Multi-Relevance: Coexisting but Distinct Notions of Scale in Large Systems*

An essential aspect of renormalization group (RG) methods is the cutoff scheme, which specifies how collective variables are integrated out. In most applications, such as in field theory, momentum is used as a cutoff scale, and RG produces effective low-momentum theories by averaging over high-momentum variables. Recently, RG methods have seen use in problems at the boundaries of statistical physics, biology, and computer science, where the models are complicated distributions over high-dimensional spaces. These models are frequently not analogous to traditional many-body systems, making it difficult to specify what precisely is meant by "scale". This makes RG hard to implement and interpret. Here, we present recent theoretical progress on both of these fronts. First, we show that non-perturbative RG is well-suited for models with finitely many degrees of freedom, and demonstrate a simple calculation. Next, we introduce a method of calculating the cutoff scheme to be used based on the structure of the model at hand. In doing so, we demonstrate that some models support multiple notions of scale, and term this property "multi-relevance". Finally, we examine how multi-relevance appears in problems relevant to fields that interface with statistical physics.