Departement Mathematik

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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Frühjahrssemester 2016

Datum / Zeit Referent:in Titel Ort
24. Februar 2016
17:15-18:15 		Gregory Lawler
University of Chicago
Details

Seminar on Stochastic Processes

Titel Loop-erased random walk, SLE(2) and natural parametrization
Referent:in, Affiliation Gregory Lawler, University of Chicago
Datum, Zeit 24. Februar 2016, 17:15-18:15
Ort Y27 H 25
Loop-erased random walk, SLE(2) and natural parametrization
Y27 H 25
2. März 2016
17:15-18:15 		Bastien Mallein
Université Paris-Sorbonne
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Seminar on Stochastic Processes

Titel Branching random walk with selection
Referent:in, Affiliation Bastien Mallein, Université Paris-Sorbonne
Datum, Zeit 2. März 2016, 17:15-18:15
Ort Y27 H 25
Abstract A branching random walk is a particle system on the real line that evolves as follows. At each generation, each particle dies giving birth to children that are independently positioned at random around their parent. In a branching random walk with selection, the position of an particle is understood as its fitness. Only a portion of the population, chosen according to their fitness, survives to reproduce. I will introduce some of the main conjectures on this process as well as some existing results.
Branching random walk with selectionread_more
Y27 H 25
9. März 2016
17:15-19:00 		Khalil Chouk
Humboldt Universität, Berlin
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Seminar on Stochastic Processes

Titel Random operator with singular potential
Referent:in, Affiliation Khalil Chouk, Humboldt Universität, Berlin
Datum, Zeit 9. März 2016, 17:15-19:00
Ort Y27 H 25
Abstract I will present in this talk a general framework to study Elliptic (or parabolic) operator with Schwartz distributional potential. Namely the two main example are the Schro ̈dinger operator H = −∆ + ξ with white noise potential on a box [− L , L ]d for d ≤ 3 and the generator operator 22 G = ∂t + 1∆ + v∇ 2 associate to the SDE dxt =v(t,xt)dt+dbt where b is a d-dimensional Brownian motion and v : [0, T ] × Rd → Rd is a continuous function in time and singular in space as for example a singular Gaussian field or the Gradient of a KPZ type equation.
Random operator with singular potentialread_more
Y27 H 25
16. März 2016
17:15-18:15 		Ioan Manolescu
Université de Fribourg
Details

Seminar on Stochastic Processes

Titel Scaling limits and influence of the seed graph in preferential attachment trees
Referent:in, Affiliation Ioan Manolescu, Université de Fribourg
Datum, Zeit 16. März 2016, 17:15-18:15
Ort Y27 H 25
Abstract We investigate two aspects of large random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees. Starting with a given tree (called the seed), a random sequence of trees is built by adding vertices one by one, connecting them to one of the existing vertices chosen randomly with probability proportional to its degree. Bubeck, Mossel and Racz conjectured that the law of the trees obtained after adding a large number of vertices still carries information about the seed from which the process started. We confirm this conjecture using an observable based on the number of ways of embedding a given (small) tree in a large tree obtained by preferential attachment. Next we study scaling limits of such trees. Since the degrees of vertices of a large preferential attachment tree are much higher than its diameter, a simple scaling limit would lead to a non locally compact space that fails to capture the structure of the object. Yet, for a planar version of the model, a much more convenient limit may be defined via its loop tree. The limit is a new object called the Brownian tree, obtained from the CRT by a series of quotients. Based on join work with Nicolas Curien, Thomas Duquesne and Igor Kortchemski.
Scaling limits and influence of the seed graph in preferential attachment treesread_more
Y27 H 25
23. März 2016
17:15-18:15 		Hendrik Weber
University of Warwick
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Seminar on Stochastic Processes

