Introduction to renormalization group of tensor networks

I've been recently thinking what it would take to rigorously prove the existence of a critical renormalization group fixed point for a tensor RG transformation which is a kind of RG transformation of 2D lattice models that I will introduce. There is great numerical evidence for convergence of tensor network RG maps, but can we turn this into a proof? The problem is more amenable than for block-spin RG maps, but not fully trivial since one needs to work with infinite-dimensional tensors, satisfy the Hilbert-Schmidt condition, and involve "disentangling."  After explaining these issues, I will describe some rigorous results about how the approach to high and low temperature fixed points can be handled without truncation. I will also describe some numerical results about Newton method acceleration of tensor network RG. Joint work with Nikolay Ebel and Tom Kennedy.