Boundary Condition dependent Universality Classes on a Hyperbolic Lattice

We study the ferromagnetic Ising model on finite hyperbolic tessellations of Euclidean AdS 2 with open and wired boundary conditions. Because a finite fraction of spins remains at the boundary as the lattice grows, these conditions select distinct thermodynamic behaviours. In the 5,4 tessellation, Monte-Carlo simulations using efficient worm algorithms on open boundaries (OBC) yield a transition near β c J=0.632(3), with susceptibility data collapsing under scaling by the total number of spins N. The fitted finite-size exponents, 1/ ν 0.162 and γ/ ν 0.743, differ substantially from the mean-field volume scaling. Wired boundaries (WBC), which correlate boundary spins, instead show a transition around β c J=0.348(4) to an ordered phase consistent with the mean-field exponents. While the mean-field criticality with WBC is consistent with previous studies and with the suppression of independent boundary fluctuations, the OBC exponents hint at the presence of a new universality class. Results from other tessellations support the robustness of the observed scaling with OBC. We discuss a possible route to interpolate between the different boundary conditions. Our findings show that boundary dynamics must be specified when characterizing critical behaviour on hyperbolic lattices.