Finite size scaling for the core of large random hypergraphs

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=n. 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 .>.c to a positive fraction of n for .<.c, with a transition window size .(n.1/2) around .c>0. Analyzing the corresponding .leaf removal. algorithm, we determine the associated finite-size scaling behavior. In particular, if . is inside the scaling window (more precisely, .=.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.