An Invariance Principle for the Multi-slice, with Applications

Given an alphabet size m inN thought of as a constant, and vec{k}=(k1,.., km}) whose entries sum of up n, the vec{k-multi-slice is the set of vectors x in[m] n in which each symbol i in[m] appears precisely ki times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space ([m] n, μ n) in which μ(i)=ki}/n. This answers a question raised by [21]. As applications of the invariance principle, we show: 1)An analogue of the 'dictatorship test implies computational hardness' paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich 2-to-1 Games Conjecture. Using this analogue, we show that assuming the Rich 2-to-1 Games Conjecture, (a) there is an r-ary CSP P}r for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most 2r+12r}}+o(1) satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size o(1). 2)A reduction of the problem of studying expectations of products of functions on the multi-slice to studying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from [38] for the multi-slice. 3)In a companion paper, we show further applications of our invariance principle in extremal combinatorics, and more specifically to proving removal lemmas of a wide family of hypergraphs H called ζ-forests, which is a natural extension of the well-studied case of matchings.