Characterization of phase transitions for generalized graph recursions

Characterize which generalized graph recursion operators R produce a phase transition in the hard-core model on the recursive graph sequences G_{n+1}=R(G_n).

Background

For the specific recursion class treated in the paper, non-degeneracy and expansion imply zero-free neighborhoods and absence of phase transitions. The authors ask for a characterization in the broader recursion framework with auxiliary connecting graphs, where bounded degrees, label distances, and the induced renormalization dynamics can behave differently.

References

Is it possible to characterize for which recursion operators R: G_k→G_k as in the more general framework from , the hard-core model on the recursive sequences (G_n){n≥0}, G{n+1}=R(G_n), has a phase transition?

— Zeros of the independence polynomial on recursive sequences of graphs  (2609.35102 - Hlushchanka et al., 28 Sep 2026) in Section 1, subsection “Open questions,” subsubsection “More general recursions”