# 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

**Brownian excursions, conformal loop ensembles and critical Liouville quantum gravity**

*8th November 2021*

"Conformal Loop Ensemble (CLE) decorated Liouville quantum gravity (LQG)" is a continuum model in random planar geometry that is expected to describe the scaling limit of various discrete statistical physics models.

I will discuss a new encoding of CLE decorated critical LQG by a Brownian half-plane excursion. In particular this allows us to connect observables of interest in the CLE/LQG model with a growth fragmentation recently studied by Aidekon and Da Silva.

This talk will be based on joint work with Juhan Aru, Nina Holden and Xin Sun.

# 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

**The continuous Anderson Hamiltonian in 1d**

*29th November 2021*

I will consider the random Schrodinger operator obtained by perturbing the Laplacian with a white noise, in 1d. In infinite volume, the operator satisfies an Anderson Localization phenomenon: the spectrum is pure point and the eigenfunctions are exponentially localized (while the unperturbed operator has absolutely continuous spectrum and totally delocalized eigenfunctions). In finite volume however, the situation is much more diverse: according to the region of the spectrum one looks at, localized or delocalized eigenfunctions can be observed. Based on joint works with Laure Dumaz (ENS).

# Sara Svaluto-Ferro

**Universality of affine and polynomial processes and application to signature processes**

*6th December 2021*

Already in the well studied finite dimensional framework, affine and polynomial processes are two fascinating classes of models. This is mostly due to the so-called affine transform formula and moment formula, respectively. In several recent works we could show that many models which are at first sight not recognized as affine or polynomial can nevertheless be embedded in this framework via infinite dimensional lifts. For instance, many examples of rough stochastic volatility models in mathematical finance can be viewed as infinite dimensional affine or polynomial processes. This suggests an inherent universality of these model classes. In this talk we perform a further step in that direction, showing that generic classes of diffusion models are projections of infinite dimensional affine processes (which in this setup coincide with polynomial processes).

The final part of the talk is dedicated to applications. We first show how to apply the introduced mechanism to one-dimensional diffusion processes with analytic coefficients, and which type of formulas can be obtained in that framework. Then, we consider the so-called signature process and explain the advantages to use the obtained formulas in the context of the corresponding signature based models.

The talk is based on ongoing joint works with Christa Cuchiero, Guido Gazzani, and Josef Teichmann.

# Jakob Björnberg

**The interchange model and related spin systems on two-block graphs**

*17th January 2022*

The interchange model is a model in quantum statistical physics with close connections to random permutations. In joint (almost completed) work with Hjalmar Rosengren and Kieran Ryan we recently analysed such models in a "two-block" setting. This includes complete bipartite graphs as a special case. Our method exploits algebraic aspects of the model and allows us to obtain explicit expressions for certain limiting quantities. In the "probabilistic regime" (which includes the Heisenberg ferromagnet) our results may be interpreted in terms of the appearance of macroscopic permutation cycles. In other regimes we obtain an intriguing phase diagram.

# Richard Pymar

**On the stationary distribution of the noisy voter model**

*24th January 2022*

Given a transition matrix P indexed by a finite set V of vertices, the voter model is a discrete-time Markov chain in {0,1}^{V} where at each time-step a randomly chosen vertex x imitates the opinion of vertex y with probability P(x,y). The noisy voter model is a variation of the voter model in which vertices may change their opinions by the action of an external noise. The strength of this noise is measured by an extra parameter p. The noisy voter model has a unique stationary distribution when p > 0. In this talk, I will discuss recent results on this stationary distribution when the number of vertices tends to infinity. This is based on joint work with Nicolás Rivera.

# Lukasz Szpruch

**Exploration-exploitation trade-off for continuous-time episodic reinforcement learning with linear-convex models**

*7th February 2022*

We develop a probabilistic framework for analysing model-based reinforcement learning in the episodic setting. We then apply it to study finite-time horizon stochastic control problems with linear dynamics but unknown coefficients and convex, but possibly irregular, objective function. Using probabilistic representations, we study regularity of the associated cost functions and establish precise estimates for the performance gap between applying optimal feedback control derived from estimated and true model parameters. We identify conditions under which this performance gap is quadratic, improving the linear performance gap in recent work [X. Guo, A. Hu, and Y. Zhang, arXiv preprint, arXiv:2104.09311, (2021)], which matches the results obtained for stochastic linear-quadratic problems. Next, we propose a phase-based learning algorithm for which we show how to optimise exploration-exploitation trade-off and achieve sublinear regrets in high probability and expectation. When assumptions needed for the quadratic performance gap hold, the algorithm achieves an order O(N (ln N)^{1/2}) high probability regret, in the general case, and an order O((ln N)^{2}) expected regret, in self-exploration case, over N episodes, matching the best possible results from the literature. The analysis requires novel concentration inequalities for correlated continuous-time observations, which we derive.

# Carina Betken

**Poisson cylinder processes: Variance asymptotics and central limit theory**

*14th February 2022*

We consider the union set of a stationary Poisson process of cylinders in R^{n}, where by a cylinder we understand any set of the form X x E, where E is an m-dimensional linear subspace of R^{n} and X is a compact subset in the orthogonal complement of E. The concept jointly generalises those of a Boolean model and a Poisson hyperplane or m-flat process.

Using techniques from Malliavin-Stein method we develop a quantitative central limit theory for a broad class of geometric functionals, including volume, surface area and intrinsic volumes. In this context we analyze the asymptotic variance constant, which in contrast to the Boolean model leads to a new degeneracy phenomenon.

