  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 $m=n\rho$ vertices and $n$ hyperedges, each consisting of
the same fixed number $l\geq3$ of vertices, the size of the core exhibits for
large $n$ a first-order phase transition, changing from $o(n)$ for $\rho>\rho
_{\mathrm{c}}$ to a positive fraction of $n$ for $\rho<\rho_{\mathrm{c}}$, with
a transition window size $\Theta(n^{-1/2})$ around $\rho_{\mathrm{c}}>0$.
Analyzing the corresponding ``leaf removal'' algorithm, we determine the
associated finite-size scaling behavior. In particular, if $\rho$ is inside the
scaling window (more precisely, $\rho=\rho_{\mathrm{c}}+rn^{-1/2}$), the
probability of having a core of size $\Theta(n)$ has a limit strictly between 0
and 1, and a leading correction of order $\Theta(n^{-1/6})$. 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.
