Uniform High Order Factorial Moment Bounds for the Critical Erd\H{o}s-R\'enyi Component Process

Abstract

We give a finite $n$ enumerative derivation of the local surplus marked point process form of Aldous's critical window limit for the Erd\H{o}s-R\'enyi random graph. For $p_n=n^{-1}+\lambda n^{-4/3}$, let $\Xi_n$ place an atom at the rescaled size and surplus of each component. Exact component enumeration yields the limiting factorial correlation densities and the uniform bound \[ \mathbb E[(\Xi_n(K))_q]\le C_K^q e^{-c_Kq^3}, \qquad q\ge1, \] for every compact marked window $K$, uniformly in admissible $n$. The cubic order is optimal for the limiting factorial measures, and the estimate persists after summing over all surpluses on compact size intervals. It yields overcrowding and local exponential moment bounds, quantitative truncation of the finite $n$ Laplace functional expansion, and local point process convergence. Using the Janson-Spencer Palm description and a classical all excess estimate, we also recover ordered $\ell^2$ component size convergence.