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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Autumn Semester 2011

Date / Time Speaker Title Location
21 September 2011
17:15-18:15
Prof. Dr. Pierre Nolin
ETH Zurich, Switzerland
Event Details

Seminar on Stochastic Processes

Title Monochromatic arm exponents for 2D percolation
Speaker, Affiliation Prof. Dr. Pierre Nolin, ETH Zurich, Switzerland
Date, Time 21 September 2011, 17:15-18:15
Location HG G 43
Abstract We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents describe the probability to observe a given number of disjoint paths of the same color across annuli of large modulus. After showing that they exist, we investigate how they are related to the (now well-understood) polychromatic exponents (describing probabilites when arms of both colors are present). In particular, an interesting correlation inequality shows up. This is a joint work with Vincent Beffara.
Monochromatic arm exponents for 2D percolationread_more
HG G 43
5 October 2011
17:15-18:15
Nicholas Kistler
Bonn
Event Details

Seminar on Stochastic Processes

Title The extremal process of branching Brownian motion
Speaker, Affiliation Nicholas Kistler, Bonn
Date, Time 5 October 2011, 17:15-18:15
Location HG G 43
Abstract I will report on work with Louis Pierre Arguin and Anton Bovier on the extremal process of branching Brownian motion. The main result is an explicit construction of the limiting object in terms of a Poisson cluster process.
The extremal process of branching Brownian motionread_more
HG G 43
12 October 2011
17:15-18:15
Elie Aidekon
Paris
Event Details

Seminar on Stochastic Processes

Title Speed of the transient random walk on a Galton-Watson tree
Speaker, Affiliation Elie Aidekon, Paris
Date, Time 12 October 2011, 17:15-18:15
Location HG G 43
Abstract We consider a biased random walk on a Galton-Watson tree, the bias $\lambda$ being towards the root of the tree. When the bias is smaller than a critical value, the walk is transient. In this case, we are interested in the asymptotic distribution of the tree seen from the particle. As a consequence, we derive an expression for the speed of the walk. The results are expressed in terms of a random variable $\beta$, which stands for the quenched probability for the random walk of never returning to the root. Similar results were known in the case of the simple random walk (Lyons,Pemantle,Peres) or more recently in the recurrent (Peres,Zeitouni) and near-recurrent (Ben Arous, Hu, Olla, Zeitouni) cases.
Speed of the transient random walk on a Galton-Watson treeread_more
HG G 43
19 October 2011
17:15-18:15
Dr. Jiri Cerny
ETH Zurich, Switzerland
Event Details

Seminar on Stochastic Processes

Title Chemical distance on random interlacements
Speaker, Affiliation Dr. Jiri Cerny, ETH Zurich, Switzerland
Date, Time 19 October 2011, 17:15-18:15
Location HG G 43
Abstract We investigate the chemical (or graph) distance on the infinite occupied cluster of interlacements. We will explain that large balls in this distance obeys a full shape theorem. The main tool in proving this is a `large deviation estimate', similar to the result of Antal and Pisztora in the case of Bernoulli percolation. This is joint work with Serguei Popov.
Chemical distance on random interlacementsread_more
HG G 43
26 October 2011
17:15-18:15
Frederik Johansson Viklund
New York
Event Details

Seminar on Stochastic Processes

Title On convergence rates for random Loewner curves
Speaker, Affiliation Frederik Johansson Viklund, New York
Date, Time 26 October 2011, 17:15-18:15
Location HG G 43
Abstract The Schramm-Loewner evolution (SLE) is a family of random planar curves that appear as scaling limits of curves derived from a range of discrete lattice models from statistical physics. SLE is constructed by solving the Loewner differential equation with a Brownian motion as the so-called Loewner driving function. A first step towards proving convergence to SLE is to show that the Loewner driving function for the discrete curve converges to Brownian motion. In the talk we will discuss recent work on how to estimate the convergence rate to the SLE curve, given as input a convergence rate for the Loewner driving function.
On convergence rates for random Loewner curvesread_more
HG G 43
2 November 2011
17:15-18:15
Thomas Richthammer
Munich
Event Details

