Exact projective contraction and Gibbs uniqueness for finite-height p -SOS models on trees

We investigate finite-height p-SOS models on locally finite rooted trees characterized by inhomogeneous edge interactions and site-specific external fields. Employing the Hilbert projective metric and Birkhoff contraction for positive transfer matrices, we establish explicit field-uniform conditions for the uniqueness of splitting Gibbs measures and robust spatial mixing. The technique is applicable to any finite spin height and any positive exponent, with the conventional SOS model serving as a specific instance. In the case of finite-type trees, the criterion for uniqueness simplifies to a Perron-Frobenius spectral condition, providing clear assessments for homogeneous and periodic tree structures. In the appealing convex domain, we additionally derive a sufficient finite-type condition for the disruption of mirror symmetry. The findings offer a transfer-matrix framework for interface-type Gibbs measurements on trees.