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Fluid properties near rough surfaces are crucial in describing fundamentalsurface phenomena and modern industrial material design implementations. One ofthe most powerful approaches to model real rough materials is based on thesurface representation in terms of random geometry. Understanding the influenceof random solid geometry on the low-temperature fluid thermodynamics is acutting edge problem. Therefore this work extends recent studies bypassinghigh-temperature expansion and small heterogeneity scale. We introduce randombranching trees whose topology reflects the hierarchical properties of a randomsolid geometry. This mathematical representation allows us to obtain averagedfree energy using a statistical model of virtual clusters interacting throughrandom ultrametric pairwise potentials. Our results demonstrate that asignificant impact to fluid-solid interface energy is induced by thehierarchical structure of random geometry at low temperature. Thesecalculations coincide with direct Monte Carlo simulations. Due to the study'sinterdisciplinary nature, the developed approach can be applied to a wide rangeof quenched disorder systems on random graphs.

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