Finite-size scaling for the 4-dimensional
multicomponent hierarchical $|\phi|^4$ model
We apply and extend the renormalisation group method developed by
Bauerschmidt, Brydges and Slade to analyse near-critical finite-size
scaling for the 4-dimensional $n$-component hierarchical $|\phi|^4$
model, for all $n=1,2,3,\ldots$ In particular, we compute the universal
scaling profile of the susceptibility in a critical window around the
infinite-volume critical point. Our methods have the potential to
extend from the hierarchical to the Euclidean model. Implications for
the self-avoiding walk (the case $n=0$) are also discussed. This is
based on joint work in progress with Emmanuel Michta and Jiwoon Park.