## Macroscopic behaviour of Lipschitz random surfaces

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## Abstract

A random field is a random function φ from the square lattice ℤᵈ to some fixed standard Borel space (E, ℰ). A random surface is a random field with the extra condition that E ∈ {ℤ, ℝ} where ℰ is the standard σ-algebra. For random surfaces, one often studies the gradient ∇φ of the random function of interest. Random fields and random surfaces serve as toy models for analysing several phenomena in statistical physics: examples include percolation models, the Ising model, dimer models, the discrete Gaussian free field, and uniformly random Lipschitz functions.

We analyse the specific free energy functional for a class of random fields and for a class of random surfaces. In either case, we are interested in the nature of the minimisers of the specific free energy, and we give a new characterisation of these minimisers even when the model fails to be quasilocal. This immediately leads to a notion of free energy in the spirit of Burton and Keane. In the case of random fields, we derive a concise theory which includes several existing results, and use this theory to prove new results for the Loop O(n) model and the Griffiths singularity model.

The study of the minimisers of the specific free energy is part of a larger programme, where the ultimate goal is to derive strict convexity of the surface tension for random surface models which are monotone in boundary conditions. We prove this conjecture in the case that the model is also Lipschitz, although we also impose some very mild conditions on the representation of the model in terms of an interaction potential to guarantee well-definedness of the statistical mechanical quantities. This in contrast to the work of Sheffield, where the case for strict convexity depends strongly on special properties of the potential, namely that it is a convex nearest-neighbour potential. The results in this thesis include a large deviations principle for (simultaneously) the macroscopic shape and the microscopic statistics of the surface under consideration. Applications include models induced by submodular potentials, that is, potentials which satisfy the Fortuin-Kasteleyn-Ginibre lattice condition. This answers an open question of Sheffield in the Lipschitz case: we derive that the surface tension is strictly convex. We furthermore prove a conjecture of Menz and Tassy: we derive strict convexity of the surface tension for uniformly random graph homomorphisms from ℤᵈ to a k-regular tree, for any d, k ≥ 2. This is remarkable as the target space is not ℤ or ℝ.

Finally, we prove new results for a generalisation of the hexagonal dimer model to higher dimensions. We give a much more direct version of Sheffield's proof for strict convexity of the surface tension, tailored to the special structure of the model. The same structure implies an identity for the covariance structure of the model in terms of its random geometry. We also derive a generalised Kasteleyn theory: the partition function of the model equals the Cayley hyperdeterminant of the hypergraph which is the natural dual to the graph supporting the generalised model.