Conditional Fluctuations of Mean-Field Systems with Stable Common Noise

Source abstract

We study conditional fluctuations of smooth mean-field particle systems on a torus with endogenous symmetric stable common jumps of index $0 α/2$. Under a quantified second-order tail expansion, we identify the endogenous rank response: it shifts the conditional Gaussian mean at criticality and dominates sampling below criticality whenever the response is nonzero. We also establish a separate stable-to-Brownian transition for the rank error. Finally, a translation model with identical microscopic laws under rank and clock couplings admits a functional Gaussian limit in the former coupling but no diverging additive path normalization at the limiting center in the latter.