Department of Mathematics

Search

Seminar on stochastic processes

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

×

Modal title

Modal content

Autumn Semester 2015

Date / Time Speaker Title Location
16 September 2015
17:15-18:15 		Titus Lupu
ETH-ITS
Event Details

Seminar on Stochastic Processes

Title Convergence of two-dimensional random walk loop soup clusters to CLE
Speaker, Affiliation Titus Lupu, ETH-ITS
Date, Time 16 September 2015, 17:15-18:15
Location HG G 43
Abstract The outermost boundaries of clusters of two-dimensional Brownian loop soups are known to be equal in law to Conformal Loop Ensembles (CLE). We will consider the random walk loop soups on two-dimensional lattice and the outermost boundaries of the corresponding clusters. We will show that in the scaling limit we obtain CLE loops. The proof uses an isomorphism between the random walk loop soups and the Gaussian free field.
Convergence of two-dimensional random walk loop soup clusters to CLEread_more
HG G 43
23 September 2015
17:15-18:15 		Pascal Maillard
Université Orsay
Event Details

Seminar on Stochastic Processes

Title On trees invariant under edge contraction
Speaker, Affiliation Pascal Maillard, Université Orsay
Date, Time 23 September 2015, 17:15-18:15
Location HG G 43
Abstract Take a random rooted tree and contract each edge with probability p, where contracting an edge means removing it from the tree and identifying its head and tail. Which random trees are invariant (in law) under this transformation? I will present recent results (joint with Olivier Hénard) which characterize all trees invariant under this operation in terms of limiting measured real (continuous) trees, and under the more general operation where edges on the infinite rays are contracted with a possibly different probability q. I will also describe the relationship with (real-valued) self-similar processes and quasi-stationary distributions of linear pure death processes.
On trees invariant under edge contractionread_more
HG G 43
30 September 2015
17:15-18:15 		Juhan Aru
ETH Zürich
Event Details

Seminar on Stochastic Processes

Title Two random measures on the sphere
Speaker, Affiliation Juhan Aru, ETH Zürich
Date, Time 30 September 2015, 17:15-18:15
Location HG G 43
Abstract Most likely we will try to understand two constructions of random measures on the sphere, one of them stemming from a recent work of Duplantier, Miller & Sheffield and the other from an equally recent work by David, Kupiainen, Rhodes, Vargas. Both of these measures should describe the unit area measure of the so called Liouville Quantum Gravity (LQG) sphere. I will try to convince you that these measures are indeed the same. > This is joint work with Yichao Huang and Xin Sun.
Two random measures on the sphereread_more
HG G 43
7 October 2015
17:15-18:15 		Rob van den Berg
CWI Amsterdam
Event Details

Seminar on Stochastic Processes

Title Percolation models where large clusters are frozen
Speaker, Affiliation Rob van den Berg, CWI Amsterdam
Date, Time 7 October 2015, 17:15-18:15
Location HG G 43
Abstract Motivated by phenomena in sol-gel transitions, David Aldous (2000) introduced and analysed a fascinating dynamic percolation model on a tree where clusters stop growing as soon as they become infinite. Soon after his work, Benjamini and Schramm pointed out that such a process does not exist on the square lattice (and other planar lattices). After an introduction I will give a brief overview of results in the literature, and then discuss recent and ongoing work (with Demeter Kiss and Pierre Nolin) on processes which, intuitively, are asymptotically of the same flavour and do exist on planar lattices. We focus on the question whether, informally speaking, the giant clusters occupy a negligible, moderate or very large fraction of space.
Percolation models where large clusters are frozenread_more
HG G 43
14 October 2015
17:15-18:15 		Christophe Sabot
Université de Lyon 1
Event Details

Seminar on Stochastic Processes

Title A random Schrödinger operator associated with the edge reinforced random walk and the vertex reinforced Jump process
Speaker, Affiliation Christophe Sabot, Université de Lyon 1
Date, Time 14 October 2015, 17:15-18:15
Location HG G 43
Abstract The ERRW and the VRJP are self-interacting processes that have been shown to be related to a supersymetric sigma field investigated by Disertori, Spencer and Zirnbauer. In this talk we construct a random Schrödinger operator, with a 1-dependent potential, and show that some of its spectral properties at ground state are related to the behavior of the VRJP and ERRW. We deduce from this a functional central limit theorem for the ERRW and the VRJP in dimension d>2 at weak disorder, and recurrence of the ERRW in dimension d=2 for any choice of initial constant weights. Based on joint works with P. Tarrès and X. Zeng.
A random Schrödinger operator associated with the edge reinforced random walk and the vertex reinforced Jump processread_more
HG G 43
21 October 2015
17:15-18:15 		Elliot Paquette
Weizmann Institute
Event Details

