Timetable (SRQW01)

Lecture notes for the course are at: https://courses.helsinki.fi/sites/default/files/course-material/4594153/MathPhys2018final.pdf We discuss parts of sections 4,6,7,8,9 which are self contained

Co-authors: Ajay Chandra (Imperial College London), Hao Shen (Columbia University)

This minicourse is coordinated with the one of Lorenzo Zambotti. The joint goal is to treat the dynamical phi-4-3 model. This minicourse will provide the basic notions in regularity structures and discuss reconstruction and integration. It will mostly follow Martin Hairer's 2015 notes "Regularity structures and the dynamical phi-4-3 model''.

This minicourse is coordinated with the one of Lorenzo Zambotti. The joint goal is to treat the dynamical phi-4-3 model. This minicourse will provide the basic notions in regularity structures and discuss reconstruction and integration. It will mostly follow Martin Hairer's 2015 notes "Regularity structures and the dynamical phi-4-3 model''.

The understanding of critical phenomena via the renormalisation group approach is one of the great achievements of theoretical physics during the last half of the previous century. This series of lectures will begin with an introduction to the critical behaviour of the phi^4 lattice spin system and its supersymmetric counterpart (the weakly self-avoiding walk). The lectures will mainly be occupied with an introduction to a rigorous renormalisation group method that has been developed and used to compute critical exponents for these models in the upper critical dimension d=4, and also below the critical dimension for long-range models. The lectures are based on collaborations with David Brydges and Roland Bauerschmidt.

This should be a mini-course on the algebraic side of the theory of regularity structures.

The Flow equations in Polchinski's setting allow for a simple and transparent proof of perturbative renormalizability, cirumventing auxiliary combinatoric structures. They also permit to prove properties of perturbative Schwinger functions which are out of reach in other settings. The main problem for a nonperturbative analysis of those equations lies in their combinatoric instability, which is present already at the mean field level.

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems at and near a critical point. In our approach, we derive a spectral gap inequality, or more generally a Brascamp--Lieb inequality, for the measure recursively in terms of spectral gap or Brascamp--Lieb inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the $4$-dimensional $n$-component $|\varphi|^4$ model at the critical point and its approach from the high temperature side, and the $2$-dimensional Sine--Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz--Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the $|\varphi|^4$ model), the scaling limit of these models in equilibrium. Co-author: Thierry Bodineau