  To study electronic transport through chaotic quantum dots, there are two
main theoretical approachs. One involves substituting the quantum system with a
random scattering matrix and performing appropriate ensemble averaging. The
other treats the transport in the semiclassical approximation and studies
correlations among sets of classical trajectories. There are established
evaluation procedures within the semiclassical evaluation that, for several
linear and non-linear transport moments to which they were applied, have always
resulted in the agreement with random matrix predictions. We prove that this
agreement is universal: any semiclassical evaluation within the accepted
procedures is equivalent to the evaluation within random matrix theory.
  The equivalence is shown by developing a combinatorial interpretation of the
trajectory sets as ribbon graphs (maps) with certain properties and exhibiting
systematic cancellations among their contributions. Remaining trajectory sets
can be identified with primitive (palindromic) factorisations whose number
gives the coefficients in the corresponding expansion of the moments of random
matrices. The equivalence is proved for systems with and without time reversal
symmetry.
