# Oberseminar Wahrscheinlichkeitstheorie und andere Vorträge im Sommersemester 2017

**Organisers**: Nina Gantert (TUM), Noam Berger (TUM), Markus Heydenreich (LMU), Franz Merkl (LMU), Silke Rolles (TUM), Konstantinos Panagiotou (LMU),

**Talks**:

**Monday**, 20^{th} February 2017, 16:15, LMU, room B 133, Theresienstr. 39, Munich

Philipp Würl (LMU)

Title: Talagrand’s Inequality and Applications

Abstract: We consider probabilities of deviations for functions, which depend on multiple independent random variables, from a certain value, usually the expected value. In order to find upper bounds for these probabilities one initially used approaches depending on martingales. In 1994, however, M. Talagrand showcased a new way to prove such concentration inequalities in his paper "Concentration of measure and isoperimetric inequalities in product spaces". This meant a significant progress in this subject and in many cases also provided better results than previous methods. The paper at hand presents this approach and thereby Talagrand's convex distance inequality as well as two proofs of it. Moreover, the variety of application possibilities of Talagrand's convex distance inequality will be demonstrated with the help of several examples such as Bin Packing and the traveling salesman problem.

**Monday**, 24^{th} April 2017, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Ewa Damek (Wroclaw)

Title: Affine stochastic equation with triangular matrices

Abstract: We consider affine stochastic equation X=AX+B, where A is an upper triangular matrix, X and B are vectors, X is independent of (A,B) and the equation is meant in law. Under appropriate assumptions X has a heavy tail, but unlike the Kesten situation the tails of components X_1,..., X_d of X decay with various speed. What is more interesting not only the exponents may be different but also non trivial slowly varying functions may appear in the asymptotics.

**Monday**, 15^{th} May 2016, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Leon Ramzews (LMU)

Title: Asymptotic Enumeration of Graph Classes with Many Components

Abstract: The aim of this presentation is to introduce a framework for the asymptotic enumeration of graph classes with many components. By "many" it is meant that the number of components grows linearly in the number of nodes. Firstly, existing results from present literature covering the asymptotic enumeration of (connected) block-stable graph classes are presented. Therefore, exponential generating functions and the symbolic method are needed in order to translate combinatorial problems into analytic ones. The second half of the presentation is devoted to discussing random sampling by Boltzmann samplers, which leads to the exact asymptotic behaviour of the number of graphs with certain properties taking into consideration the number of components. More precisely, Boltzmann samplers allow for transitioning into the field of probability theory by analysing sums of i.i.d. integer-valued random variables.

**Monday**, 22^{th} May 2017, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Serguei Popov (University of Campinas)

Title: Two-dimensional random interlacements

Abstract: We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some its properties. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view. Also, we discuss a continuous version of the model, build using the conditioned (on not hitting the unit disk) Brownian motion trajectories. This is a joint work with Francis Comets and Marina Vachkovskaia.

**Monday**, 29^{th} May 2017, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Simone M. Massaccesi (LMU)

Title: How to determine if a random simple graph with a fixed degree sequence has a giant component

Abstract: Recent works on the structure of social, biological and internet networks have attracted much attention on random graphs G(D) chosen uniformly at random among all graphs with a fixed degree sequence D = (d_1,...,d_n), where the vertex i has degree d_i. On this topic, a big step forward is represented by the result achieved by Joos, Perarnau, Rautenbach and Reed (1). It determines whether such a random simple graph G(D) has a giant component or not by imposing only one condition: the sum of all degrees which are not 2 must go to infinity with n. Furthermore, if it is not the case, they show that both the probability that G(D) has a giant component and the probability that G(D) has no giant component lie between p and 1-p, for a positive constant p. In this Thesis we present their work, traveling trough the main theorems and the generalization of the previous results again and adding some missing calculations and intermediate steps in order to elucidate it completely. Furthermore, we offer some examples and direct applications of these new criteria. Finally, we attaching implementation and graphical illustrations of almost all the treated cases.

**Monday**, 12^{th} June 2017, 16:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Timo Schlüter (Universität Mainz)

Title: Fractional moment estimates for interacting diusions

Abstract: TBA

**Monday**, 12^{th} June 2017, 17:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Andrej Depperschmidt (Erlangen)

Title: Recombination as a tree-valued process along the genome

Abstract: In Moran models the genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral recombination graph encodes the genealogies at all loci. For a continuous genome we study the tree-valued process of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure spaces, we show convergence to a tree-valued process. Furthermore we discuss some mixing properties of the resulting process. This is joint work with Etienne Pardoux and Peter Pfaffelhuber.

