The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF [2] , or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability. For a uniformly chosen random hypergraph of m = nρ vertices and n hyperedges, each consisting of the same fixed number l ≥ 3 of vertices, the size of the core exhibits for large n a first-order phase transition, changing from o(n) for ρ > ρ c to a positive fraction of n for ρ < ρ c , with a transition window size (n −1/2 ) around ρ c > 0. Analyzing the corresponding "leaf removal" algorithm, we determine the associated finite-size scaling behavior. In particular, if ρ is inside the scaling window (more precisely, ρ = ρ c + rn −1/2 ), the probability of having a core of size (n) has a limit strictly between 0 and 1, and a leading correction of order (n −1/6 ). The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with n. This behavior is expected to be universal for a wide collection of combinatorial problems.
In the same paper they remarked (see REF Remark 2.6] ) that their techniques were applicable to a wide variety of properties of the core "in the scaling regime" ρ = ρ c + rn −1/2 where ρ c is the threshold for the appearance of core.
Finite size scaling for the core of large random hypergraphs
Annals of Applied Probability
Mathematics