  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.
