Phase transitions in discrete structures

Random discrete structures such as random graphs or matrices play an important role in several disciplines, including combinatorics, coding theory, computational complexity theory, and the theory of algorithms. Over the past about 20 years, non-rigorous techniques from statistical mechanics have been adapted for/applied to many of these problems. The monographs [3, 4, 8] provide an overview of random discrete structures in general, and [9] of the statistical mechanics slant in particular. From a mathematical perspective, the obvious problem is to provide a rigorous foundation for the physics “predictions”. In this lecture, I am going to give a very brief introduction into this work. The concrete problem that we are going to focus on is the 2-coloring problem in random K-uniform hypergraphs. Thus, fix an integer K ≥ 3, let N,M be a (large) integers, and set α = M/N . Recall that a K-uniform hypergraph Φ with N vertices and M edges consists of a vertex set V of size |V | = N and an edge set E of size |E| = M such that each e ∈ E is a K-element subset of V . We say that the hypergraph Φ is 2-colorable if there is a map σ : V → {0, 1} such that σ(e) = {0, 1} for all e ∈ E. In other words, if we think of 0, 1 as colors, then none of the edges is monochromatic. Given K,N,M , let Φ = ΦK(N,M) denote the uniformly random K-uniform hypergraph on V = [N ] with M edges. Throughout the lecture, we are going to be interested in the situation that N → ∞, while α = M/N remains fixed. We say that an event A occurs with high probability (‘w.h.p.’) if limN→∞ P [A] = 1. The question that we are going to be dealing with are the following.