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Cumulants of conserved charges in GGE and cumulants of total transport in GHD: exact summation of matrix elements?

We obtain the cumulants of conserved charges in Generalized Gibbs Ensemble (GGE) by a direct summation of their finite-particle matrix elements. The Gaudin determinant that describes the norm of Bethe states is written as a sum over forests by virtue of the matrix-tree theorem. The aforementioned cumulants are then given by a sum over tree-diagrams whose Feynman rules involve simple Thermodynamic Bethe Ansatz (TBA) quantities. The internal vertices of these diagrams have the interpretation of virtual particles that carry anomalous corrections to bare charges. Our derivation follows closely the spirit of recent works [1,2]. We also conjecture that the cumulants of total transport in Generalized Hydrodynamics (GHD) are given by the same diagrams up to minor modifications. These cumulants play a central role in large deviation theory and were obtained in [3] using linear fluctuating hydrodynamics at Euler scale. We match our conjecture with the result of [3] up to the fourth cumulant. This highly non-trivial matching provides a strong support for our conjecture.