High-Dimensional Lipschitz Functions are Typically Flat

A homomorphism height function on the d-dimensional torus Z_n^d is a function on the vertices of the torus taking integer values and constrained to have adjacent vertices take adjacent integer values. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. For each of these models, we consider the uniform distribution over all such functions, subject to boundary conditions. We prove that in high dimensions and with zero boundary values, the random function obtained is typically very flat, having bounded variance at any fixed vertex and taking at most C(logn)^{1/d} values with high probability. This result is sharp up to constants. Our results extend to any dimension d>=2, if one replaces Z_n^d by an enhanced version of it, the torus Z_n^d x Z_2^{d_0} for some fixed d_0. Consequently, we establish one side of a conjectured roughening transition in 2 dimensions. The full transition is established for a class of tori with non-equal side lengths. We also find that when d is taken to infinity while n remains fixed, the random function takes at most r values with high probability, where r=5 for the homomorphism model and r=4 for the Lipschitz model. Suitable generalizations are obtained when n grows with d. Our results have consequences also for the related model of uniform 3-coloring and establish that for certain boundary conditions, a uniformly sampled proper 3-coloring of Z_n^d will be nearly constant on either the even or odd sub-lattice. Our proofs are based on the construction of a combinatorial transformation and on a careful analysis of the properties of a new class of cutsets which we term odd cutsets. For the Lipschitz model, our results rely also on a bijection of Yadin. This work generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini, Yadin and Yehudayoff and answers a question of Benjamini, Haggstrom and Mossel.