Nonlinear Projection for Ballistic Correlation Functions: a Formula in Terms of Minimal Connected Covers

In many-body systems, the dynamics is governed, at large scales of space and time, by the hydrodynamic principle of projection onto the conserved densities admitted by the model. This is formalised as local relaxation of fluctuations in the Ballistic Macroscopic Fluctuation Theory, and is a nonlinear version of the Boltzmann-Gibbs principle. We use it to derive a projection formula, expressing n-point connected correlation functions (cumulants) of local observables at different space-time points, in terms of those of conserved densities. This applies in every $d\ge 1$ spatial dimensions and under the ballistic scaling of space and time, both in and out of equilibrium. It generalises the well-known linear-response principle for 2-point functions. For higher-point functions, one needs to account for nonlinear fluctuations of conserved densities and, correspondingly, higher derivatives of local averages. Using Malyshev's formula for the cumulant expansion, and keeping the leading order, the result is a nonlinear projection, expressed as a sum of products of correlation functions of conserved densities with equilibrium multivariances as coefficients. The sum is combinatorially organised via certain covers of the set of space-time points, which we call "minimal connected covers". We use this in order to get general, explicit formulas for two- and three-point functions in stationary states, expressed in terms of thermodynamic and Euler-scale data.