Invariance Principle on the Slice

Abstract

The non-linear invariance principle of Mossel, O’Donnell, and Oleszkiewicz establishes that if f ( x 1 ,… , x n ) is a multilinear low-degree polynomial with low influences, then the distribution of if f ( b 1 ,…, b n ) is close (in various senses) to the distribution of f ( G 1 ,…, G n ), where B i ∈ R {-1,1} are independent Bernoulli random variables and G i ∼ 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 ( X 1 ,… , X n ),( Y 1 ,… , Y n ) such that (i) Matching moments : X i and Y i have matching first and second moments and (ii) Independence : the variables X 1 ,… , X n are independent, as are Y 1 ,…, Y n . The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions X 1 ,… , X n 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 ( x 1 ,…, x n )∈{0,1} n 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 ( X 1 ,…, X n ) is the uniform distribution on a slice ( [ n ] pn and ( Y 1 ,… , Y n ) 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 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.