MATHEMATICS

Pinned gradient measures of SOS model associated with $H_A$-boundary laws on Cayley trees

Farhod H. Haydarov^ab, Risolat A. Ilyasova^cb, Khudoyor S. Mamayusupov<sup>b

a</sup> V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan
^b New Uzbekistan University, Tashkent, Uzbekistan
^c National University of Uzbekistan, Tashkent, Uzbekistan

Abstract: This paper investigates pinned gradient measures for SOS (Solid-On-Solid) models associated with $H_A$-boundary laws of period two, a class that encompasses all 2-height periodic gradient Gibbs measures corresponding to a spatially homogeneous boundary law. While previous research has predominantly focused on a spatially homogeneous boundary law and corresponding GGMs on Cayley trees, this study extends the analysis by providing a comprehensive characterization of such measures. Specifically, it demonstrates the existence of pinned gradient measures on a set of $G$-admissible configurations and precisely quantifies their number under certain temperature conditions.

Keywords: SOS model, gradient configuration, $G$-admissible configuration, spin values, Cayley tree, gradient measure, gradient Gibbs measure, two periodic boundary law.

Received: 15.11.2024
Revised: 12.02.2025
Accepted: 18.02.2025

Language: English

DOI: 10.17586/2220-8054-2025-16-2-154-163