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

Harmonicity and invariance on slices of the Boolean cube