Tensor networks for rigorous RG
In this talk, I will discuss how tensor networks enable the rigorous study of the renormalization group for statistical systems in two and three spatial dimensions. First, we will discuss the tensor network representation of classical spin systems: how it helps define renormalization group maps and what challenges arise within this framework. Then, I will introduce the “2×1” RG map and define hat-tensors and hat-RG maps, which allow us to rigorously track finite-dimensional “bounding boxes” for infinite-dimensional tensors. We will see that, by tracking the flow of these bounding boxes, one can show the stability of the high-temperature fixed point, estimate the size of its basin of attraction, and compute the free energy density of any particular lattice system within that basin with rigorous error bars. Finally, we will discuss how tensor RG techniques can be applied to three-dimensional systems to show the stability of the high-temperature fixed point. This talk is based on [2506.03247] and [2401.04229].