Wall repulsion and mutual interface repulsion: a harmonic crystal model in high dimensions

We consider two independent lattice harmonic crystals in dimension d⩾3 constrained to live in the upper half-plane and to lie one above the other in a large region. We identify the leading order asymptotics of this model, both from the point of view of probability estimates and of pathwise behavior: this gives a rather complete picture of the phenomenon via a detailed analysis of the underlying entropy–energy competition. From the technical viewpoint, with respect to earlier work on sharp constants for harmonic entropic repulsion, this model is lacking certain monotonicity properties and the main tool that allows to overcome this difficulty is the comparison with suitable rough substrate models.