, 17:15-18:15
Location Y27 H 25
Abstract A Coulomb gas, also named $beta$ ensemble in random matrix theory, consists in N charged particules on the real line, trapped in a potential and interacting mutually by logarithmic repulsion, with inverse temperature $beta$. When N is large, heuristic arguments suggest that the partition function and the n-point correlation functions have a 1/N expansion when the particles condensate on one segment (one-cut regime), whereas oscillatory terms in N appears when they condensate on several disconnected segments. I will present a result stating the existence of the 1/N expansion for any fixed beta, under appropriate assumptions corresponding to the one-cut regime. The coefficients can be effectively computed to all orders. As an application, I will describe predictions of the all-order asymptotic tails of Tracy-Widom beta laws. This talk is based on joint works with A. Guionnet, and B. Eynard, S.N. Majumdar and C. Nadal.
All order finite size corrections in Coulomb gasesread_more
Y27 H 25
* 1 March 2012
15:15-16:15
Ross Pinsky
Technion Haifa
Event Details

Seminar on Stochastic Processes

Title Probabilistic and combinatorial aspects of the card-cyclic to random insertion shuffle
Speaker, Affiliation Ross Pinsky, Technion Haifa
Date, Time 1 March 2012, 15:15-16:15
Location HG D 3.2
Abstract Consider a permutation $\sigma\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $\sigma_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on $S_n$ defined by removing and then randomly reinserting each of the $n$ cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features.
Probabilistic and combinatorial aspects of the card-cyclic to random insertion shuffleread_more
HG D 3.2
7 March 2012
17:15-18:15
Rajat Subhra Hazra
Universität Zürich
Event Details

Seminar on Stochastic Processes

Title Patterned Random Matrices
Speaker, Affiliation Rajat Subhra Hazra, Universität Zürich
Date, Time 7 March 2012, 17:15-18:15
Location Y27 H 25
Abstract In this talk we focus on some recent results on large symmetric random matrices like Wigner, Toeplitz, Hankel, Circulant, Reverse Circulant etc. We show that they have some structural pattern which allows the limiting spectral distribution to exist. We also show that the joint convergence of different kinds of random matrices in the limit give rise to different notions of independence on non-commutative probability spaces. We point out some results and conjectures on the edge behavior of these matrices.
Patterned Random Matricesread_more
Y27 H 25
* 13 March 2012
17:15-18:15
Adela Svejda
Universität Bonn
Event Details

Seminar on Stochastic Processes

Title Convergence to extremal processes in random environments
Speaker, Affiliation Adela Svejda, Universität Bonn
Date, Time 13 March 2012, 17:15-18:15
Location Y27 H 12
Abstract We present general criteria for convergence of clock processes of random dynamics in random environments to extremal processes. Using this, we extend recent results on aging in mean field spin glasses on short time scales, obtained by G. Ben Arous and O. Gun in law wrt the environment, to results that hold almost surely, respectively in probability, wrt the environment. It is based on joint work with A. Bovier and V. Gayrard.
Convergence to extremal processes in random environmentsread_more
Y27 H 12
* 14 March 2012
16:00-17:00
Sylvia Serfaty
Université Paris 6
Event Details

Seminar on Stochastic Processes

Title 2D Classical Coulomb gas and the renormalized energy
Speaker, Affiliation Sylvia Serfaty, Université Paris 6
Date, Time 14 March 2012, 16:00-17:00
Location Y27 H 12
Abstract In joint work with Etienne Sandier, we study the statistical mechanics of a two-dimensional classical Coulomb gas, a particular case of which also correspond to the Ginibre ensemble, a random matrix model. We connect the problem to the "renormalized energy" W, a Coulombian interaction for an infinite set of points in the plane that we introduced in connexion to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the "Abrikosov" triangular lattice. We obtain a next order asymptotic expansion of the partition function, an various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero we show the system tends to "crystallize" to a minimizer of W.
2D Classical Coulomb gas and the renormalized energyread_more
Y27 H 12
14 March 2012
17:15-18:15
Antti Knowles
Harvard University
Event Details

