Fr, 09.02.2018, 15:15
I. Pinning model, universality and rough paths II. Coexistence of competing first-passage percolation on hyperbolic graphs.

Referent: Prof. Francesco Caravenna / Dr. Elisabetta Candellero
Veranstalter: AG Stochastik
Raum: S2 | 07 Raum 167

I. 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.

II.  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.)



Fachbereich Mathematik
Technische Universität Darmstadt

Schlossgartenstraße 7
64289 Darmstadt

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