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In this manuscript, in honour of L. Kadanoff, we present recent progressobtained in the description of finite dimensional glassy systems thanks to theMigdal-Kadanoff renormalisation group (MK-RG). We provide a critical assessmentof the method, in particular discuss its limitation in describing situations inwhich an infinite number of pure states might be present, and analyse the MK-RGflow in the limit of infinite dimensions. MK-RG predicts that the spin-glasstransition in a field and the glass transition are governed by zero-temperaturefixed points of the renormalization group flow. This implies a typical energyscale that grows, approaching the transition, as a power of the correlationlength, thus leading to enormously large time-scales as expected fromexperiments and simulations. These fixed points exist only in dimensions largerthan $d_L>3$ but they nevertheless influence the RG flow below it, inparticular in three dimensions. MK-RG thus predicts a similar behavior forspin-glasses in a field and models of glasses and relates it to the presence ofavoided critical points.

Real Space Migdal-Kadanoff Renormalisation of Glassy Systems: Recent Results and a Critical Assessment