Mark Kac Seminar

April 3, 2009






Location: Utrecht, KNG80, room 230 (NOTE: different room than usually)


speaker: Franz Merkl (München)

title: What are crystals? (2)


This is the second lecture from a series of three lectures, starting March 6.


abstract (for the series): 


As is well-known from everyday knowledge, interacting molecules at thermal equilibrium at low temperature form cristals. At higher temperature, they undergo a melting transition to a liquid phase. From the viewpoint of statistical mechanics, these well-known facts are by far not understood.


The talks are concerned with the notion of spontaneous symmetry breaking in interacting particle systems. On the one hand, I will present a (over-)simplified model for a two-dimensional continuum particle system that spontaneously breaks rotational symmetry. On the other hand, by the famous Mermin-Wagner phenomenon, two-dimensional particle systems frequently show preservation of continuous symmetry. For example, Richthammer has recently shown that translational symmetry is preserved in two-dimensional hard-core particle systems.


The Mermin-Wagner phenomenon plays also a role in the recent understanding of linearly edge-reinforced random walks. There, absence of spontaneous breaking of a certain scaling symmetry plays an essential role.


(Joint work with Silke Rolles.).



speaker: Jürgen Gärtner (Berlin)

title: The Parabolic Anderson Model


The parabolic Anderson model is the spatially discrete heat equation with a random potential. As time evolves, the randomness of the potential may lead to a very irregular spatial structure of the solution consisting of islands of high peaks located far from each other. We explain how to investigate this effect of intermittency indirectly in terms of annealed Lyapunov exponents.

Our focus will be on recent progress for potentials which may be interpreted as moving catalysts in an autocatalytic reaction-diffusion model. We consider catalysts modeled by independent random walks, exclusion dynamics, and voter model dynamics in equilibrium. We show that the annealed Lyapunov exponents display an interesting dependence on the dimension and the diffusion coefficient and exhibit a kind of universal behavior in the limit of large diffusion.

(Joint work with Gregory Maillard and Frank den Hollander.)


Mark Kac Seminar 2008-2009


last updated: 18 dec 2008 by Markus