Branching random graph model of rough surfaces describes thermal properties of the effective molecular potential.

Fluid properties near rough surfaces are crucial in describing fundamental surface phenomena and modern industrial material design implementations. One of the most powerful approaches to model real rough materials is based on the surface representation in terms of random geometry. Understanding the influence of random solid geometry on the low-temperature fluid thermodynamics is a cutting-edge problem. Therefore, this work extends recent studies bypassing high-temperature expansion and small heterogeneity scale. We introduce random branching trees whose topology reflects the hierarchical properties of a random solid geometry. This mathematical representation allows us to obtain averaged free energy using a statistical model of virtual clusters interacting through random ultrametric pairwise potentials. Our results demonstrate that a significant impact to fluid-solid interface energy is induced by the hierarchical structure of random geometry at low temperature. These calculations coincide with direct Monte Carlo simulations. Due to the study's interdisciplinary nature, the developed approach can be applied to a wide range of quenched disorder systems on random graphs.