# Sarah-Jean Meyer

**The Forward Backward Approach to sine-Gordon up to 6π**

*30th September 2024*

I will present a novel approach for a rigorous construction of the sine-Gordon quantum field theory on the full-space in the first few regions. The approach relies on a scale decomposition of the QFT and the observation that the dynamics of moving between scales is described by a forward-backward stochastic differential equation (FBSDE). Using a simplified version of the problem, I will explain the general set-up and issues that arise in the analysis of the FBSDE. Once the FBSDE can be controlled, it serves as a powerful tool to show several properties of the resulting QFT. I will give an overview over results such as the verification of the Osterwalder-Schrader axiom, decay of correlations, as well as a variational problem and a resulting large deviations principle for the semi-classical limit.

# Tommaso Rosati

**Lower bounds to energy dissipation in passive scalar advection**

*7th October 2024*

We consider a passive scalar advected by a stochastic velocity field. Under some non-degeneracy assumptions on the noise, we prove a lower bound on the energy dissipation that is quantitative in the diffusivity of the scalar. This partially addresses a conjecture by Miles and Doering and proves a first lower bound to the so-called Batchelor scale. The proof is based on dynamics of energy level sets, and a refined short- time and high-frequency expansion. Joint work with M. Hairer, S. Punshon-Smith and J. Yi.

# Sophie Laruelle

**Urn Models and Stochastic Approximation : From Pólya Urns to Nonlinear Randomized Urns**

*14th October 2024*

This talk will begin with the presentation of the classic model of Pólya urns and the results obtained to introduce the notations used subsequently. We will then look at variants where we modify the deterministic replacement matrix (without replacement, adding a different color or several colors).

We will then study the case of stochastic replacement matrix with uniform drawing and balanced urn. This has been motivated by applications in clinical trials and finance, and can be solved more efficiently using stochastic algorithms on the dynamics of the proportions of balls in the urn.

We will end this talk with a final generalisation: the drawing rule will no longer be uniform, but will be modified by a convex or concave function. With 2 colors, this will allow us to exhibit a phase transition (in the convex case) passing from one equilibrium to several (the number of which depends on the function). For the number of balls greater than 2, work is in progress, but we have some new results for 3 colors and some ideas we will discuss for going further.

# James Foster and Dáire O'Kane

**On the convergence of adaptive approximations for SDEs**

and

**Simulating underdamped Langevin dynamics: A third order convergent method**

*21st October 2024* **at 14:15**

**On the convergence of adaptive approximations for SDEs**

When using ordinary differential equations (ODEs), numerical solutions are often approximated and propagated in time via discrete step sizes. For a large variety of ODE problems, performance can be improved by making these step sizes “adaptive” – that is, adaptively changed based on the state of system. However, for stochastic differential equations (SDEs), adaptive numerical methods can be difficult to study and even fail to converge due to the rough nature of Brownian motion.

In this talk, we will show that convergence does indeed occur, provided the underlying Brownian motion is discretized in an adaptive but “martingale-like” fashion. Whilst this prevents adaptive steps from skipping over time points (which we show can prevent convergence), we believe our convergence theory is the first that is applicable to standard SDE solvers. We will discuss the key ingredients in this analysis – including martingale convergence, rough path theory and the approximation of Brownian motion by polynomials.

Based on our theory, we also modify an adaptive “Proportional-Integral” (PI) step size controller for use in the SDE setting. Unlike those used for ODEs, this new PI controller is designed to revisit time points where the Brownian motion was previously sampled. Finally, we conclude with a numerical experiment showing that SDE solvers can achieve an order of magnitude more accuracy with adaptive step sizes than with constant step sizes.

Joint work with Andraž Jelinčič.

**Simulating underdamped Langevin dynamics: A third order convergent method**

Underdamped Langevin dynamics (ULD) is a stochastic differential equation of great interest to those in the molecular dynamics and machine learning communities. Interest from the latter is due to ergodic properties of ULD, meaning that under assumptions on the potential function, the process can be used as proxy for generating samples from an unnormalized (log-concave) target density. Due to the presence of a non-linear term, one cannot simulate ULD exactly and must apply a numerical discretisation.
In this talk, we introduce the numerical method “QUICSORT”, which is a third order convergent method for simulating ULD. Our method is based off discretising the “Shifted ODE” method of Foster, Lyons and Oberhauser, where Brownian motion is replaced by a piecewise linear path matching higher order stochastic integrals. During the talk we will motivate the construction of QUICSORT, discuss the key ideas in the proof of third order convergence (in the 2-Wasserstein metric) and highlight the key role played by numerical contractivity. To the best of our knowledge, this is the first method to achieve third order convergence over the infinite-time horizon for strongly convex MCMC problems. We conclude with a numerical experiment (courtesy of Andraž Jelinčič) illustrating the performance of our method against the current state of the art sampler for high dimensional problems.

Joint work with James Foster.

# Rivka Mitchell

**Discrete Snakes with Globally Centered Displacements**

*28th October 2024*

We consider certain size-conditioned critical Bienaymé trees, in which each vertex is endowed with a spatial location that is a random displacement away from their parent’s location. By construction, the positions along each vertex’s lineage form a random walk. It is convenient to encode the genealogy and spatial locations using a path-valued process called the discrete snake. We prove that under a global finite variance and a tail behaviour assumption on the displacements, any globally centered discrete snake on a Bienaymé tree whose offspring distribution is critical and admits a finite third moment has the Brownian snake driven by a normalised Brownian excursion as its scaling limit. Our proof relies on two perspectives of Bienaymé trees. To prove convergence of finite dimensional distributions we rely on a line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin. To prove tightness, we adapt a method used by Haas and Miermont in the context of height functions of Markov branching trees.

This is based on joint work in progress with Louigi Addario-Berry, Serte Donderwinkel, and Christina Goldschmidt.

# Gady Kozma

**A reduction of the theta(p_c) problem to a conjectured inequality**

*4th November 2024*

A famous open question asks if critical percolation has an infinite cluster or not. This question has been resolved in dimension 2 and in sufficiently high dimensions, but is open in intermediate dimensions. We will discuss a reduction of this problem to a conjectured FKG-like inequality. Joint work with Shahaf Nitzan.

# Nikos Zygouras

**TBA**

*11th November 2024*

Abstract to appear.

# Giuseppe Cannizzaro

**TBA**

*11th November 2024*

Abstract to appear.

# Alexandre Stauffer

**TBA**

*25th November 2024*

Abstract to appear.

# Matthew Jenssen

**TBA**

*2nd December 2024*

Abstract to appear.

# Shuo Qin

**TBA**

*4th December 2024 (Note: Wednesday!)*

Abstract to appear.

# Thomas Powell

**TBA**

*9th December 2024*

Abstract to appear.