Uncountably many extremal Series--Sinai states for the Ising model on Lobachevsky lattices

We construct an uncountable family of extremal Gibbs states of the low temperature Ising model on hyperbolic lattices embedded in the hyperbolic plane $\mathbb{H}_2$ whose interfaces are complete geodesics of $\mathbb{H}_2$.

These states are extracted from the states constructed by D'Achille, Coquille and Le Ny in arXiv:2504.19553v2 by considering path in the dual lattice at close enough distance from geodesics of $\mathbb{H}_2$ thanks to the Morse--Mostow lemma.