Winter term 2023/2024
Friday 19 January 2024 at LMU Munich
Titles and abstracts:
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).
Friday 1 December 2023 in Augsburg
Titles and abstracts:
Pierre Calka: Close-up on random convex interfaces
In this talk, we investigate random convex interfaces which are generated as convex hulls of random point sets. We are interested in their asymptotic behavior when the size of the input goes to infinity. In a first part, we mainly identify average and maximal fluctuations in the radial and longitudinal directions through precise convergence results. We observe that the model shares some common features with the famous KPZ universality class of certain growth processes, including the scaling of type 1:2:3 or the appearance of a limit distribution similar to the Tracy-Widom distribution. In a second part, we consider the peeling procedure which consists in iterating the construction of the convex hull of the point set. The so-called layers are asymptotically governed by a deterministic analytical model and we study the geometric characteristics of each of the first layers. The talk is based on several joint works with Joe Yukich and Gauthier Quilan.
Vitali Wachtel: Harmonic measure in a multidimensional gambler's problem
We consider a random walk in a truncated cone K_N , which is obtained by slicing cone K by a hyperplane at a growing level of order N. We study the behaviour of the Green function in this truncated cone as N increases. Using these results we also obtain the asymptotic behaviour of the harmonic measure. The obtained results are applied to a multidimensional gambler’s problem studied by Diaconis and Ethier (2022). In particular we confirm their conjecture that the probability of eliminating players in a particular order has the same exact asymptotic behaviour as for the Brownian motion approximation. We also provide a rate of convergence of this probability towards this approximation.
Monday 30 October 2023 at LMU Munich
Titles and abstracts:
Ecatarina Sava-Huss: Abelian Sandpile Markov chains
The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G. The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed as a random walk on a finite group. Then it is natural to ask about the stationary distribution and the speed of convergence to stationarity, and how do these quantities depend on the underlying graph . I will report on some recents results on Abelian sandpiles on fractal graphs, and state some open questions concerning the critical exponents for such processes. The talk is based on joint works with Nico Heizmann, Robin Kaiser and Yuwen Wang.
Dirk Erhard: The tube property for the swiss cheese problem
In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation event is for the underlying Brownian motion to behave like a swiss cheese: stay most of the time inside a ball of subdiffusive size, visit most of the points but leave some random holes. They moreover conjectured that to do so the Brownian motion behaves like a Brownian motion in a drift field given by a function of the maximizer of the variational problem.
In this talk I will talk about the corresponding problem for the random walk and will explain that conditioned to having a small range its properly defined empirical measure is indeed close to the maximizer of the above mentioned variational problem.
This is joint work with Julien Poisat.
Summer term 2023
Friday 21 July 2023 in Augsburg
Titles and abstracts:
Elia Bisi: Non-intersecting path constructions for inhomogeneous TASEP and the KPZ fixed point
Abstract: The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, alternative approach to the KPZ fixed point, based on combinatorial structures and non-intersecting path constructions, which also allows studying inhomogeneous interacting particle systems. More specifically, we consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters, starting from an arbitrary initial configuration. We provide an explicit, step-by-step route from the very definition of the model to a Fredholm determinant representation of the joint distribution of the particle positions in terms of certain random walk hitting probabilities. Our tools include the combinatorics of the Robinson-Schensted-Knuth correspondence, intertwining relations, non-intersecting lattice paths, and determinantal point processes.
Stefan Adams: Scaling limits for non-convex interaction
Abstract: We introduce a currently hot topic in probability theory, the theory of scaling limits for random fields of gradients in all dimensions. The random fields are a class of model systems arising in the studies of random interfaces, random geometry, Euclidean field theory, the theory of regularity structures, and elasticity theory. After explaining how non-convex energy terms can influence the scaling limit, we outline our result on the scaling limit to the continuum Gaussian Free Field in dimension d=2,3 for a class of non-convex interaction energies. We show that the Hessian of the free energy governs the continuum Gaussian Free Field. The second result concern the Gaussian decay of correlations. All our results hold in the low-temperature regime and moderate boundary tilts. We outline how multi-scale/renormalisation group methods provide means of proving our statements. if time permits we discuss isomorphism theorems for the model.