Probability Theory seminars

Optimal transport supports clustering, dictionary learning and domain adaptation, but high-dimensional data often force projections that discard information. Charles Bouveyron develops an alternative for Gaussian distributions with parsimonious covariance structure and low intrinsic dimension. A low-dimensional decomposition of empirical covariance matrices gives an efficient, robust closed form for the 2-Wasserstein distance without projecting the data. An explicit Monge map uses exact covariance square roots and inverse square roots. Numerical comparisons demonstrate analytical and computational benefits, including competitiveness with model-free approaches for non-Gaussian data and applications to unsupervised domain adaptation for classification.

Classical optimal transport assumes fully specified probability distributions. This talk relaxes that requirement by representing partially known marginals with belief functions. It examines joint belief functions with fixed marginals and minimizes a Choquet integral of transport cost. Preferences use an α-maxmin criterion, with α between zero and one measuring pessimism. When the cost is a ground metric, the construction gives a Choquet–Wasserstein pseudo-distance for comparing belief functions on a common space. The same framework describes games under ambiguity through a Stackelberg–Cournot–Nash equilibrium with Dempster–Shafer uncertainty and α-maxmin preferences. An application considers a continuum of investors whose type distribution is only partly known: investors choose portfolios of assets while a market maker sets prices.

The Korteweg–de Vries equation is a nonlinear third-order partial differential equation central to wave theory and commonly studied through inverse scattering. This seminar addresses the less extensively developed stochastic setting, focusing on the Cauchy problem for a generalized equation with noise in its nonlinear term. The noise is expressed through a symmetric integral with an arbitrary continuous integrator. This construction generalizes both Stieltjes integration and Stratonovich integration against a Wiener process. The proposed numerical methods extend Strang splitting and BCH schemes to stochastic Korteweg–de Vries equations. The presentation also examines existence, uniqueness and stability of solutions, and formulates a change-point problem for detecting shifts in the underlying regime from observations.

How do elementary biological events give rise to ecological community patterns, epidemic behaviour and collective population dynamics? This seminar develops a mechanistic view using Markov jump processes for birth, death, competition, dispersal, infection and recovery. These models connect individual events to biodiversity gradients, species abundance distributions and disease persistence. It examines the roles of demographic stochasticity, environmental variation, spatial heterogeneity and species interactions, highlighting mathematical similarities and differences across ecological and epidemiological systems. The inverse problem is also considered: inferring underlying mechanisms and interaction structures from macroscopic fluctuations and time-series data. Together, stochastic modelling and statistical inference provide a framework for studying ecological complexity and community stability.

Ramon van Handel surveys recent progress on constructing graphs, manifolds and groups with optimal spectral gaps, which control how rapidly random walks explore a structure. The talk describes a polynomial method that proves optimality in new settings and connects these developments to longstanding questions about the prevalence of optimal gaps.

Antonio Ocello tests the idea that heavy-tailed noise improves diffusion and flow-based generative models by matching heavy-tailed data and encouraging diverse samples. Replacing Gaussian noise also changes the estimation task. A combined theoretical and experimental study derives sampling-error bounds for representative heavy- and light-tailed diffusion models, finding that heavy-tailed noise makes statistical estimation harder and produces less favourable bounds. Experiments with synthetic and real data recover the predicted trade-off. The results question whether heavy-tailed initialization reliably improves exploration of rare regions. This is joint work with Hamza Cherkaoui and Hélène Halconruy. Online via Microsoft Teams. Friday 2 October 2026 at 15:30 CEST / 13:30 UTC / 14:30 BST (Europe/Paris). Follow the Microsoft Teams link beneath this occurrence on the seminar page; the public route offers browser or app access. Ocello is affiliated with ENSAE Paris. Organized by GAMEX — Generative AI Modeling for Extreme Events, with GLE²N and CIRCE support; the series welcomes scientific exchange beyond the network.

A Markov chain can sample effectively from a random or data-informed start even when worst-case initialization mixes slowly. This talk uses weak Poincaré inequalities to analyse that regime, with applications to simulated annealing for spherical spin glasses and to mixtures of log-concave distributions.

Aldous' spectral-gap theorem links the interchange process to a random walk. This talk presents a related result for the Kipnis–Marchioro–Presutti model on weighted hypergraphs, where the correct comparison involves a two-particle dual process. The result resolves a conjecture of Alon and Puder and draws on joint work with Pietro Caputo and Matteo Quattropani.

Continuum Glauber dynamics samples Gibbs point processes with repulsive interactions, including hard spheres. This talk develops continuous spectral-independence and localization tools to establish spectral gaps and efficient sampling for arbitrary-range repulsive potentials. The approach improves fixed-density packing thresholds and works directly in the continuum rather than discretizing the model.

Classical isoperimetric inequalities on the Boolean hypercube can extend beyond independent product measures. The talk proves KKL, Talagrand and Eldan–Gross inequalities under a Dobrushin-type weak-dependence condition, including a class of Ising models. The method combines semigroup arguments, discrete Bakry–Émery theory and gradient estimates.

Correlation decay for graph colourings is central to efficient sampling but remains difficult to prove near conjectured thresholds. This talk establishes strong spatial mixing on trees of maximum degree Delta for at least Delta plus three colours, when Delta is at least three. It then obtains optimal Glauber-dynamics mixing for fixed degree and sufficiently large girth under the same colour bound.

A refinement of the product replacement algorithm aims to draw nearly uniform elements from a finite group. This talk studies how many steps are needed for the output distribution to approach uniformity, using spectral-gap estimates and computer-assisted calculations in joint work with Michał Marcinkowski.

Salvatore Torquato gives a public lecture in the Simons Foundation's 2026 Randomness series, connecting the geometry of complex systems with random structures and applications across mathematics, computer science, physics and chemistry.

Sampling uniformly from a convex body links randomized volume computation to the Kannan–Lovász–Simonovits conjecture. This talk identifies geometric quantities controlling sampler mixing and introduces stochastic localization as a way to bound them, including a simple argument yielding a polylogarithmic, though nonoptimal, bound.

Quantum Gibbs samplers generalize classical thermalization dynamics such as Glauber and Metropolis processes. This talk shows that one-dimensional, short-range quantum Hamiltonians admit samplers with an optimal spectral gap independent of system size at every finite temperature.

Field dynamics provides a continuous analogue of the down-up walk on simplicial complexes. This talk introduces the process and explains applications to discrete sampling, including independent sets on graphs of unbounded degree up to the critical point.

Entropic independence replaces variance with entropy in an analogue of spectral independence. This talk introduces that property and shows how it can yield sharp bounds on the time required for Markov chains to approach equilibrium.

The Knabe method provides lower bounds on spectral gaps. This talk reviews the technique and applies it to random unitary circuits arranged in a one-dimensional brickwork pattern. For qubits, the resulting estimate is optimal up to a constant factor.

A unitary-group analogue of Aldous' spectral-gap problem randomizes coordinate subspaces associated with weighted hyperedges. Its spectrum contains those of related particle processes and symmetric-group walks. This talk reduces the search for the gap to zero-weight sectors of balanced mixed tensor powers, then proposes equality with a two-particle KMP gap. Proofs cover mean-field and codimension-one cases.

Localization schemes offer a framework for proving functional inequalities governing Markov chains. This talk introduces the Chen–Eldan approach and shows how local-to-global arguments from high-dimensional expansion fit within it. Examples illustrate the connection and its use in analysing stochastic processes.