Invariance Principle on the Slice

The non-linear invariance principle of Mossel, O’Donnell and Oleszkiewicz establishes that if fpx1, . . . , xnq is a multilinear low-degree polynomial with low influences then the distribution of fpB1, . . . ,Bnq is close (in various senses) to the distribution of fpG1, . . . ,Gnq, where Bi PR t ́1, 1u are independent Bernoulli random variables and Gi „ Np0, 1q are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans–Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO’s invariance principle works for any two vectors of hypercontractive random variables pX1, . . . ,Xnq, pY1, . . . ,Ynq such that (i) Matching moments: Xi and Yi have matching first and second moments, (ii) Independence: the variables X1, . . . ,Xn are independent, as are Y1, . . . ,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions pX1, . . . ,Xnq in which the individual coordinates are not independent. A common example is the uniform distribution on the slice `rns k ̆ which consists of all vectors px1, . . . , xnq P t0, 1u with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdős–Ko–Rado theorems) and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which pX1, . . . ,Xnq is the uniform distribution on a slice `rns pn ̆ and pY1, . . . ,Ynq consists either of n independent Berppq random variables, or of n independent Npp, pp1 ́ pqq random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain’s tail theorem, a version of the Kindler–Safra structural theorem, and a stability version of the t-intersecting Erdős–Ko–Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.