Seminar on Stochastic Processes

Title The spectral gap of the interchange process - a proof of Aldous' conjecture
Speaker, Affiliation Thomas Richthammer, Munich
Date, Time 2 November 2011, 17:15-18:15
Location HG G 43
Abstract We consider a finite, connected, undirected graph with weighted edges. Labeled particles are assigned to its vertices (one to each vertex), and any two particles connected by an edge may interchange their positions at a rate given by the corresponding edge weight. The resulting continuous time Markov chain is called interchange process. The spectral gap of this process determines its rate of convergence to the uniform distribution. In about 1990 David Aldous conjectured that the spectral gap of the interchange process is the same as that of the random walk on the same graph, but so far this has been proved only in some special cases. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, where the edge weights play the role of conductances. The main technical obstacle is to show that this reduction decreases the total energy of the network corresponding to the (weighted) Cayley graph of the group of vertex permutations, generated by edge transpositions. Our method gives a proof of Aldous' conjecture in full generality. As a consequence we are able to relate the spectral gaps of various processes (such as exclusion processes) to that of the random walk on the same graph. This is joint work with Pietro Caputo (Roma Tre) and Tom Liggett (UCLA).
The spectral gap of the interchange process - a proof of Aldous' conjecture read_more
HG G 43
9 November 2011
17:15-18:15
Stefan Adams
Warwick
Event Details

Seminar on Stochastic Processes

Title Random field of gradients
Speaker, Affiliation Stefan Adams, Warwick
Date, Time 9 November 2011, 17:15-18:15
Location HG G 43
Abstract Random fields of gradients are a class of model systems arising in the studies of random interfaces, random geometry, field theory, and elasticity theory. These random objects pose challenging problems for probabilists as even an a priori distribution involves strong correlations. Gradient fields are likely to be an universal class of models combining probability, analysis and physics in the study of critical phenomena. They emerge in the following three areas, effective models for random interfaces, Gaussian Free Fields (scaling limits), and mathematical models for the Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear elasticity. The latter class of models requires that interaction energies are non-convex functions of the gradients. Open problems over the last decades include unicity of Gibbs measures and strict convexity of the free energy. We present in the talk the first break through for the free energy at low temperatures using Gaussian measures and rigorous renormalisation group techniques yielding an analysis in terms of dynamical systems. The key ingredient is a finite range decomposition for parameter dependent families of Gaussian measures. We outline the connection to the Cauchy-Born rule which states that the deformation at the atomistic level is locally given by an affine deformation at the boundary. Finally we discuss the behaviour under scaling limits.
Random field of gradientsread_more
HG G 43
16 November 2011
17:15-18:15
Amandine Veber
Palaiseau
Event Details

Seminar on Stochastic Processes

Title Large-scale behaviour of the spatial Lambda-Fleming-Viot process
Speaker, Affiliation Amandine Veber, Palaiseau
Date, Time 16 November 2011, 17:15-18:15
Location HG G 43
Abstract The SLFV process is a population model in which individuals live in a continuous space. Each of them also carries some heritable genetic type or allele. We shall describe the long-term behaviour of this measure-valued process and that of the corresponding genealogical process of a sample of individuals in two cases : one that mimics the evolution of nearest-neighbour voter model (but in a spatial continuum), and one that allows some individuals to send offspring at very large distances. This is a joint work with Nathanaël Berestycki and Alison Etheridge.
Large-scale behaviour of the spatial Lambda-Fleming-Viot processread_more
HG G 43
21 November 2011
17:15-18:15
Jian Ding
Stanford
Event Details

Seminar on Stochastic Processes

Title Cover times, blanket times, and Gaussian free fields
Speaker, Affiliation Jian Ding, Stanford
Date, Time 21 November 2011, 17:15-18:15
Location HG G 19.2
Abstract In this talk, I will report some recent progresses on the cover time of a random walk, with the emphasis on its connection to the maximum of the discrete Gaussian free field. Among other things, below are the two main results to be presented. 1, We show that both cover times and blanket times are equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on $G$, scaled by the number of edges in $G$. This yields a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). Also, this positively resolves the blanket time conjecture of Winkler and Zuckerman (1996). The best previous approximation factor for both these problems was $O((\log \log n)^2)$ for $n$-vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000). This result is a joint work with James R. Lee and Yuval Peres (2010). 2, We show that on bounded degree graphs and general trees the above-mentioned universal constant is asymptotically 1, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time. This will be a survey type of talk (very light in technical details). I will try to convey the conceptual connections between cover times and GFFs as well as the resistance metric, and to highlight several main tools/methods in the proof including Talagrand's majorizing measure theory, Dynkin's Isomorphism theorem and the sprinkling method.
Cover times, blanket times, and Gaussian free fieldsread_more
HG G 19.2
23 November 2011
14:40-18:00
Event Details

