Random Graph-Homomorphisms and Logarithmic Degree

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function psi(n) tending to infinity, the range of a typical homomorphism of C_{n,k} is super-constant for k = 2 log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n)

High-Dimensional Lipschitz Functions are Typically Flat

A homomorphism height function on the $d$-dimensional torus $\mathbb{Z}_n^d$ is a function taking integer values on the vertices of the torus with consecutive integers assigned to adjacent vertices. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. In each model, we consider the uniform distribution over such functions, subject to boundary conditions. We prove that in high dimensions, with zero boundary values, a typical function is very flat, having bounded variance at any fixed vertex and taking at most $C(\log n)^{1/d}$ values with high probability. Our results extend to any dimension $d\ge 2$, if $\mathbb{Z}_n^d$ is replaced by an enhanced version of it, the torus $\mathbb{Z}_n^d\times\mathbb{Z}_2^{d_0}$ for some fixed $d_0$. This establishes 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, a typical 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 apply also to the related model of uniform 3-coloring and establish, for certain boundary conditions, that a uniformly sampled proper 3-coloring of $\mathbb{Z}_n^d$ will be nearly constant on either the even or odd sub-lattice. Our proofs are based on a combinatorial transformation and on a careful analysis of the properties of a 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, H\"aggstr\"om and Mossel.Comment: 63 pages, 5 figures (containing 10 images). Improved introduction and layout. Minor correction

Delocalization of the height function of the six-vertex model

We show that the height function of the six-vertex model, in the parameter range $\mathbf a=\mathbf b=1$ and $\mathbf c\ge1$, is delocalized with logarithmic variance when $\mathbf c\le 2$. This complements the earlier proven localization for $\mathbf c>2$. Our proof relies on Russo--Seymour--Welsh type arguments, and on the local behaviour of the free energy of the cylindrical six-vertex model, as a function of the unbalance between the number of up and down arrows.Comment: 54 pages, 16 figure

Three lectures on random proper colorings of Zd\mathbb{Z}^d

A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer lattice $\mathbb{Z}^d$. Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension $d$ and the number of colors $q$. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.Comment: 53 pages, 10 figures; Based on lectures given at the workshop on Random Walks, Random Graphs and Random Media, September 2019, Munich and at the school Lectures on Probability and Stochastic Processes XIV, December 2019, Delh