Seminar on Stochastic Processes

Title Universality and non-universality in deformed random matrix ensembles
Speaker, Affiliation Antti Knowles, Harvard University
Date, Time 14 March 2012, 17:15-18:15
Location Y27 H 25
Abstract Large random matrices exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. In the first part of the talk, I review recent results on the spectral universality of random matrices. The second part of the talk is devoted to deformed random matrix models, which exhibit an intriguing phase transition associated with the creation or annihilation of outlier eigenvalues. (Joint work with Jun Yin.)
Universality and non-universality in deformed random matrix ensemblesread_more
Y27 H 25
28 March 2012
17:15-18:15
Peter Pfaffelhuber
Universität Freiburg i.Br., Deutschland
Event Details

Seminar on Stochastic Processes

Title Tree-valued Markov processes arising in population genetics
Speaker, Affiliation Peter Pfaffelhuber, Universität Freiburg i.Br., Deutschland
Date, Time 28 March 2012, 17:15-18:15
Location Y27 H 25
Abstract We construct a tree-valued Markov process describing the evolution of genealogical relationships in populations of constant size. This process is an extension of the Fleming-Viot superprocess, which is frequently studied via its dual coalescent process. After introducing the topology on the spaces of trees and construct the tree-valued process by a well-posed martingale problem. As an application we state some path properties of the resulting process. This is a joint work with Andrej Depperschmidt, Andreas Greven and Anita Winter.
Tree-valued Markov processes arising in population geneticsread_more
Y27 H 25
4 April 2012
17:15-18:15
Adam Harper
Cambridge University, UK
Event Details

Seminar on Stochastic Processes

Title Suprema of certain Gaussian processes
Speaker, Affiliation Adam Harper, Cambridge University, UK
Date, Time 4 April 2012, 17:15-18:15
Location Y27 H 25
Abstract I will describe a number-theoretic problem that can be attacked using sharp quantitative estimates for the supremum of a certain Gaussian process. Then I will explain how one can obtain such estimates, using a two step "conditioning and comparison" procedure. In particular, this approach allows one to identify the expectation of the supremum up to second order terms. If time allows I will also describe applications to some other Gaussian processes.
Suprema of certain Gaussian processesread_more
Y27 H 25
* 18 April 2012
16:00-17:00
Takashi Kumagai
Kyoto University
Event Details

Seminar on Stochastic Processes

Title Quenched invariance principle for random walks and random divergence forms in random media on cones
Speaker, Affiliation Takashi Kumagai, Kyoto University
Date, Time 18 April 2012, 16:00-17:00
Location Y27 H 12
Abstract We will consider the following two models and establish quenched invariance principles; 1. Simple random walks on the infinite clusters for super critical percolations on half and quarter planes in d-dimensional Euclidean spaces. 2. Uniform elliptic divergence forms with random stationary coefficients on cones in Euclidean spaces. Note that because of the lack of translation invariance, we cannot apply the method of the 'corrector'. Instead, we make full use of the heat kernel estimates and Dirichlet form techniques to resolve the problem. This is a joint work with Z.Q. Chen (Seattle) and D.A. Croydon (Warwick).
Quenched invariance principle for random walks and random divergence forms in random media on conesread_more
Y27 H 12
18 April 2012
17:15-18:15
Rob van den Berg
CWI Amsterdam
Event Details

Seminar on Stochastic Processes

Title Extensions of the BK inequality
Speaker, Affiliation Rob van den Berg, CWI Amsterdam
Date, Time 18 April 2012, 17:15-18:15
Location Y27 H 25
Abstract The BK inequality, proved by van den Berg and Kesten (1985), says that for product measures on {0,1}^n, the probability that two increasing events `occur disjointly' is smaller than or equal to the product of the two individual probabilities. This result is often used in percolation and interacting particle systems. Their conjecture that the inequality even holds for all events was proved by Reimer in 1994. In spite of Reimer's work, several natural, fundamental problems in this area remained open. In this talk I will discuss recent progress, in particular an extension of the BK inequality to randomly drawn subsets of fixed size (joint work with Johan Jonasson) and more recent extensions for the ferromagnetic Ising model and the antiferromagnetic Curie-Weiss model (joint work with Alberto Gandolfi).
Extensions of the BK inequalityread_more
Y27 H 25
25 April 2012
17:15-18:15
Sylvie Méléard
Palaiseau, France
Event Details

