Bounded-size rules: The barely subcritical regime

Bounded-size rules are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. One starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed $K\geq 1$ are accorded the same treatment. Writing $\BS(t)$ for the state of the system with nt/2 edges, Spencer and Wormald proved that for such rules, there exists a critical time t_c such that when t< t_c the size of the largest component is of order $\log{n}$ while for $t> t_c$, the size of the largest component is of order $n$. In this work we obtain upper bounds (that hold with high probability) of order $n^{2\gamma} \log ^4 n$, on the size of the largest component, at time instants $t_n = t_c-n^{-\gamma}$, where $\gamma\in (0,1/4)$. This result for the barely subcritical regime forms a key ingredient in the study undertaken in \cite{amc-2012}, of the asymptotic dynamic behavior of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space $\Rbold_+\times \cD([0,\infty):\NNN_0)$ where $\cD([0,\infty):\NNN_0)$ is the Skorohod $D$-space of functions that are right continuous and have left limits equipped with the usual Skorohod topology. The coupling construction also gives an alternative characterization (than the usual explosion time of the susceptibility function) of the critical time $t_c$ for the emergence of the giant component in terms of the operator norm of integral operators on certain $L^2$ spaces.