Seminar on Stochastic Processes

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Autumn Semester 2012

Date / Time Speaker Title Location
19 September 2012
17:15-18:15
Alexandre Gaudillière
Université Aix-Marseille
Event Details

Seminar on Stochastic Processes

Title Looking for large cliques through spin glasses
Speaker, Affiliation Alexandre Gaudillière, Université Aix-Marseille
Date, Time 19 September 2012, 17:15-18:15
Location HG G 43
Abstract The search problem for the largest cliques in a given graph is an NP-hard problem. Numerical simulations have proven the high efficiency of a recent algorithm for this problem: the cavity algorithm that was introduced by Iovanella, Scoppola and Scoppola. This is a conservative version of a probabilistic cellular automata built on statistical mechanics methods introduced in the study of spin glasses. We will analyze quantitatively the algorithm efficiency for graphs that are generally considered among the more challenging for the largest cliques search problem: Erdös random graphs. We will then have to understand the dynamics of a small cloud of particles in a disordered environment.
Looking for large cliques through spin glassesread_more
HG G 43
3 October 2012
17:15-18:15
Dmitry Panchenko
Texas A & M University
Event Details

Seminar on Stochastic Processes

Title The Sherrington-Kirkpatrick model: an overview
Speaker, Affiliation Dmitry Panchenko, Texas A & M University
Date, Time 3 October 2012, 17:15-18:15
Location HG G 43
Abstract The goal of this talk is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model.
The Sherrington-Kirkpatrick model: an overviewread_more
HG G 43
10 October 2012
17:15-18:15
Alan Hammond
University of Oxford
Event Details

Seminar on Stochastic Processes

Title Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE
Speaker, Affiliation Alan Hammond, University of Oxford
Date, Time 10 October 2012, 17:15-18:15
Location HG G 43
Abstract Consider a large number of Brownian particles in R^d, each carrying a mass which determines its diffusion rate; the particles interact in pairs at close range, coagulating to form a new particle whose mass is the sum of two colliding particles. The range of interaction for this collision is chosen to ensure a kinetic limit, under which a typical particle experiences a bounded number of interactions per unit time, as the initial particle number is taken to infinity. I will discuss joint work with Fraydoun Rezakhanlou in which a coupled system of PDE, the Smoluchowski PDE, is shown to model the evolution of particle densities. The passage to the limit involves a non-trivial computation of a certain macroscopic coagulation propensity; I will give a probabilistic explanation for the form of this coefficient and explain how stochastic control theory arguments may be used in particle correlation results needed in the proofs of the kinetic limit.
Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDEread_more
HG G 43
17 October 2012
17:15-18:15
Laure Dumaz
ENS, Paris
Event Details

Seminar on Stochastic Processes

Title Some properties of the "true self repelling motion"
Speaker, Affiliation Laure Dumaz, ENS, Paris
Date, Time 17 October 2012, 17:15-18:15
Location HG G 43
Abstract In this talk, I will first recall the definition of the "true self repelling motion" (TSRM), a rather unusual one-dimensional process that has been shown by Balint Toth and Wendelin Werner to be the continuous counterpart of certain self repelling random walks. Its construction uses a family of coalescing Brownian motions now called the "Brownian Web" which is interesting for its own sake and has been largely studied since. I will then show how to derive properties of the TSRM such as large deviation estimates or local fluctuations, using properties of this Brownian web. With a more analytical approach, it is also possible to find explicit formulas for the marginal densities of the process. It turns out that the density function of the position of the TSRM at a fixed time is rather suprising for such an intrinsically continuous and symmetric process: Its derivative is discontinuous at the origin (this last result is a joint work with Balint Toth).
Some properties of the "true self repelling motion"read_more
HG G 43
24 October 2012
17:15-18:15
Vladas Sidoravicius
IMPA, Rio de Janeiro
Event Details