Titel The dynamic $Phi^4$ model - Scaling limits and global existence
Referent:in, Affiliation Hendrik Weber, University of Warwick
Datum, Zeit 23. März 2016, 17:15-18:15
Ort Y27 H 25
Abstract In this talk I will discuss some recent progress on the dynamic $\Phi^4$ model, which is formally given by a non-linear stochastic PDE which is driven by space-time white noise. Due to the irregularity of the noise for spatial dimension $d \geq 2$ solutions are distribution valued and a renormalisation procedure has to be performed to interpret the non-linear term. In the first part of the talk I will discuss how this equation arises of scaling limit of a suitably rescaled dynamic Kac-Ising model. I will show in particular that the renormalisation procedure corresponds to a shift of the inverse temperature on the level of the particle model. In the second part of the talk I will discuss a very recent result on the non-explosion of this SPDE over the three dimensional torus. This is joint work with J.C. Mourrat (Lyon).
The dynamic $Phi^4$ model - Scaling limits and global existenceread_more
Y27 H 25
6. April 2016
17:15-18:15 		Torben Krüger
IST Austria
Details

Seminar on Stochastic Processes

Titel Local eigenvalue statistics for random matrices with short range correlations
Referent:in, Affiliation Torben Krüger, IST Austria
Datum, Zeit 6. April 2016, 17:15-18:15
Ort Y27 H 25
Abstract The statistics of eigenvalues of random matrix ensembles often exhibit universal behaviors as the sizes of the matrices grow to infinity. By this we mean that statistical quantities (e.g. k-point correlation functions of eigenvalues, fluctuations of eigenvalues around their expected positions, distributions of gap sizes between neighboring eigenvalues, etc.) do not depend on most of the details of the model. We prove such a universality statement for non-centered random matrices with short range correlations and entries of comparable sizes ('mean field' regime). Our analysis shows that the resolvent G(z) = 1/(H-z) of H approaches a deterministic limit as long as the spectral smoothing Im[z] is larger than the typical eigenvalue spacing. This limit satisfies a matrix equation which only depends on the first and second moments of the entries of H.
Local eigenvalue statistics for random matrices with short range correlationsread_more
Y27 H 25
13. April 2016
17:15-18:15 		Perla Sousi
University of Cambridge
Details

Seminar on Stochastic Processes

Titel Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
Referent:in, Affiliation Perla Sousi, University of Cambridge
Datum, Zeit 13. April 2016, 17:15-18:15
Ort Y27 H 25
Abstract A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on the integer points in [0,n) without seeing each other. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, Y. Peres, R. Peretz and P. Winkler).
Hunter, Cauchy Rabbit, and Optimal Kakeya Setsread_more
Y27 H 25
20. April 2016
17:15-18:15 		Béatrice de Tilière
LAMA, UFR des Sciences et Technologie, Paris
Details

Seminar on Stochastic Processes

Titel Aspects of the Z-invariant Ising model
Referent:in, Affiliation Béatrice de Tilière, LAMA, UFR des Sciences et Technologie, Paris
Datum, Zeit 20. April 2016, 17:15-18:15
Ort Y27 H 25
Aspects of the Z-invariant Ising model
Y27 H 25
27. April 2016
17:15-18:15 		Matthias Winkel
University of Oxford
Details

Seminar on Stochastic Processes

Titel Recursive construction of CRTs and a binary embedding of the stable tree
Referent:in, Affiliation Matthias Winkel, University of Oxford
Datum, Zeit 27. April 2016, 17:15-18:15
Ort Y27 H 25
Abstract We introduce a general recursive method to construct continuum random trees (CRTs) from i.i.d. copies of a string of beads, that is, any random interval equipped with a random discrete measure. We prove the existence of these CRTs as a new application of the fixpoint method formalised in high generality by Aldous and Bandyopadhyay. We apply this recursive method to “embed” Duquesne and Le Gall’s stable tree into a binary compact CRT in a way that solves an open problem posed by Goldschmidt and Haas. We also express this binary CRT as a tree built by replacing all branch points of a stable tree by rescaled i.i.d. copies of a Ford CRT. Some of these developments are carried out in a space of ∞-marked metric spaces generalising Miermont’s notion of a k-marked metric space. This is joint work with Franz Rembart.
Recursive construction of CRTs and a binary embedding of the stable treeread_more
Y27 H 25
4. Mai 2016
17:15-18:15 		Assaf Naor
Princeton University
Details

