Abstract
The first lecture began with an introduction to the renormalization group. This is a ubiquitous approach in statistical physics, imported from field theory
where it is used to overcome short-range divergences. Although the ideas behind renormalization are fairly straightforward, their implementation is often technically quite complicated. The aim of this series of lectures has been to explain how this approach leads to the computation of critical behaviors characteristic of second-order phase transitions, and how the notion of universality follows quite naturally from it. The main idea of the renormalization group is to try to relate the large-scale properties of a system of size L in dimension d, i.e. of volume Ld, for certain parameter choices, to those of a smaller system, of size L/b and therefore of volume (L/b)d, with modified parameter values. More precisely, we seek to find for which choices T′, h′, ρ′ of parameters such as temperature T, magnetic field h, density ρ, the large-scale properties of a system of size L/b are similar to those of a system of size L for a choice T, h, ρ of these parameters. We can see that if we manage to find the function Rb that relates the renormalized parameters T′, h′, ρ′... to the starting parameters T, h, ρ..
(T′, h′, ρ′...) = Rb(T, h, ρ...)
we can iterate the renormalization transformation and thus link the system of size L to systems of size L/b, L/b2... L/bn. The difficulty lies in finding the renormalization transformation Rb. This first lecture has shown how to determine this function Rb in an approximate way for three examples: percolation, the liquid-gas transition and the transition to chaos by period doubling. Once the renormalization transformation Rb is known, the attractive fixed points correspond to the possible phases, while the hyperbolic fixed points represent the transitions. In particular, critical surfaces are given by stable varieties of hyperbolic fixed points, which helps explain the universality of critical behaviors: different starting points on this stable variety converge to the same fixed point and thus have the same large-scale behaviors. As for the critical exponents, they can be calculated from the unstable eigenvalues of these hyperbolic fixed points.