Seminar on Stochastic Processes

Title Greedy Motions and Coffman - Gilbert Conjecture
Speaker, Affiliation Vladas Sidoravicius, IMPA, Rio de Janeiro
Date, Time 24 October 2012, 17:15-18:15
Location HG G 43
Abstract During my talk I will discuss several models where a particle (walker, or server) is moving in a random environment and is driven by a greedy strategy. Environment is changing in time, interacting with the particle, and the question is what is the long term behavior of the particle. These models were introduced in physics and queuing theory literature, and are interesting and complicated due to their self-repelling character. One of the classical examples is the greedy server on the unit circle, which moves to the closest customer with velocity v>0, and serves it during random time, which has distribution \mu. Coffman and Gilbert in 1986 conjectured that if customers arrival is distributed as Poisson process on the circle with intensity \nu, which is stochastically smaller than \mu, then regardless of the velocity of the server, the system is stationary. This conjecture inspired a lot of attempts to solve it, generating large body of works. At the second half of my talk. I will present the complete proof of this conjecture (joint with L. Rolla).
Greedy Motions and Coffman - Gilbert Conjectureread_more
HG G 43
25 October 2012
17:15-18:15
Vladas Sidoravicius
IMPA, Rio de Janeiro
Event Details

Seminar on Stochastic Processes

Title Special Minicourse: Scheduling of random walks, Lipschitz embeddings and rough isometries of random sequences. Part I
Speaker, Affiliation Vladas Sidoravicius, IMPA, Rio de Janeiro
Date, Time 25 October 2012, 17:15-18:15
Location Y27 H 46
Abstract In the late eighties Peter Winkler introduced the following problem: consider two independent (discrete time) random walks, A and B, on the complete graph with N vertices. If the trajectories of A and B are given, can one, knowing all the future steps of the walks, and by changing jump times only, keep A and B apart forever (with positive probability)? It became well known as Clairvoyant Demon Problem. Soon after, Noga Alon observed that this question is equivalent to the existence of a phase transition (in N) in a planar dependent percolation model. Remarkably, several other interesting questions such as Lipschitz embeddings of binary sequences and rough (quasi-)isometries between one dimensional random objects also could be reduced to a similar type of percolation. During three lectures I will explain the multiscale approach suitable for dependent percolative systems and prove two Theorems which provide positive answer to the above questions. The course is based on joint works with H.Kesten and M.Vares, and with R. Basu and A.Sly.
Special Minicourse: Scheduling of random walks, Lipschitz embeddings and rough isometries of random sequences. Part Iread_more
Y27 H 46
31 October 2012
17:15-18:15
Pierre Tarrès
University of Oxford
Event Details

Seminar on Stochastic Processes

Title Edge reinforced random walks, Vertex reinforced jump process, and the SuSy hyperbolic sigma model
Speaker, Affiliation Pierre Tarrès, University of Oxford
Date, Time 31 October 2012, 17:15-18:15
Location HG G 43
Abstract Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. Then we prove that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure, which we interpret as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer. This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010). (Joint work with Christophe Sabot).
Edge reinforced random walks, Vertex reinforced jump process, and the SuSy hyperbolic sigma modelread_more
HG G 43
1 November 2012
17:15-18:15
Vladas Sidoravicius
IMPA, Rio de Janeiro
Event Details

Seminar on Stochastic Processes

Title Special Minicourse: Scheduling of random walks, Lipschitz embeddings and rough isometries of random sequences. Part II
Speaker, Affiliation Vladas Sidoravicius, IMPA, Rio de Janeiro
Date, Time 1 November 2012, 17:15-18:15
Location Y27 H 46
Abstract In the late eighties Peter Winkler introduced the following problem: consider two independent (discrete time) random walks, A and B, on the complete graph with N vertices. If the trajectories of A and B are given, can one, knowing all the future steps of the walks, and by changing jump times only, keep A and B apart forever (with positive probability)? It became well known as Clairvoyant Demon Problem. Soon after, Noga Alon observed that this question is equivalent to the existence of a phase transition (in N) in a planar dependent percolation model. Remarkably, several other interesting questions such as Lipschitz embeddings of binary sequences and rough (quasi-)isometries between one dimensional random objects also could be reduced to a similar type of percolation. During three lectures I will explain the multiscale approach suitable for dependent percolative systems and prove two Theorems which provide positive answer to the above questions. The course is based on joint works with H.Kesten and M.Vares, and with R. Basu and A.Sly.
Special Minicourse: Scheduling of random walks, Lipschitz embeddings and rough isometries of random sequences. Part IIread_more
Y27 H 46
7 November 2012
17:15-18:15
Prof. Dr. Andreas Kyprianou
University of Bath, UK
Event Details