# Barbara Dembin

**The time constant for Bernoulli percolation is Lipschitz continuous strictly above p_c**

*21st February 2022*

We consider the standard model of i.i.d. first passage percolation on Z^{d} given a distribution G on [0,+∞] (+∞ is allowed). When G([0,+∞))>p_c(d), it is known that the time constant μ_G exists. We are interested in the regularity properties of the map G --> μ_G. We study the specific case of distributions of the form G_p=pδ_1+(1-p)δ_∞ for p>p_c(d). In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter p. We prove that the function p --> μ_{G_p} is Lipschitz continuous on every interval [p_0,1], where p_0>p_c(d). This is a joint work with Raphaël Cerf.

# Serte Donderwinkel

**Random trees have height O(√n)**

*28th February 2022*

I will discuss a recent work with Louigi Addario-Berry, in which we resolve several conjectures on the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaymé trees. The proof is based on a new bijection between trees and sequences that we introduced in a joint work with Mickaël Maazoun and James Martin. In the second half of the talk, I will present a stochastic domination result on the height of uniformly random trees with a given degree sequence, which informally implies that binary trees are stochastically the tallest trees.

# Julio Backhoff

**On the martingale projection of a Brownian motion given initial and terminal marginals**

*14th March 2022*

In one of its dynamic formulations, the optimal transport problem asks to determine the stochastic process that interpolates between given initial and terminal marginals and is as close as possible to the constant particle. Typically, the answer is a stochastic process with constant-speed trajectories. We explore the analogue problem in the setting of martingales, and ask: what is the martingale that interpolates between given initial and terminal marginals and is as close as possible to the constant volatility particle? The answer is a process called 'stretched Brownian motion', a generalization of the well-known Bass martingale. After introducing this process and discussing some of its properties, I will present current work in progress (with Mathias Beiglböck, Walter Schachermayer and Bertram Tschiderer) concerning the structure of stretched Brownian motions.

# Matthias Winkel

**The Aldous diffusion**

*21st March 2022*

Motivated by an up-down Markov chain on cladograms, David Aldous conjectured in 1999 that there exists a "diffusion on continuum trees" whose mass partitions at any finite number of branchpoints evolve as certain Wright-Fisher diffusions with some negative mutation rates, until some branchpoint disappears. We construct this conjectured process via a consistent system of stationary evolutions of binary trees with k labelled leaves and edges decorated with diffusions on a space of interval partitions. This pathwise construction allows us to study and compute path properties of this "Aldous diffusion," including evolutions of projected masses and distances between branch points. We establish the (simple) Markov property and path-continuity, and we discuss the failure of the strong Markov property for the process on continuum trees. This is joint work, partly in progress, with Noah Forman, Soumik Pal and Douglas Rizzolo.

# Daniel Valesin

**The contact process on random d-regular graphs, static and dynamic**

*4th April 2022*

We consider the contact process on random d-regular graphs, briefly presenting earlier work on the case where the graph is static, and focusing on more recent work where the graph evolves simultaneously with (and independently of) the contact process. In both cases, the analysis involves fixing the infection rate of the contact process and the graph parameters, taking the number of vertices N to infinity and studying the asymptotic behavior of τ_N, the time it takes for the infection to disappear. Concerning the static graph, we prove that this behavior undergoes a phase transition: there is a value λ_c such that τ_N is of order log(N) if λ < λ_c, whereas τ_N grows exponentially with N if λ > λ_c. The latter situation is called the metastable regime. Turning to the dynamic graph setting, our choice of graph evolution is a Markovian edge-switching mechanism, whose rate is chosen so that the evolving local landscape seen by a fixed vertex approaches a limiting dynamic graph process. We again show the existence of a metastable regime for the contact process on these graphs, and notably, we show that this regime occurs for values of λ that would be subcritical in the static graph. Joint work with Jean-Christophe Mourrat (static graphs) and with Gabriel Baptista da Silva and Roberto Imbuzeiro Oliveira (dynamic graphs).

# Marcelo Hilário

**Percolation on randomly stretched lattices**

*7th April 2022 (Note: Thursday in 3E 2.2)*

We study the existence/absence of percolation on a class of planar graphs. These graphs are random dilute versions of the square lattice defined as follows: Starting with all the sites in Z^{2} we remove every horizontal edge connecting nearest neighbors. At this stage, the graph consists of infinitely many copies of the Z lattice parallel to one another, hence disconnected. We now insert only the vertical edges connecting nearest neighbors lying on {X_i} x Z, where X_1, X_2, X_3,... is an integer-valued renewal process. We perform Bernoulli percolation on the resulting graph and relate the question of whether it undergoes a non-trivial phase transition to the moments of the interarrivals of the renewal process. This is a joint work with Marcos Sá, Remy Sanchis and Augusto Teixeira.

# Benoît Dagallier

**Log-Sobolev inequality for the continuum phi42 and phi43 models**

*16th May 2022*

The continuum phi4 model is one of the simplest models of field theory, introduced at least 50 years ago in the physics community. It can be thought of as a continuous analogue of the Ising model. In particular, it exhibits a phase transition, separating a weakly correlated and a strongly correlated regime. The analysis of this phase transition is made particularly subtle due to the fact that the continuum limit of the continuum phi4 model is ill-defined, in the sense that a certain renormalisation procedure is needed to make sense of it.

In this talk, I will report on a joint work with Roland Bauerschmidt in which we analyse the Langevin dynamics associated with the continuum phi4 model. The presence of a phase transition should imply a dramatic difference in how much time it takes for the dynamics to approach its steady state, from fast convergence above the critical point, to slow convergence diverging with the system size below it. We characterise the relaxation to the steady state by establishing a logarithmic Sobolev inequality (LSI) with constant bounded under optimal assumptions. The proof makes use of a very general LSI criterion developed in 2019 by Roland Bauerschmidt and Thierry Bodineau.

In the talk, I will introduce the phi4 model and LSI inequalities, then try to explain the main ideas as non-technically as possible.