# Renormalisation group analysis of the $n$-component $|\varphi|^4$ spin model and the weakly self-avoiding walk in dimension four

We give a series of three lectures to present elements of

a rigorous renormalisation group method and its application, from recent joint

work with Roland Bauerschmidt and David Brydges. The applications are to

two interesting models of critical phenomena in dimension four:

the weakly self-avoiding walk and the $n$-component $|\varphi|^4$

spin model for all $n \ge 1$. The results include statements concerning

scaling limits and critical exponents. For example, the susceptibility

has a logarithmic correction to mean field scaling, with exponent

$\frac{n+2}{n+8}$ for the logarithm in the $|\varphi|^4$ model, and

with exponent $\frac{1}{4}$ for the weakly self-avoiding walk.

The lectures will present the general setting for the work, present

the principal results, and discuss aspects of their proof.