Seminar on Stochastic Processes

Title Censored Stable Processes
Speaker, Affiliation Prof. Dr. Andreas Kyprianou, University of Bath, UK
Date, Time 7 November 2012, 17:15-18:15
Location HG G 43
Abstract We look at a general two-sided jumping strictly alpha-stable process where alpha is in (0,2). By censoring its path each time it enters the negative half line we show that the resulting process is a positive self-similar Markov Process. Using Lamperti's transformation we uncover an underlying driving Lévy process and, moreover, we are able to describe in surprisingly explicit detail the Wiener-Hopf factorization of the latter. Using this Wiener-Hopf factorization together with a series of spatial path transformations, it is now possible to produce an explicit formula for the law of the original stable processes as it first *enters* a finite interval, thereby generalizing a result of Blumenthal, Getoor and Ray for symmetric stable processes from 1961. This is joint work with Juan Carlos Pardo and Alex Watson.
Censored Stable Processesread_more
HG G 43
8 November 2012
17:15-18:15
Vladas Sidoravicius
IMPA, Rio de Janeiro, Brazil
Event Details

Seminar on Stochastic Processes

Title Special Minicourse: Scheduling of random walks, Lipschitz embeddings and rough isometries of random sequences. Part III
Speaker, Affiliation Vladas Sidoravicius, IMPA, Rio de Janeiro, Brazil
Date, Time 8 November 2012, 17:15-18:15
Location Y27 H 46
Abstract In the late eighties Peter Winkler introduced the following problem: consider two independent (discrete time) random walks, A and B, on the complete graph with N vertices. If the trajectories of A and B are given, can one, knowing all the future steps of the walks, and by changing jump times only, keep A and B apart forever (with positive probability)? It became well known as Clairvoyant Demon Problem. Soon after, Noga Alon observed that this question is equivalent to the existence of a phase transition (in N) in a planar dependent percolation model. Remarkably, several other interesting questions such as Lipschitz embeddings of binary sequences and rough (quasi-)isometries between one dimensional random objects also could be reduced to a similar type of percolation. During three lectures I will explain the multiscale approach suitable for dependent percolative systems and prove two Theorems which provide positive answer to the above questions. The course is based on joint works with H.Kesten and M.Vares, and with R. Basu and A.Sly.
Special Minicourse: Scheduling of random walks, Lipschitz embeddings and rough isometries of random sequences. Part IIIread_more
Y27 H 46
14 November 2012
17:15-18:15
Hao Wu
Université Paris-Sud
Event Details

Seminar on Stochastic Processes

Title Conformally Invariant Growing Mechanism in CLE$_4$ and Couplings between GFF and CLE$_4$
Speaker, Affiliation Hao Wu, Université Paris-Sud
Date, Time 14 November 2012, 17:15-18:15
Location HG G 43
Abstract In this talk, I first recall the definitions and a short history of SLE, CLE and GFF which are three important planar random objects in probability, statistic physics and quantum field theory. Second, I show that there is a conformally invariant growing mechanism in CLE for the special case $\kappa=4$ (constructed by Werner and me). This growing process give a time parameter to each loop in the CLE$_4:$ $((\gamma_j,t_j): j\in J)$ where $(\gamma_j,j\in J)$ is a sample of CLE$_4$. Third, I will show that, for the moment, there are two different couplings between GFF and CLE$_4$. The first coupling is given by Sheffield and Miller and the second coupling is given by Sheffield, Watson and me by use of the time parameter in CLE$_4.$ I will also show the relation between these two couplings.
Conformally Invariant Growing Mechanism in CLE$_4$ and Couplings between GFF and CLE$_4$read_more
HG G 43
21 November 2012
17:15-18:15
Jason Miller
Massachusetts Institute of Technology
Event Details