Seminar on Stochastic Processes

Title Adaptive Dynamics in a stochastic multi-resources chemostat model
Speaker, Affiliation Sylvie Méléard, Palaiseau, France
Date, Time 25 April 2012, 17:15-18:15
Location Y27 H 25
Abstract We study a model of population with competition for resources through a chemostat-type model where species consume the common resources that are constantly supplied. Our interest is to understand the adaptation of bacterias to a moving environment. We assume a fast deterministic dynamics for the supply of the resources and a slow stochastic dynamics for the bacterias which are characterized by a continuous trait. We prove that starting from a stochastic birth and death process model, that, when advantageous mutations are rare and mutational steps are not too large, the population behaves at the mutational time scale as a jump process moving between evolutionary equilibria. In the small mutation steps limit, this process gives rise to a differential equation in the phenotype space referred in the biologist literature as the canonical equation. Then we prove a rigorous characterization of the evolutionary branching points. It's a joint work with Pierre-Emmanuel Jabin and Nicolas Champagnat.
Adaptive Dynamics in a stochastic multi-resources chemostat modelread_more
Y27 H 25
2 May 2012
17:15-18:15
Christina Goldschmidt
University of Oxford
Event Details

Seminar on Stochastic Processes

Title The scaling limit of the minimum spanning tree of the complete graph
Speaker, Affiliation Christina Goldschmidt, University of Oxford
Date, Time 2 May 2012, 17:15-18:15
Location Y27 H 25
Abstract Consider the complete graph on $n$ vertices with independent and identically distributed edge-weights having some absolutely continuous distribution. The minimum spanning tree (MST) is simply the spanning subtree of smallest weight. It is straightforward to construct the MST using one of several natural algorithms. Kruskal's algorithm builds the tree edge by edge starting from the globally lowest-weight edge and then adding other edges one by one in increasing order of weight, as long as they do not create any cycles. At each step of this process, the algorithm has generated a forest, which becomes connected on the final step. In this talk, I will explain how it is possible to exploit a connection between the forest generated by Kruskal's algorithm and the Erdös-Rényi random graph in order to prove that $M_n$, the MST of the complete graph, possesses a scaling limit as $n \to \infty$. In particular, if we think of $M_n$ as a metric space (using the graph distance) and rescale edge-lengths by $n^{-1/3}$, then $M_n$ converges in distribution in the sense of the Gromov-Hausdorff distance to a certain random real tree. This is joint work with Louigi Addario-Berry (McGill), Nicolas Broutin (INRIA Paris-Rocquencourt) and Grégory Miermont (Orsay).
The scaling limit of the minimum spanning tree of the complete graphread_more
Y27 H 25
16 May 2012
17:15-18:15
Nicolas Curien
Ecole Normale Supérieure, Paris
Event Details

Seminar on Stochastic Processes

Title A panoramic view of the Uniform Infinite Planar Quadrangulation
Speaker, Affiliation Nicolas Curien, Ecole Normale Supérieure, Paris
Date, Time 16 May 2012, 17:15-18:15
Location Y27 H 25
Abstract The purpose of this talk is to present and study the Uniform Infinite Planar Quadrangulation (UIPQ), which is a model of random discrete planar geometry consisting in a cell decomposition of the plane into quadrangles, chosen "uniformly at random" among all homeomorphically distinct possibilities. We will present a construction of the UIPQ based on well-labeled trees, study some of its amazing properties, and link this object to the Brownian Map of Le Gall and Miermont. Based on joint works with Itai Benjamini, Jean-François Le Gall, Laurent Ménard & Grégory Miermont.
A panoramic view of the Uniform Infinite Planar Quadrangulationread_more
Y27 H 25
23 May 2012
17:15-18:15
Joaquin Fontbona
University of Santiago de Chile
Event Details

