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

The Critical 2d Stochastic Heat Flow and Critical SPDEs
11th November 2024
Thanks to the theories of Regularity Structures and Paracontrolled Distributions we now have a complete theory of singular SPDEs, which are “sub-critical” in the sense of renormalisation. Recently, there have been efforts to approach the situation of “critical” SPDEs and statistical mechanics models. A first such treatment has been through the study of the two-dimensional stochastic heat equation, which has revealed a certain phase transition and has led to the construction of the novel object called the Critical 2d Stochastic Heat Flow. In this talk we will present some aspects of this model and its construction and some first steps towards an approach to critical SPDEs.

Most of the talk will be based on joint works with Caravenna and Sun.

Giuseppe Cannizzaro

Superdiffusive Central Limit Theorem for the critical Stochastic Burgers Equation
11th November 2024
The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a way to encode the fluctuations of driven diffusive systems with one conserved quantity (e.g. ASEP). In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to a Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. In the present talk, we pin down the logarithmic superdiffusivity exactly by identifying the limit of the so-called diffusion coefficient and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem.

Joint work with Q. Moulard and F. Toninelli.

Matthew Jenssen

Sphere packing in high dimensions
2nd December 2024
The classical sphere packing problem asks: what is the densest possible arrangement of identical, non-overlapping spheres in R^d? Over the past century, this question has been intensely studied by mathematicians and physicists alike. In this talk I will discuss some different perspectives on this problem along with some recent progress. In particular, I will sketch a proof that there exists a sphere packing in R^d with density at least

(1+o(1)) d log d / 2^{d+1}

This improves upon previous bounds by a factor of order log d and is the first improvement by more than a constant factor to Rogers’ bound from 1947.

This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.

Shuo Qin

Recurrence and transience of multidimensional elephant random walks
4th December 2024 (Note: Wednesday!)
We prove a conjecture by Bertoin that the multidimensional elephant random walk on Z^d is transient in dimensions d ≥ 3. We show that it undergoes a phase transition in dimensions d = 1, 2 between recurrence and transience at p = (2d + 1)/(4d). We also generalize these results to the step-reinforced random walk under mild moments conditions.

Thomas Powell

Quantitative results for stochastic processes
9th December 2024
It is an elementary fact from real analysis that any monotone bounded sequence of real numbers converges. It turns out that the monotone convergence theorem can be given an equivalent finitary formulation: roughly that any sufficiently long monotone bounded sequence experiences long regions where the sequence is metastable. This so-called "finite convergence principle" is carefully motivated and discussed by Terence Tao in a 2007 blog post ('Soft analysis, hard analysis, and the finite convergence principle'), but was already known to proof theorists, where the use of logical methods to both finitize infinitary statements and provide uniform quantitative information for the finitary versions plays a central role in the so-called proof mining program.

The goal of my talk is to discuss how methods from proof theory can be applied to the theory of stochastic processes, an area of mathematics hitherto unexplored from this perspective. I will begin by discussing the simple monotone convergence principle, and will then focus on how everything I mentioned above can be generalised to the analogous result in the stochastic setting: Doob's martingale convergence theorems. This then sets the groundwork for new applications of proof theory in stochastic optimization, where almost-supermartingales play a crucial role, and I will give a high level overview of some work in progress in this direction.

Avi Mayorcas

Large deviations for the Φ43 measure via Stochastic Quantisation 
6th January 2025
The Φ43 measure is one of the easiest non-trivial examples of a Euclidean quantum field theory (EQFT) whose rigorous construction in the 1970’s has been one of the celebrated achievements of the constructive QFT community. In recent years, progress in the field of singular stochastic PDEs, initially by the theory of regularity structures, has allowed to construct the Φ43 EQFT as the invariant measure of a previously ill-posed Langevin dynamics—a strategy originally proposed by Parisi and Wu (’81) under the name stochastic quantisation. In this talk, I will demonstrate that the same idea also allows for the transference of large deviation principles for the Φ43 dynamics, obtained by Hairer and Weber (’15), to the corresponding EQFT. Our strategy is inspired by earlier work of Sowers (’92) and Cerrai and Röckner (’05) for non-singular dynamics and potentially also applies to other EQFT measures. The talk is based on joint work with Tom Klose (University of Oxford).

Sefika Kuzgun

Time-dependent averages of a critical long-range stochastic heat equation
13th January 2025
In this talk we will introduce a critical long-range linear stochastic heat equation. We will discuss the solution theory which dates back to 2004 work of Mueller and Tribe. We will then talk about the time-dependent spatial averages and the limiting behavior on different scales. The latter part of this talk is based on a joint work with Ran Tao.