Seminar on Stochastic Processes

Title Imaginary Geometry and the Gaussian Free Field
Speaker, Affiliation Jason Miller, Massachusetts Institute of Technology
Date, Time 21 November 2012, 17:15-18:15
Location HG G 43
Abstract The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been one of the most exciting areas in probability theory over the last decade because Schramm's curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion, and how this perspective can be used to study the sample path behavior of SLE. Based on joint works with Scott Sheffield.
Imaginary Geometry and the Gaussian Free Fieldread_more
HG G 43
28 November 2012
00:00-00:00
Event Details

Seminar on Stochastic Processes

Title Swiss Probability Seminar
Speaker, Affiliation
Date, Time 28 November 2012, 00:00-00:00
Location Universität Bern, Sidlerstrasse 5, 3012 Bern, ExWi B78
Swiss Probability Seminar
Universität Bern, Sidlerstrasse 5, 3012 Bern, ExWi B78
5 December 2012
17:15-18:15
Pierre-Loïc Méliot
Universität Zürich
Event Details

Seminar on Stochastic Processes

Title The cut-off phenomenon for Brownian motions on compact Lie groups and symmetric spaces
Speaker, Affiliation Pierre-Loïc Méliot, Universität Zürich
Date, Time 5 December 2012, 17:15-18:15
Location HG G 43
Abstract For random shuffles of cards, a celebrated result of Diaconis ensures that the convergence towards the uniform law exhibits a cut-off: the total variation distance between the marginal law of the process and the uniform law on the symmetric group S_n is very close to 1 before t = 3 log n / log 4, and very close to 0 just after (hence, one needs precisely 7 shuffles of a deck of 52 cards to obtain a uniform permutation). This phenomenon is not restricted to discrete-time processes on discrete spaces: during this presentation, we shall see that the same happens for Brownian motions traced on compact Lie groups (unitary groups, orthogonal groups, symplectic groups) and on compact symmetric spaces (spheres, projective spaces, Grassmannian manifolds). The proof of this result mixes standard techniques of harmonic analysis, various probabilistic tools (method of moments, stochastic differential equations), and some computer-assisted calcutations.
The cut-off phenomenon for Brownian motions on compact Lie groups and symmetric spacesread_more
HG G 43
12 December 2012
17:15-18:15
Steffen Rohde
University of Washington
Event Details

Seminar on Stochastic Processes

Title Conformal geometry and random 2d structures
Speaker, Affiliation Steffen Rohde, University of Washington
Date, Time 12 December 2012, 17:15-18:15
Location HG G 43
Abstract The Brownian map is a highly non-smooth random metric space that is homeomorphic to the sphere (Le Gall, Paulin, Miermont) and arises for instance as the scaling limit of random triangulations. Motivated by the attempt to understand its conformal geometry, we will discuss conformal representations of the Uniform Infinite Planar Triangulation, the Continuum Random Tree, and the Conformal Loop Ensemble. Such representations are well-understood for their deterministic counterparts (fractal surfaces, dendrites, and Sierpinski carpets), and we will also discuss these analogies as well as some open questions.
Conformal geometry and random 2d structuresread_more
HG G 43
* 19 December 2012
16:15-17:15
Jiri Cerny
Universität Wien
Event Details

Seminar on Stochastic Processes

Title Directed random walk on an oriented percolation cluster
Speaker, Affiliation Jiri Cerny, Universität Wien
Date, Time 19 December 2012, 16:15-17:15
Location HG G 19.2
Abstract We prove the quenched CLT for the directed random walk on the infinite percolation cluster generated by supercritical oriented percolation, and explain how this problem relates to spatial embeddings of ancestral lineages in locally regulated populations.
Directed random walk on an oriented percolation clusterread_more
HG G 19.2

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

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