Seminar on Stochastic Processes

Title Swiss Probability Seminars
Speaker, Affiliation
Date, Time 23 November 2011, 14:40-18:00
Location Universität Bern, Sidlerstrasse 5, A6
Swiss Probability Seminars
Universität Bern, Sidlerstrasse 5, A6
30 November 2011
17:15-18:15
Jean-François Le Gall
Orsay
Event Details

Seminar on Stochastic Processes

Title The continuous limit of large random planar maps
Speaker, Affiliation Jean-François Le Gall, Orsay
Date, Time 30 November 2011, 17:15-18:15
Location HG G 43
Abstract Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they have also significant geometrical applications. Particular cases of planar maps are p-angulations, where each face (meaning each component of the complement of edges) has exactly p adjacent edges. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all p-angulations with n vertices. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. Both in the case p=3 and when p>3 is even, we prove that the resulting random metric spaces converge as n tends to infinity to a universal object called the Brownian map. This convergence holds in the sense of the Gromov-Hausdorff distance between compact metric spaces. In the particular case of triangulations (p=3), this result solves an open problem stated by Oded Schramm in his 2006 ICM paper. As a key tool, we use bijections between planar maps and various classes of labeled trees.
The continuous limit of large random planar mapsread_more
HG G 43
7 December 2011
17:15-18:15
Nicolas Fournier
Creteil
Event Details

Seminar on Stochastic Processes

Title One-dimensional forest-fire models
Speaker, Affiliation Nicolas Fournier, Creteil
Date, Time 7 December 2011, 17:15-18:15
Location HG G 43
Abstract We consider the forest fire process on Z: on each site, seeds and matches fall at random, according to some independent Poisson processes. When a seed falls on a vacant site, a tree immediately grows. When a match falls on an occupied site, a fire destroys immediately the corresponding occupied connected component. We are interested in the asymptotics of rare fires. We prove that, under space/time re-scaling, the process converges (as matches become rarer and rarer) to a limit forest fire process. Next, we consider the more general case where seeds and matches fall according to some independent stationary renewal processes (not necessarily Poisson). According to the tail distribution of the law of the delay between two seeds (on a given site), there are 4 possible scaling limits. We finally introduce some related coagulation fragmentation equations, of which the stationary distribution can be more or less explicitely computed and of which we study the scaling limit.
One-dimensional forest-fire modelsread_more
HG G 43
14 December 2011
17:15-18:15
Martin Barlow
Vancouver
Event Details

Seminar on Stochastic Processes

Title Energy of cutoff functions and heat kernel upper bounds
Speaker, Affiliation Martin Barlow, Vancouver
Date, Time 14 December 2011, 17:15-18:15
Location HG G 43
Abstract I will discuss heat kernel upper bounds (on a manifold or graph, for example), particularly the type of 'sub-Gaussian' bounds which hold on regular fractals. I will explain how these are equivalent to the existence of low energy cutoff functions in annuli. This new characterisation of the bounds enables us to show that they are stable under various perturbations of the space, including rough isometries. This is a joint work with Sebastian Andres.
Energy of cutoff functions and heat kernel upper boundsread_more
HG G 43
21 December 2011
17:15-18:15
Gady Kozma
Weizmann Institute
Event Details

Seminar on Stochastic Processes

Title Disorder, entropy and harmonic functions
Speaker, Affiliation Gady Kozma, Weizmann Institute
Date, Time 21 December 2011, 17:15-18:15
Location HG G 43
Abstract We show that a supercritical percolation cluster does not support a non-constant sublinear harmonic function. The proof uses the entropy argument of Avez (originally developed to study bounded harmonic functions on groups). Joint work with Itai Benjamini, Hugo Duminil-Copin and Ariel Yadin. All terms will be explained in the talk.
Disorder, entropy and harmonic functionsread_more
HG G 43

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Organizers: Erwin Bolthausen, Jiri Cerny, Ashkan Nikeghbali, Martin Schweizer, Alain-Sol Sznitman

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