Random Graph-Homomorphisms and Logarithmic Degree

Abstract

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 1 product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function ψ(n) tending to infinity, the range of a typical homomorphism of C n,k is super-constant for k = 2 log(n) − ψ(n), and is 3 for k = 2 log(n) + ψ(n).

Grounded Lipschitz functions on trees are typically flat

“…An integer-valued function f on the vertices of a graph is called a graph homomorphism (or a homomorphism height function) if |f (u) − f (v)| = 1 for every pair u, v of adjacent vertices. A lower bound on the typical range of random graph homomorphisms on general graphs was established in [2]. Results for this model were obtained for: tree-like graphs [1], the hypercube [4,5], and finite boxes in the integer lattice Z d for large d [6].…”

Delocalization of Two-Dimensional Random Surfaces with Hard-Core Constraints

“…If T 2 n is replaced by a triangle graph on 3 vertices then the analogous quantity to Z T 2 n ,0,U is zero when, say, {x : U (x) < ∞} = [−3, −2] ∪ [2,3]. However, the above argument can be easily modified to work for all graphs if {x : U (x) < ∞} contains an interval around 0.…”

“…. , n − 1, n} 2 (1.1) where the vertices and edges of T 2 n are denoted by V (T 2 n ) and E(T 2 n ) respectively, dϕ v denotes Lebesgue measure on ϕ v , δ 0 is a Dirac delta measure at 0 and Z T 2 n ,0,U is a normalization constant. For this definition to make sense the potential U needs to satisfy additional requirements.…”

Diffusivity of a random walk on random walks

“…. , z (K+1) ) ∈ Z K+1 , ∀i ∈ [1, K] , |z (i+1) − z (i) | = 1} As in the case of a simple random walk, the rescaled trajectory of a walker, say Z (1) , will converge in law to a Brownian motion. However, it is interesting to note that the constraint between each coordinate only slows down the walk by decreasing its variance.…”