**Monday**, 19^{th} June 2017, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Dr. Asja Fischer (Universität Bonn)

Title: "Towards biologically plausible deep learning"

Abstract: "In recent years (deep) neural networks got the most prominent models for supervised machine learning tasks. They are usually trained based on stochastic gradient descent where backpropagation is used for the gradient calculation. While this leads to efficient training, it is not very plausible from a biological perspective. We show that Langevin Markov chain Monte Carlo inference in an energy-based model with latent variables has the property that the early steps of inference, starting from a stationary point, correspond to propagating error gradients into internal layers, similar to backpropagation. Backpropagated error gradients correspond to temporal derivatives with respect to the activation of hidden units. These lead to a weight update proportional to the product of the presynaptic firing rate and the temporal rate of change of the postsynaptic firing rate. Simulations and a theoretical argument suggest that this rate-based update rule is consistent with those associated with spike-timing-dependent plasticity. These ideas could be an element of a theory for explaining how brains perform credit assignment in deep hierarchies as efficiently as backpropagation does, with neural computation corresponding to both approximate inference in continuous-valued latent variables and error backpropagation, at the same time."

**Monday**, 3^{rd} July 2017, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Peter Gracar (University of Bath)

Title: Spread of infection by random walks - Multi-scale percolation along a Lipschitz surface

Abstract: A conductance graph on Z^d is a nearest-neighbor graph where all of the edges have positive weights assigned to them. In this talk, we will consider the spread of information between particles performing continuous time simple random walks on a conductance graph. We do this by developing a general multi-scale percolation argument using a two-sided Lipschitz surface that can also be used to answer other questions of this nature. Joint work with Alexandre Stauffer.

**Monday**, 10^{th} July 2017, 16:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Sebastian Andres (University of Cambridge)

Title: Diffusion processes on branching Brownian motion

Abstract: Branching Brownian motion (BBM) is a classical process in probability, describing a population of particles performing independent Brownian motion and branching according to a Galton Watson process. In this talk we present a one-dimensional diffusion process on BBM particles which is symmetric with respect to a certain random martingale measure. This process is obtained by a time-change of a standard Brownian motion in terms of the associated positive continuous additive functional. In a sense it may be regarded as an analogue of Liouville Brownian motion which has been recently constructed in the context of a Gaussian free field. This is joint work with Lisa Hartung.

**Monday**, 10^{th} July 2017, 17:00, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Perla Sousi (University of Cambridge)

Title: Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

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).

**Monday**, 17^{th} July 2017, 16:30, LMU, room B 251, Theresienstr. 39, Munich

Robert Bildl (LMU)

Title: Intrinsic arm exponents of critical percolation on high-dimensional lattices

Abstract: siehe Link

**Monday**, 17^{th} July 2017, 17:30, LMU, room B 251, Theresienstr. 39, Munich

Prof. Dr. Stefan Tappe (Universität Hannover)

Title: Invariant manifolds in the space of tempered distributions

Abstract: In this presentation we deal with the existence of solutions to stochastic partial differential equations in scales of Hilbert spaces, and show how this is related to the existence of invariant manifolds. As a particular example, we will treat an equation in the space of tempered distributions; here the Hilbert scales are given by Hermite-Sobolev spaces.

**Monday**, 24^{th} July 2017, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Prof. Dr. Aernout van Enter (University of Groningen)

Title: One-sided versus two-sided points of view

Abstract: Finite-state, discrete-time Markov chains coincide with Markov fields on Z, (which are nearest-neighbour Gibbs measures in one dimension). That is, the one-sided Markov property and the two-sided Markov property are equivalent. We discuss to what extent this remains true if we try to weaken the Markov property to the almost Markov property, which is a form of continuity of conditional probabilities. The generalisation of the one-sided Markov measures leads to the so-called "g-measures" (aka "chains with complete connection", "uniform martingales",..), whereas the two-sided generalisation leads to the class of Gibbs or DLR measures, as studied in statistical mechanics. It was known before that there exist g-measures which are not Gibbs measures. It is shown here that neither class includes the other. We consider this issue in particular on the example of long-range, Dyson model, Gibbs measures. (Work with R.Bissacot,E.Endo and A. Le Ny)

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