Abstracts
Summer school in Probability, King's College London, June - July 2026.
Summer school in Probability, King's College London, June - July 2026.
After introducing the main tools of Malliavin Calculus/Analysis on Wiener spaces, we describe how to derive smoothness of laws of random variables and give classic applications to existence and smoothness of laws of solutions of stochastic differential equations. Finally we describe quasi-sure analysis on Wiener spaces and some recent application to dispersive partial differential equations.
Gibbs distributions provide a broad framework for models arising in probability, statistical mechanics, combinatorics, and computer science. Classical examples include the Ising and Potts models, graph colourings, independent sets, and constraint satisfaction problems (e.g., k-SAT). Understanding Gibbs distributions is closely connected to some of the most fundamental questions in these areas, including phase transitions, efficient sampling algorithms, and the tractability of learning problems.
Recent breakthroughs have led to substantial progress in our understanding of Gibbs distributions and the behaviour of classical Markov chain methods. In this lecture series, we will review some of these developments in sampling and learning, with an emphasis on the key ideas behind spectral independence and related techniques. Along the way, we will highlight connections with phase transitions, discuss applications in worst-case and average-case settings, and conclude with some open directions.
We present an explicit numerical approximation scheme for the effective simulation of solutions to a multivariate stochastic differential equation (SDE) with a superlinearly growing dissipative drift, driven by a multiplicative heavy-tailed Lévy process. The scheme combines the well-known Euler method with a Lie-Trotter-type splitting technique. The specific ordering of the splitting terms enables the approximation to capture all finite moments of the true solution. In the special case of SDEs driven solely by Brownian motion, our numerical scheme preserves the solution's superexponential moments. We prove strong convergence of approximations and determine the order of convergence. This talk is based on the following works:
O. Aryasova, O. Kulyk, and I. Pavlyukevich, arXiv:2504.07255
I. Pavlyukevich, O. Aryasova, A. Chechkin, and O. Kulyk, Chaos 35(12), 123139, 2025. arXiv:2508.07339
In these lectures, I will focus on models of statistical mechanics defined on planar lattices whose variables, attached to the vertices of the lattice, take values either in the integers (the so-called height functions) or on the unit circle (the so-called abelian continuous spin models). It is by now classical that both types of models undergo a special phase transition that does not occur in either one dimension or higher dimensions.
On the height-function side, this is the transition between the localized regime, in which the value of the variable at the origin remains bounded as the size of the system grows, and the delocalized regime, in which this value fluctuates on increasingly large scales as the system grows. This should be compared with the behavior of the random walk bridge, whose value at the midpoint of a long interval is never tight (that is, always delocalized), as well as with the absence of delocalization for height functions defined on higher-dimensional lattices.
On the spin-model side, the Berezinskii–Kosterlitz–Thouless transition manifests itself as a change in the decay rate of the two-point function: from exponential decay in the high-temperature phase to power-law decay in the low-temperature phase. The existence of this transition was first established rigorously by Fröhlich and Spencer in 1982.
Moreover, the two types of models and their respective phase transitions are known to be related through a (Fourier-type) duality transformation.
These phenomena have recently attracted renewed attention in the mathematical literature. I will discuss some of the new ideas and approaches that have emerged in this context.
Liouville quantum gravity (LQG) is a canonical model of a random two-dimensional geometry, indexed by a parameter γ, which is intimately related to Liouville conformal field theory. A central conjecture in modern probability theory posits that LQG surfaces arise as the scaling limits of discrete surfaces known as random planar maps, with different universality classes of maps corresponding to different values of γ. In this course we will survey this circle of ideas, including an overview of the probabilistic theory of LQG, and its deep connections with a model of random loops called conformal loop ensembles (CLE). We will then turn to a powerful tool for making connections with random planar maps rigorous: the so-called "mating of trees theory” which is a surprising result that allows one to encode LQG surfaces using pairs of correlated planar Brownian motions.
The specific focus of the course will be the "critical case" γ=2, corresponding to a parameter κ=4 for CLE, which is special in several respects: it marks the boundary between the dilute and dense phases of the CLE, and is also the transition point where the LQG metric becomes singular with respect to Euclidean distance. Since several key tools break down in this regime, it requires substantially different techniques than for other values of γ. We will describe how the mating-of-trees theory adapts to this setting, and survey recent progress on a particular random planar map model in the corresponding universality class. Specifically, we will describe a recent article that identifies γ=2 LQG, decorated with CLE(4), as the scaling limit of FK-decorated random planar maps at the “critical" parameter q=4.
Based on joint works with Juhan Aru, Nina Holden & Xin Sun, and William Da Silva, Xingjian Hu & Mo Dick Wong.
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