Critical Phenomena and the Renormalisation Group (David Brydges and Gordon Slade)
The subject of phase transitions and critical phenomena in physics has had a major influence on mathematics for over half a century, especially in probability theory but also in other disciplines. In fact the influence on mathematics is now greater than ever before.
The subject is mainly focussed on the study of various specific models. This course will include an introduction to some of the models of greatest interest: the Ising and $|\varphi|^4$ spin models and self-avoiding walks. One of the fascinating features of these models is their dependence on the spatial dimension, and the course will provide a survey of what is rigorously known, and of what is predicted but not yet rigorously known, in the different dimensions. Emphasis will be placed on the existence of phase transitions and the accompanying universal critical behaviour, characterised by universal critical exponents.
Recent joint work with R. Bauerschmidt for self-avoiding walk and spin models in dimension d = 4 uses a rigorous implementation of Wilson’s renormalisation group method to prove the existence of logarithmic corrections to mean-field scaling. The renormalisation group method will be introduced and developed in the course. The application of the method to the self-avoiding walk makes use of a functional integral representation involving anti-commuting (fermionic) variables. This representation will be discussed in the course; such representations are becoming increasingly useful in probability theory.
The level of the course will be suitable for graduate students in probability without previous background in statistical mechanics. Necessary background will be provided, and prior specialised knowledge will not be assumed. Tutorials will used to further develop aspects presented in the lectures.
Please click here to download the preliminary information for lectures by David Brydges.
Tutorials for the Critical Phenomena and the Renormalisation Group course (Alexandre Tomberg)
The tutorials are meant to complement the Critical Phenomena and the Renormalisation Group course by providing further details into a number of selected topics. We will discuss the proof of the integral representation of weakly self-avoiding walks, Gaussian integrals and fields, and develop norms to implement Wilson's renormalisation group ideas in our setting. We will also derive the renormalisation group flow equations and illustrate how logarithmic corrections can be obtained from them.
Topics in random interfaces (Tadahisa Funaki)
Randomly fluctuating interfaces, which arise to separate two distinct phases, are studied in several different settings such as Ising model, effective interface models like $\nabla \varphi$-interface model or dynamic Young diagrams, sharp interface limit for stochastic Allen-Cahn equation (sometimes called time-dependent Ginzburg-Landau model), Kardar-Parisi-Zhang equation and others.
This course focuses on two different approaches, that is, scaling limits for$\nabla \varphi$-interface model (static theory) and sharp interface limits for stochastic Allen-Cahn equation (dynamic theory). If time permits, I will touch dynamic models of 2D Young diagrams or KPZ equation.
Assuming that the interface is represented as a height function measured from a fixed reference discretized hyperplane, the system is governed by the Hamiltonian of the height function. This is called the $\nabla \varphi$-interface model. I will discuss the scaling limits for Gaussian (or non-Gaussian) random fields with a pinning effect under the situation that the rate functional of the corresponding large deviation principle has non-unique minimizers (joint works with E. Bolthausen and others).
Sharp interface limit for Allen-Cahn equation, that is a reaction-diffusion equation with bistable reaction term, leads to a motion of mean curvature for the interface. Its stochastic perturbation will be discussed.
If the values of the height function in the $\nabla \varphi$-interface model are discrete and monotone, then it forms a Young diagram. One can define dynamics of Young diagrams in a natural way, and study its hydrodynamic behavior and fluctuations (joint works with M. Sasada and others).
KPZ equation describes a fluctuation of growing interfaces, and recently attracts a lot of attentions. This is an ill-posed stochastic partial differential equation and requires a renormalization. I will discuss its invariant measures (joint work with J. Quastel).
Please click here to download the lecture notes.