Critical-flow behavior for nonlinear block-spin transformations

Determine the scaling limit of the nonlinear $2\times2\to1$ block-spin renormalization group transformations $T_\alpha$ for the nearest-neighbour Ising model at the critical inverse temperature $\beta=\beta_c$ when $\alpha\in[0,1/4)$, in particular whether the flow converges to a non-trivial fixed-point measure.

Background

For the family of symmetry-preserving, monotone $2\times2\to1$ transformations, the paper proves the expected high- and low-temperature limiting behavior for $\alpha\in(0,1/4)$ away from criticality, while transformations with $\alpha\geq1/4$ exhibit different behavior. The critical case for $\alpha\in[0,1/4)$ is not resolved because the correlation estimates used away from criticality fail there. Resolving it is important for determining whether these transformations can produce the non-trivial fixed point expected at criticality.