Invariance Principle on the Slice

Abstract

© 2018 ACM. The non-linear invariance principle of Mossel, O'Donnell, and Oleszkiewicz establishes that if f (x1, . . . , xn) is a multilinear low-degree polynomial with low influences, then the distribution of f (B1, . . . , Bn ) is close (in various senses) to the distribution of f (G1, . . . , Gn ), where Bi ∈R{-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many applications 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 (x1, . . . , xn ), (Y1, . . . ,Yn ) such that (i) Matching moments: Xi and Yi have matching first and second moments and (ii) Independence: the variablesx1, . . . , 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 (x1, . . . , xn ) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice([n]k which consists of all vectors (x1, . . . , xn ) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdös-Ko-Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, . . . ,Xn ) is the uniform distribution on a slice([n]pn) and (Y1, . . . ,Yn ) consists either of n independent Ber(p) random variables, or of n independent N(p,p(1 - p)) 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 Erdös-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.