Seminar on Stochastic Processes

Title The law of fractional logarithm in the GUE minor process
Speaker, Affiliation Elliot Paquette, Weizmann Institute
Date, Time 21 October 2015, 17:15-18:15
Location HG G 43
Abstract Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry. The distribution of the largest eigenvalue of the upper-left NxN submatrix is well understood, as this is the GUE (Gaussian unitary ensemble). We show that if one lets N vary over all natural numbers, then the sequence of largest eigenvalues satisfies a 'law of fractional logarithm,' in analogy with the classical law of iterated logarithm for simple random walk. This is joint work with Ofer Zeitouni.
The law of fractional logarithm in the GUE minor processread_more
HG G 43
28 October 2015
17:15-18:15 		Laurie Field
EPF Lausanne
Event Details

Seminar on Stochastic Processes

Title SLE curves with a random marked point
Speaker, Affiliation Laurie Field, EPF Lausanne
Date, Time 28 October 2015, 17:15-18:15
Location HG G 43
Abstract Schramm-Loewner evolution (SLE) curves can be produced in a number of configurations that correspond to particular boundary conditions or constraints in physical models. If we take chordal SLE curves configured to pass through a marked point and average over all such points, the resulting random curve is related to unconstrained chordal SLE by a bias factor which is exactly the length of the curve in the natural, geometric parametrization. We will discuss this result, what it says about the shape of the curve, and an analogy to decompositions of Brownian motion.
SLE curves with a random marked pointread_more
HG G 43
4 November 2015
17:15-18:15 		Dr. Ron Rosenthal
ETH Zurich, Switzerland
Event Details

Seminar on Stochastic Processes

Title Simplicial branching random walks
Speaker, Affiliation Dr. Ron Rosenthal, ETH Zurich, Switzerland
Date, Time 4 November 2015, 17:15-18:15
Location HG G 43
Abstract We will discuss a new stochastic process on general simplicial complexes which allows to study their spectral and homological properties. Some results for random walks on graphs will be shown to hold in this general setting. In addition, we will discuss transience/recurrence classification for the process on high-dimensional analogues of regular trees, and show how to construct solutions to the high-dimensional Dirichlet problem.
Simplicial branching random walksread_more
HG G 43
11 November 2015
17:15-18:15 		Gourab Ray
Cambridge University
Event Details

Seminar on Stochastic Processes

Title Parabolic and hyperbolic unimodular random maps
Speaker, Affiliation Gourab Ray, Cambridge University
Date, Time 11 November 2015, 17:15-18:15
Location HG G 43
Abstract We show that for unimodular random rooted maps, various well known notions of parabolicity and hyperbolicity are equivalent. Joint work with Omer Angel, Tom Hutchcroft and Asaf Nachmias.
Parabolic and hyperbolic unimodular random mapsread_more
HG G 43
18 November 2015
17:15-18:15 		Mathieu Merle
Université Paris 7
Event Details

Seminar on Stochastic Processes

Title Self-organized criticality for coagulation/gelation dynamics
Speaker, Affiliation Mathieu Merle, Université Paris 7
Date, Time 18 November 2015, 17:15-18:15
Location HG G 43
Abstract In 99, D.Aldous studied a model of frozen percolation on a regular tree and proved it exhibits self-organized criticality : past first gelation event, typical unfrozen clusters are critical trees. He conjectured that this behavior could also occur for a model of frozen percolation on the complete graph in the asymptotics of a large number of vertices. In this model, edges between any two vertices appear independently at rate 1 and clusters are frozen when their size exceeds a given threshold $\alpha(n)$. As B.Rath, and N.Fournier and P.Laurençot have shown in 09, the $n \to \infty$ asymptotics of this model can be related to Smoluchowski equation with multiplicative kernel. Such equation was known to exhibit a gelation phenomenon : past a certain time, some mass is lost. The work of Rath further suggested Aldous' conjecture was true, which was confirmed in a recent work with R. Normand for a convenient choice of the threshold : past the first gelation time, asymptotic distribution of typical unfrozen clusters is that of critical trees. In this talk we will focus on a similar model, but with limited aggregations : $n$ vertices are initially given a number of half-edges $a_1,...,a_n$, that are successively linked together to form edges. When a cluster size exceeds a given threshold $\alpha(n)$ it becomes frozen. The model of coagulation with limited aggregations was introduced by J.Bertoin in 09, and linked before gelation to Smoluchowski's equation with limited aggregations. A remarkable feature of this equation is that it is completely solvable, even after gelation, as Bertoin, then Normand and L.Zambotti have shown. Choosing to freeze clusters above a given threshold allows to have the discrete model converge to Smoluchowski's equation with limited aggregations even past gelation time. In a joint work with Normand, for convenient choices of the number of initial half-edges and of the threshold, we are further able to find an explicit limit for the concentration and the degree measure of unfrozen vertices. In particular we find that the asymptotic distribution of a typical unfrozen cluster is that of a subcritical Galton-Watson tree before the first gelation time, and of a critical Galton-Watson tree afterwards.
Self-organized criticality for coagulation/gelation dynamicsread_more
HG G 43
* 25 November 2015
17:15-18:15 		Volker Betz
TU Darmstadt
Event Details