Ajay Chandra

Non-commutative singular SPDE
3rd February 2025
In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of non-commutative probability theory - examples will include the stochastic quantization of Fermionic quantum field theories and also the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.

Daniel Heydecker

The Porous Medium Equation: large deviations and gradient flow with degenerate and unbounded diffusion
10th February 2025
We consider a rescaling of the zero-range process, which converges in the limit to the porous medium equation ∂_t u = Δu^α , and study the dynamical large deviations. The degeneracy of the diffusion close to u=0, as well as the unboundedness as u → ∞, mean that the usual proof of the key superexponential estimate cannot be applied; instead, we give a new argument based on pathwise regularity. As a consequence, we exhibit a formulation of the porous medium equation as the gradient flow of the entropy in a degenerate Wasserstein geometry through the entropy-dissipation equality.

Léonie Papon

Interface scaling limit for the critical planar Ising model perturbed by a magnetic field
17th February 2025
In this talk, I will consider the interface separating +1 and -1 spins in the critical planar Ising model with Dobrushin boundary conditions perturbed by an external magnetic field. I will prove that this interface has a scaling limit. This result holds when the Ising model is defined on a bounded and simply connected subgraph of δZ^2, with δ > 0. I will show that if the scaling of the external field is of order δ^15/8 then, as δ→0, the interface converges in law to a random curve whose law is conformally covariant and absolutely continuous with respect to SLE 3. This limiting law is a massive version of SLE 3 in the sense of Makarov and Smirnov and I will give an explicit expression for its Radon-Nikodym derivative with respect to SLE 3. I will also prove that if the scaling of the external field is of order δ^15/8 g(δ) with g(δ) → 0, then the interface converges in law to SLE_3. In contrast, I will show that if the scaling of the external field is of order δ^15/8 f(δ) with f(δ) → ∞, then the interface degenerates to a boundary arc.

Yoan Tardy

Collisions of the supercritical Keller-Segel particle system
24th February 2025
We study a particle system naturally associated to the 2-dimensional Keller-Segel equation. It consists of N Brownian particles in the plane, interacting through a binary attraction in θ/(Nr), where r stands for the distance between two particles. When the intensity θ of this attraction is greater than 2, this particle system explodes in finite time. We assume that N>3θ and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of N and θ, here is one: at explosion, a cluster consisting of precisely k0 particles emerges, for some deterministic k0≥7 depending on N and θ. Just before explosion, there are infinitely many (k0−1)-ary collisions. There are also infinitely many (k0−2)-ary collisions before each (k0−1)-ary collision. And there are infinitely many binary collisions before each (k0−2)-ary collision. Finally, collisions of subsets of 3,…,k0−3 particles never occur. The other scenario is similar except that there are no (k0−2)-ary collisions.

Vianney Brouard

How power-law mutation rates shape supercritical branching populations
3rd March 2025
Motivated by a better understanding of cancer evolution, we consider an individual-based model representing a cell population where cells divide, die, and mutate along the edges of a finite directed graph, representing the genetic trait space. The process starts with only one wild-type cell. Following typical parameter values in cancer cell populations, we study the model under power-law mutation rates, where mutation probabilities are parametrised by negative powers of a scaling parameter n, and the typical sizes of the population of interest are positive powers of n.

A wide variety of models with power-law mutation rates have been studied in the adaptive dynamics and population genetics literatures. A common result in this context is the convergence of the logarithmic frequencies of the size of the different traits when time is scaled by a log n factor. Under a non-increasing growth rate condition, corresponding to neutral and/or deleterious evolution, we succeed in describing not only the logarithmic frequencies but also the actual first-order asymptotics of the size of each subpopulation on the log n time scale, as well as in the random time scale at which the wild-type subpopulation, respectively the total population, reaches the size n^t. These results allows for an exact characterisation of evolutionary pathways.

Selective mutations present a far greater mathematical challenge, as the approach developped for dealing with neutral/deleterious mutations typically fails in this case. Together with Hélène Leman, we developed a novel methodology to determine the first-order asymptotics for the first selective mutant trait.

Alexandre Stauffer

Non-monotone phase transition in interacting particle systems
10th March 2025
In this talk we will discuss a reaction-diffusion particle system which has a non-monotone phase transition. I will explain the techniques used to analyze monotone models and how they can be refined to analyze non-monotone particle systems.

