19.01.2024

Joscha Prochno: The probabilistic behavior of lacunary sums

It is known through classical works of Kac, Salem, Zygmund, Erdös and Gal that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem or a law of the iterated logarithm. We present some recent results on their large deviation behavior, which show that on this scale, contrary to the scale of the CLT or the LIL, the LDP is sensitive to the arithmetic properties of the underlying Hadamard gap sequence. If time allows, we shall briefly discuss some recent results regarding moderate deviations and the optimality of Diophantine conditions in the law of the iterated logarithm for lacunary systems.

Volker Betz: The polaron problem

The Fröhlich polaron models a charged quantum particle interactiong with a polar cystal. Since the moving particle has to drag along a ‘cloud’ of polarization, it appears heavier than it would be without the interaction. An old conjecture of Landau and Pekar states that this so-called effective mass scales as the fourth power of the coupling constant, with a precisely predicted pre-factor. While the conjecture is now 75 years old, only very recently significant progress has been made on the mathematical side. Part of it uses probabilistic methods, and I will report mostly on these aspects: starting with a Feynman-Kac representation, the task is to study the mean square displacement of a Brownian motion perturbed by an attractive pair interaction. This model and its connection to the polaron is known for a long time, will be the starting point of my talk, and can be understood without reference to the quantum model.
I will then report on two different recent methods which allow to actually estimate the mean square displacement. This is based on joint work with Steffen Polzer (Geneva), Tobias Schmidt (Darmstadt) and Mark Sellke (Harvard).