Seminar on Stochastic Processes

Title Spatial Random Permutations
Speaker, Affiliation Volker Betz, TU Darmstadt
Date, Time 25 November 2015, 17:15-18:15
Location HG E 33.3
Abstract Spatial random permutations are implemented by probability measures on permutations of a set with spatial structure; these measures are made so that they favor permutations that map points to nearby points. The strength of this effect is encoded in a parameter alpha > 0, where larger alpha means stronger bias toward short jumps. I will introduce some variants of the model, and explain the connections to the theory of Bose-Einstein condensation. Then I will present a few older results, as well as very recent progress made jointly with Lorenzo Taggi (TU Darmstadt) for the regime of large alpha. Finally, I will discuss two conjectures suggested by numerical simulation: in two dimensions, the model appears to exhibit a Kosterlitz-Thouless phase transition, and there are reasons to believe that in the phase of algebraic decay of correlations, long cycles are Schramm-Löwner curves, with parameter between 4 and 8 depending on alpha.
Spatial Random Permutationsread_more
HG E 33.3
2 December 2015
17:15-18:15 		Alain Rouault
Versailles
Event Details

Seminar on Stochastic Processes

Title Sum rules via large deviations
Speaker, Affiliation Alain Rouault , Versailles
Date, Time 2 December 2015, 17:15-18:15
Location HG G 43
Abstract In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. I will give a short historical introduction, from Szegö until Killip and Simon and descendants. These last authors proved in 2003 a quite surprising sum rule for measures dominating the semicircular distribution on $[-2,2]$. This sum rule includes a contribution of the atomic part of the measure away from $[-2,2]$. It is posssible to recover this sum rule by using large deviations on random matrices and on their extremal eigenvalues. This method allows to obtain new sum rules, when the reference measure is the Marchenko-Pastur or Kesten-McKay distribution. These formulas include a contribution of the atomic part appearing away from the support of the reference measure; (joint work with F. Gamboa and J. Nagel).
Sum rules via large deviationsread_more
HG G 43
9 December 2015
17:15-18:15 		Ronen Eldan
Weizmann Institute
Event Details

Seminar on Stochastic Processes

Title On Talagrand's convolution conjecture in Gaussian space: regularization of $L_1$ functions under the Ornstein-Uhlenbeck operator
Speaker, Affiliation Ronen Eldan, Weizmann Institute
Date, Time 9 December 2015, 17:15-18:15
Location HG G 43
Abstract We consider the Ornstein-Uhlenbeck convolution operator in Gaussian space, $f \to P_t[f]$. An easy fact is that every function, when convoluted with a small Gaussian noise, becomes $C_\infty$ smooth. This raises the question: is there any quantitative way of characterizing how quickly smoothing occurs under convolution? One natural way to quantify this is the so-called hypercontractivity property of the operator $P_t$: for every $t>0$ and $p>1$ there exists $q>p$ such that $P_t$ is a contraction from $L_p$ to $L_q$. This property, which is equivalent to a Log-Sobolev inequality has turned out to be extremely useful in several fields such as analysis of PDE's and quantum information theory. However, this is only meaningful when one has some a-priori bound the $L_p$ norm of the initial function, for some $p>1$. In a 1989 paper, Talagrand conjectured that for any non-negative function $f$ normalized to have integral $1$ over Gaussian space, the function $P_t[f]$ becomes smooth in the sense that the Gaussian measure of the set ${P_t[f](x) > \alpha}$ has Gaussian measure $o(1/\alpha)$, hence $P_t[f]$ satisfies an improved Markov inequality (this is dual to a certain isoperimetric-type bound). We prove this conjecture (this is joint work with James Lee).
On Talagrand's convolution conjecture in Gaussian space: regularization of $L_1$ functions under the Ornstein-Uhlenbeck operatorread_more
HG G 43

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.

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

JavaScript has been disabled in your browser