Based on a joint work with Leandro Chiarini.

Marco Seiler

The Offended Voter Model
17th March 2025
In this talk we discuss a variant of the voter model on a co-evolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability q. Otherwise, with probability 1-q, both individuals become offended in the sense that they never interact again, i.e. the corresponding edge is removed from the underlying network. Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of "segregation", otherwise of "consensus".

We show that segregation and a weaker form of consensus both occur with positive probability for every q ϵ (0,1) and that the segregation probability tends to 1 as q → 0. Furthermore, we establish that, if q → 1 fast enough, with high probability the population reaches consensus while the underlying network is still densely connected. Furthermore, we briefly discuss results from simulations to assess the obtained results.

This talk is based on joint work with Raphael Eichhorn and Felix Hermann.

Vittoria Silvestri

Branching Internal DLA
24th March 2025
Internal DLA is a random aggregation process in which the growth of discrete clusters is governed by the harmonic measure seen from an internal point. That is, a simple random walk is released from inside the cluster, and its exit location is added to it. The asymptotic shape of IDLA on Z^d starting from a single seed has long been known to be a Euclidean ball, with very small fluctuations. In this talk I will discuss a natural variant of IDLA, namely Branching IDLA, in which the particles that drive the process perform critical branching random walks rather than simple random walks. We will show that BIDLA has a strikingly different phenomenology, namely we prove a phase transition from macroscopic fluctuations in low dimension to the existence of a shape theorem in higher dimension. Based on a joint work with Amine Asselah (Paris-Est Créteil) and Lorenzo Taggi (Rome La Sapienza).

Sebastien Martineau

Arithmetic percolation (or visibility in a crystal)
31st March 2025
Plant a thin tree at each vertex of the square lattice. Then, pick a tree uniformly at random in a huge box, and replace it with a lamp. A tree is lit if the segment joining it to the lamp does not contain any other tree; otherwise, it is shady. The structure of this lit/shady colouring involves both probability and arithmetic. Regarding probability, this is no surprise. As for arithmetic, it is rather expected to play a role as well because the definition of "lit" can naturally be phrased by using the word "multiple": a tree is lit if and only if the vector going from the lamp to this tree cannot be expressed as a multiple of a shorter vector with integer coordinates.

In this talk, we will be interested in the percolative properties of this random colouring. If we only keep the lit trees, how many infinite connected components are there? if we keep the shady trees instead? what happens if we work in other dimensions than 2? And before all that: can we really make sense of a limit distribution when the size of the "huge box" tends to infinity?

This is joint work with Samuel Le Fourn and Mike Liu.

Hugo Vanneuville

Noise sensitivity of percolation events for high temperature Ising model
22nd April 2025
Unusual room: 6W 1.2 Consider critical Bernoulli percolation on the triangular lattice, i.e. color each site independently black or white with probability 1/2. This model is noise sensitive: if you resample a small proportion of the colors then you obtain an almost independent percolation picture. This was proven by Benjamini, Kalai and Schramm in 1999. In this talk, we would like to explain why this result still holds if the colors are sampled according to the Ising measure at high temperature. More generally, we would like to present a 'robust' approach to prove noise sensitivity that, contrary to the previous works, is not based on spectral tools and is rather inspired by the study of pivotal points by Kesten.

To study the question of noise sensitivity, we will ask more general questions about the pair (Spin configuration at time 0, Spin configuration at time t), that we will see as a spin configuration in itself. Do you expect that this pair still satisfies important properties of the static Ising model such as spatial Markov, finite-energy or FKG?

This is joint work with Vincent Tassion.

Franco Severo

Cutsets, percolation and random walk
28th April 2025
Which graphs G admit a percolating phase (i.e. p_c(G)<1)? This seemingly simple question is one of the most fundamental ones in percolation theory. A famous argument due to Peierls implies that if the number of minimal cutsets of size n from a vertex to infinity in the graph grows at most exponentially in n, then p_c(G)<1. Our first theorem establishes the converse of this statement. This implies, for instance, that if a (uniformly) percolating phase exists, then a "strongly percolating" one also does. In a second theorem, we show that if the simple random walk on the graph is uniformly transient, then the number of minimal cutsets is bounded exponentially (and in particular p_c<1). Both proofs rely on a probabilistic method that uses a random set to generate a random minimal cutset whose probability of taking any given value is lower bounded exponentially on its size.

Based on a joint work with Philip Easo and Vincent Tassion.

Tom Klose

TBA
19th May 2025
Abstract to appear.