  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.
