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Probability seminar abstracts

Our seminars are usually held on Mondays at 1.15pm. In-person talks are usually in room 4W 1.7, and online talks will be held via Zoom.

Lukas Gonon

Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality
11th October 2021 - Online
In this talk we consider a supervised learning problem, in which the unknown target function is the solution to a Kolmogorov partial (integro-)differential equation associated to a Black-Scholes model or a more general exponential Lévy model. We analyze the learning performance of random feature neural networks in this context. Random feature neural networks are single-hidden-layer feedforward neural networks in which only the output weights are trainable. This makes training particularly simple, but (a priori) reduces expressivity. Interestingly, this is not the case for Black-Scholes type PDEs, as we show here. We derive bounds for the prediction error of random neural networks for learning sufficiently non-degenerate Black-Scholes type models. A full error analysis is provided and it is shown that the derived bounds do not suffer from the curse of dimensionality. We also investigate an application of these results to basket options and validate the bounds numerically.

Benedikt Jahnel

Phase transitions and large deviations for the Boolean model of continuum percolation for Cox point processes
18th October 2021 - Online
In this talk, I consider the Boolean model with random radii based on Cox point processes, i.e., Poisson point processes in random environment. Under a condition of stabilization for the random environment, our results establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points, also including their moments. The second part of the talk will address rates of convergence of the percolation probability for the Cox--Boolean model with fixed radii in a variety of asymptotic regimes. This is based on joint work with Christian Hirsch, András Tóbiás and Élie Cali.

Pierre-François Rodriguez

Critical exponents for a three-dimensional percolation model
1st November 2021
We will report on recent progress regarding the near-critical behavior of certain statistical mechanics models in dimension 3. Our results deal with the second-order phase transition associated to two percolation problems involving the Gaussian free field in 3D. In one case, they determine a unique "fixed point" corresponding to the transition, which is proved to obey one of several scaling relations. Such laws are classically conjectured to hold by physicists on the grounds of a corresponding scaling ansatz.

Ellen Powell

8th November 2021
Abstract to appear.

Sunil Chhita

GOE Fluctuations for the maximum of the top path in ASMs
15th November 2021
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling. This talk is based on joint work with Arvind Ayyer and Kurt Johansson.

Cyril Labbé

29th November 2021
Abstract to appear.

Sara Svaluto-Ferro

6th December 2021
Abstract to appear.

Jakob Björnberg

17th January 2021
Abstract to appear.

Richard Pymar

24th January 2021
Abstract to appear.