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Ising Spins on Randomly Multi-Branched Husimi Square Lattice: Thermodynamics and Phase Transition in Cross-dimensional Range

An inhomogeneous random recursive lattice was constructed from the multi-branched Husimi square lattice. The number of repeating units connected on one vertex was randomly set to be 2 or 3 with a quenched ratio $P_2$ or $P_3$ with $P_2+P_3=1$. The model was designed to describe complex thermodynamic systems with variable coordinating neighbors, e.g. the cross-dimensional range around the surface of a bulk materials. Classical ferromagnetic spin-1 Ising model was solved on the lattice to achieve an annealed solution via the local exact calculation technique. The model exhibits distinct spontaneous magnetization similar to the deterministic system, with however rigorous thermal fluctuations and significant singularities on the entropy behavior around the critical temperature, indicating a complex superheating and supercooling frustration in the cross-dimensional range induced by the stochasticity. The critical temperature was found to be exponentially correlated to the structural ratio $P$ with the coefficient fitted as 0.53187, while the ground state energy presents linear relation to $P$, implying a well-defined average property according to the structural ratio.