  In a recent work with Kindler and Wimmer we proved an invariance principle
for the slice for low-influence, low-degree functions. Here we provide an
alternative proof for general low-degree functions, with no constraints on the
influences. We show that any real-valued function on the slice, whose degree
when written as a harmonic multi-linear polynomial is $o(\sqrt{n})$, has
approximately the same distribution under the slice and cube measure.
  Our proof is based on a novel decomposition of random increasing paths in the
cube in terms of martingales and reverse martingales. While such decompositions
have been used in the past for stationary reversible Markov chains, ours
decomposition is applied in a non-reversible non-stationary setup. We also
provide simple proofs for some known and some new properties of harmonic
functions which are crucial for the proof.
  Finally, we provide independent simple proofs for the facts that 1) one
cannot distinguish between the slice and the cube based on functions of $o(n)$
coordinates and 2) Boolean symmetric functions on the cube cannot be
approximated under the uniform measure by functions whose sum of influences is
$o(\sqrt{n})$.
