Critical Scaling of Finite-Size Fluctuations around Marginal Stability in Long-Range Hamiltonian Systems

Finite size fluctuations are a crucial ingredient in kinetic theory of long-range interacting collisionless systems. In this Letter, we introduce a phenomenological theory which predicts an anomalous scaling close to marginal stability for these fluctuations. It also pinpoints the critical window inside which the fluctuations are anomalous, and outside which they are Gaussian. Shrinking very slowly as $N^{-1\mathrm{/}5}$, this critical window encompasses a wide region around marginal stability. We confirm our predictions through extended numerical simulations on two different simplified models.