Seminar on Stochastic Processes

Titel Metric X_p inequalities
Referent:in, Affiliation Assaf Naor, Princeton University
Datum, Zeit 4. Mai 2016, 17:15-18:15
Ort Y27 H 25
Abstract The purpose of this talk is to explain how a moment inequality for sums of symmetrically exchangeable random variables of Johnson, Maurey, Schechtman and Tzafriri (1979) inspired the definition of a new metric invariant that completes the search for bi-Lipschitz invariants that certify when L_q fails to embed into L_p. These investigations include various extensions of the classical moment inequality of Johnson, Maurey, Schechtman and Tzafriri, including its generalization to noncommutative settings as well as to Rademacher chaos of arbitrary order.
Metric X_p inequalitiesread_more
Y27 H 25
11. Mai 2016
17:15-18:15 		Alessandro Giuliani
Università di Roma Tre
Details

Seminar on Stochastic Processes

Titel Height fluctuations and universality relations in interacting dimer models
Referent:in, Affiliation Alessandro Giuliani, Università di Roma Tre
Datum, Zeit 11. Mai 2016, 17:15-18:15
Ort Y27 H 25
Abstract Two-dimensional dimer models are popular models, which are used to describe either the liquid phase of dense anisotropic molecules or, thanks to a well-known mapping between dimer configurations and discrete height functions, the rough phase of fluctuating random surfaces. In this talk, we consider a system of *interacting* dimers on the two-dimensional square lattice. By constructive renormalization group techniques, we compute the multipoint dimer correlationsand all the moments of the height function. In particular, we rigorously establish the asymptotic equivalence between the height function and the massless Gaussian free field (GFF). The variance K=K(\lambda) of the GFF is shown to be a non trivial analytic function of the interaction strength \lambda between dimers. We also prove one of the Haldane relations adapted to the present context, namely that K *equals* the dimer-dimer critical exponent X_+. Joint work with V. Mastropietro and F. Toninelli.
Height fluctuations and universality relations in interacting dimer modelsread_more
Y27 H 25
18. Mai 2016
17:15-18:15 		Margherita Disertori
Universität Bonn
Details

Seminar on Stochastic Processes

Titel History dependent stochastic processes and nonlinear sigma models
Referent:in, Affiliation Margherita Disertori, Universität Bonn
Datum, Zeit 18. Mai 2016, 17:15-18:15
Ort Y27 H 25
Abstract Edge reinforced random walk (ERRW) and vertex reinforced jump processes (VRJP) are history dependent stochastic processes, where the particle tends to come back more often on sites it has already visited in the past. For a particular scheme of reinforcement these processes are random walks in a random environment (mixing of reversible Markov chains) whose mixing measure can be related to a nonlinear sigma model introduced in the context of random matrix models for quantum diffusion. This relation allows to prove, in particular, transience for weak reinforcement of both processes, in d larger or equal to 3. I will give a review of the problem and some recent results.
History dependent stochastic processes and nonlinear sigma modelsread_more
Y27 H 25
1. Juni 2016
17:15-18:15 		Roland Bauerschmidt
Harvard University
Details

Seminar on Stochastic Processes

Titel Rigidity of one-component plasma in 2D
Referent:in, Affiliation Roland Bauerschmidt, Harvard University
Datum, Zeit 1. Juni 2016, 17:15-18:15
Ort Y27 H 25
Abstract The one-component plasma is a Coulomb gas of N equal negatively charged particles (in the continuum) confined by a potential. In two dimensions, for a special temperature, it is integrable as a determinantal point process, and the system can be understood is much detail. However, for other temperatures, our understanding is rather limited, and basic properties of its behaviour remain not understood. I will discuss a proof that it is rigid, in the sense that fluctuations of its linear statistics are much smaller than for a Poisson process, at all temperatures. This is joint work with Paul Bourgade, Miika Nikula, and Horng-Tzer Yau.
Rigidity of one-component plasma in 2Dread_more
Y27 H 25

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