Seminar on Stochastic Processes

Title A trajectorial interpretation of entropy dissipation and a non-intrinsic Bakry-Emery criterion
Speaker, Affiliation Joaquin Fontbona, University of Santiago de Chile
Date, Time 23 May 2012, 17:15-18:15
Location Y27 H 25
Abstract We develop a pathwise description of the dissipation of general convex entropies for continuous time Markov processes, based on simple backward martingales and convergence theorems with respect to the tail sigma field. The entropy is in this setting the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit its Doob-Meyer decomposition, thereby providing a stochastic analogue of the well known entropy dissipation formula, valid for general convex entropies (including total variation). Under additional regularity assumptions, and using Itô calculus and some ideas of Arnold, Carlen and Ju, we obtain a new Bakry Emery criterion which ensures exponential convergence of the entropy to 0. This criterion is non-intrinsic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide an example where the classic Bakry Emery criterion fails, but our non-intrinsic criterion ensuring exponential convergence to equilibrium applies without modifying the law of the diffusion process. Joint work with Benjamin Jourdain (Cermics ENPC, Paris).
A trajectorial interpretation of entropy dissipation and a non-intrinsic Bakry-Emery criterionread_more
Y27 H 25
30 May 2012
17:15-18:15
Jean-Christophe Mourrat
EPF Lausanne
Event Details

Seminar on Stochastic Processes

Title On the homogenization of the heat equation with random coefficients
Speaker, Affiliation Jean-Christophe Mourrat, EPF Lausanne
Date, Time 30 May 2012, 17:15-18:15
Location Y27 H 25
Abstract We consider the heat equation with random coefficients on Z^d. The randomness of the coefficients models the inhomogeneous nature of the medium where heat propagates. We assume that the distribution of these coefficients is invariant under spatial translations, and has a finite range of dependence. It is known that if a solution to the heat equation is rescaled diffusively, then it converges to the solution of a heat equation with constant coefficients. In probabilistic terms, this convergence corresponds to the fact that the associated random walk satisfies a central limit theorem. I will present recent progress on the estimation of the speed of this convergence, based on the random walk representation.
On the homogenization of the heat equation with random coefficientsread_more
Y27 H 25
6 June 2012
17:15-18:15
Mike Cranston
University of Irvine
Event Details

Seminar on Stochastic Processes

Title On the overlap of the Brownian polymer
Speaker, Affiliation Mike Cranston, University of Irvine
Date, Time 6 June 2012, 17:15-18:15
Location Y27 H 25
On the overlap of the Brownian polymer
Y27 H 25
* 2 July 2012
17:15-18:15
Alexander Drewitz
ETH Zürich
Event Details

Seminar on Stochastic Processes

Title Effective Polynomial Ballisticity Condition for Random Walk in Random Environment in all Dimensions
Speaker, Affiliation Alexander Drewitz, ETH Zürich
Date, Time 2 July 2012, 17:15-18:15
Location HG G 19.2
Abstract The conditions $(T)_\gamma,$ $\gamma \in (0,1),$ which have been introduced by Sznitman in 2002, have had a significant impact on research in random walk in random environment. They require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature. We show that in all relevant dimensions (i.e., $d \ge 2$), in order to establish the conditions $(T)_\gamma$, it is actually enough to check a corresponding condition $(\mathcal{P})$ of polynomial type on a finite box. In particular, this extends the conjectured equivalence of the conditions $(T)_\gamma,$ $\gamma \in (0,1),$ to all relevant dimensions. Joint work with N. Berger and A.F. Ramirez.
Effective Polynomial Ballisticity Condition for Random Walk in Random Environment in all Dimensionsread_more
HG G 19.2

Notes: events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location and if you want you can subscribe to the iCal/ics Calender.

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

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