An invariance principle for the multi-slice, with applications

Abstract

Given an alphabet size $m\in N$ thought of as a constant, and $\overrightarrow{k}=(k_1,\dots ,k_m)$ whose entries sum of up n, the $\overrightarrow{k}$-multi-slice is the set of vectors $x\in {\left[m\right]}^n$ in which each symbol $i\in \left[m\right]$ appears precisely $k_i$ times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space $({\left[m\right]}^n,{\mu }^n)$ in which $\mu \left(i\right)=k_i/n$. This answers a question raised by [23].

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 $\frac{2r+1}{2^r}+o\left(1\right)$ satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size $o\left(1\right)$.
• 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 [42] 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.