Numerical evidence suggests that the Random Field Ising Model loses
Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4
and 5 dimensions, while a related model of branched polymers retains these
features in any d. These notes give a leisurely introduction to a recent
theory, developed jointly with A. Kaviraj and E. Trevisani, which aims to
explain these facts. Based on the lectures given in Cortona and at the IHES in
2022.Comment: 55 pages, 11 figures; v2 - minor changes, mentioned forthcoming work
by Fytas et a