# Rhein-Main Kolloquium Stochastik

## TU Darmstadt / Goethe-Universität Frankfurt / Gutenberg-Universität Mainz

### Termine im Wintersemester 2017/2018

Freitag, 19.01.2018

Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05-432 (Hilbertraum)

15:15 Uhr: Steffen Dereich (Münster): Extended notions of local convergence for sequences of random graphs

Local convergence also known as Benjamini-Schramm convergence is an analogue of the Palm measure concept for growing sequences of finite random graphs $(G_n)_{n\in\mathbb N}$. Here one centers the graph around a uniformly chosen vertex and considers distributional convergence (in an appropriate local sense) to a random rooted and connected graph $(G,o)$  (in most cases a tree). At least on an informal level the local limit provides many insights about the structure of large graphs. For instance, the relative size of the largest component typically converges in probability to the probability that the graph $G$ is infinite.

In this talk we introduce new extended notions of local convergence that incorporate additional information on the graph localised around a uniformly chosen vertex. In one variant we keep information on the shortest weight-path to a second uniformly chosen vertex which can be interpreted as the path along which an infection occurs. A second variant keeps track of the local visits of a random walk run on the random graph.

16:45 Uhr: Peter Mörters (Köln): Metastability of the contact process on evolving scale-free networks

We study the contact process in the regime of small infection rates on scale-free networks evolving by stationary dynamics. A parameter allows us to interpolate between slow (static) and fast (mean-field) network dynamics. For two paradigmatic classes of networks we investigate transitions between phases of fast and slow extinction and in the latter case we analyse the density of infected vertices in the metastable state.

This is joint work with Emmanuel Jacob (ENS Lyon) and Amitai Linker (Universidad de Chile).

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Freitag, 26.01.2018

Universität Frankfurt, Institut für Mathematik, Robert-Mayer-Str. 8, Hilbertraum (302) (Achtung: Raumänderung!)

15:15 Uhr: Wolfgang König (WIAS / TU Berlin): The principal part of the spectrum of random Schrödinger operators in large boxes

We consider random Schr\"odinger operators of the form $\Delta+\xi$, where $\Delta$ is the lattice Laplacian on $\mathbb Z^d$ and $\xi$ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of $\mathbb Z^d$. We show that, for $\xi$ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where $\xi$ takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Our proofs are largely independent of existing  methods for controlling Anderson localization and they permit a rather explicit description of the shape of the potential and the eigenfunctions.

(Joint work with M. Biskup)

16:45 Uhr: Götz Kersting (Frankfurt): Laws of large numbers for general Lambda-coalescents

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Freitag, 09.02.2018

TU Darmstadt, Gebäude S2|07, Hörsaal 167 (Physik), Hochschulstraße 6

15:15 Elisabetta Candellero (University of Warwick): Coexistence of competing first-passage percolation on hyperbolic graphs

We consider two first-passage percolation processes FPP_1 and FPP_{\lambda}, spreading with rates 1 and \lambda > 0 respectively, on a non-amenable hyperbolic graph G with bounded degree.
FPP_1 starts from a single source at the origin of G, while the initial con figuration of FPP_{\lambda} consists of countably many seeds distributed according to a product of iid Bernoulli random variables of parameter \mu > 0 on V (G)\{o}. Seeds start spreading FPP_{\lambda} after they are reached by either FPP_1 or FPP_{\lambda}. We show that for any such graph G, and any fixed value of \lambda > 0 there is a value \mu_0 = \mu_0(G,\lambda ) > 0 such that for all 0 < \mu < \mu_0 the two processes coexist with positive probability. This shows a fundamental difference with the behavior of such processes on Z^d.

(Joint with Alexandre Stauffer.)

16:45 Francesco Caravenna (University of Milano-Bicocca): Pinning model, universality and rough paths

One of the simplest, yet challenging exampes of "disordered systems" in statistical mechanics is the so-called pinning model. This can be roughly described as a random walk which interacts with a random medium (the "disorder") concentrated along a line. In a suitable weak-disorder regime, this model admits a continuum scaling limit, which can be characterized through the solution of a singular stochastic equation, driven by a Brownian motion. In this talk, we present a robust analysis of this equation, using ideas from rough paths. This sheds light on the effect of disorder and leads naturally to universality results.

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### Wegbeschreibungen

zur Anreise an die Uni Mainz finden Sie unter https://www.mathematik.uni-mainz.de/anfahrt/ , zur Anreise an die Uni Frankfurt unter https://www.uni-frankfurt.de/38074653/campus_bockenheim und https://www.uni-frankfurt.de/38093742/Campus_Bockenheim-pdf.pdf, zur Anreise an die TU Darmstadt unter http://www3.mathematik.tu-darmstadt.de/fb/mathe/wir-ueber-uns/adresse-und-lageplan/anreise.html

### Termine in früheren Semestern finden Sie hier.

Last update: 12.12.2017, S. Grün, gruen@mathematik.uni-mainz.de