The classical invariance principle of Mossel, O’Donnell and Oleszkiewicz states that the distribution of low-degree, low-influence multilinear polynomials under Bernoulli random variables is similar to their distribution under Gaussian random variables with the same expectation and variance.
We prove an invariance principle for functions on the slice (all vectors in the Boolean cube having a fixed Hamming weight). The main difficulty is that the variables are no longer independent.
As corollaries, we prove a version of majority is stablest, a Bourgain tail bound, and a weak version of the Kindler–Safra theorem. The Kindler–Safra theorem implies a stability result for t-intersecting families along the lines of Friedgut.
In follow-up work, we improve the invariance principle by removing the condition of low influences (when appropriate).