The course will start with a rapid introduction to the Log-Sobolev inequality and some of its consequences. I will then introduce our recent approach to proving Log-Sobolev inequality for spin systems based on Polchinski's continuous renormalisation group. As an application, I will present the Log-Sobolev inequality for the continuum Sine-Gordon model. The lectures are based on joint work with Thierry Bodineau.

Log‐Sobolev Inequality for the Continuum Sine‐Gordon Model

In this course, we will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be proven for the λϕ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

Marginal triviality of the scaling limits of critical 4D Ising and ϕ^4_4 models

The Two-Dimensional Ising Model

Probability and Statistical Physics in Two and More Dimensions

Conformal invariance of lattice models

Lace expansion for dummies

A simple convergence proof for the lace expansion

The continuous-time lace expansion

Quantum gauge theories are the most successful models for the description of nature at its fundamental level. Unfortunately, despite decades of intense effort, their mathematical foundations remain murky. In these lectures, we will start by introducing the basic concepts behind them and see what this all has to do with probability theory. We will then proceed to formulate a number of precise mathematical questions and conjectures. The remainder of the lectures will be devoted to a presentation of joint work with A. Chandra, I. Chevyrev and H. Shen that makes some small steps towards a quantum Yang-Mills theory, at least in space-time dimension 3.

Langevin dynamic for the 2D Yang-Mills measure - Ajay Chandra, Ilya Chevyrev, Martin Hairer, Hao Shen

Yang-Mills measure on compact surfaces - Thierry Levy

Stochastic quantization of an Abelian gauge theory - Hao Shen

Yang-Mills measure on the two-dimensional torus as a random distribution - Ilya Chevyrev

Geometric stochastic heat equations - Yvain Bruned, Franck Gabriel, Martin Hairer, Lorenzo Zambotti

Singular SPDEs are stochastic equations that are ill posed because of the interplay of very irregular noise and nonlinearities. Their mathematical theory was developed in the past years based on Hairer’s regularity structures and related theories. These approaches are all based on a pathwise calculus, we freeze the realisation of the noise and then perform deterministic analysis. This is orthogonal to the Markovian viewpoint based on semigroup, infinitesimal generator and martingale problem. The Markovian viewpoint is an important tool for studying the convergence to Markov processes and for understanding their long time behaviour, so it would be interesting to study singular SPDEs from this perspective. In general that is a too difficult problem at the moment, but for some particularly nice equations we can indeed construct the infinitesimal generator and this is indeed useful. I will explain this on the example of the conservative stochastic Burgers equation and indicate how to extend the results to some other equations with similar structure.

The infinitesimal generator of the stochastic Burgers equation

Energy solutions of KPZ are unique

Video Lectures on "Stochastic Partial Differential Equations: Classical and New".