A Shared Observation Shields Collective Fluctuations while Preserving Local Independence

Abstract

As a liquid approaches its glass transition, its dynamics turns heterogeneous: mobile and immobile regions coexist, and the four-point susceptibility $\chi_4$ that quantifies this heterogeneity grows sharply. Interpreting that growth is subtle, because the collective signals experiments record, such as a tagged particle's trajectory, an overlap function, or a mean field, are generated by the same particles they describe. Here we compute exactly what conditioning on such a shared record does to the population that produced it, for a broad class of stochastically observed systems; the guiding example is a tagged particle and the cage of $z$ neighbors that drives its force history. Using a Girsanov path transformation, we prove that the conditioning multiplies the independent joint law of the $z$ trajectories by exactly one term: a centered-square penalty along the single collective direction the record can see. Any fixed pair of particles stays nearly independent, with covariance falling as $O(z^{-1})$ and mutual information as $O(z^{-2})$, the property known as propagation of chaos, yet the $z(z-1)$ weak pair correlations add coherently into a finite suppression of collective fluctuations, the Schur shield $D - C = -C^2(aI + C)^{-1} \preceq 0$. An exactly solvable Brownian model calibrates the construction. The physical consequence is a calculable baseline for dynamical heterogeneity: conditioning itself contributes a computable, nonpositive amount to the susceptibility of a conditioned ensemble, so the genuine cooperative signal is the excess of the measured $\chi_4$ over this baseline rather than over zero, a comparison that existing simulation data can already perform.