LetXandYbe two real-valued random variables. Let (X1,Y1), (X2,Y2),... be independent identically distributed copies of (X,Y). Suppose there are two players A and B. Player A has access toX1,X2,... and player B has access toY1,Y2,.... Without communication, what joint probability distributions can players A and B jointly simulate? That is, ifk,mare fixed positive integers, what probability distributions on {1,...,m}2are equal to the distribution of (f(X1,...,Xk),g(Y1,...,Yk)) for somef,g: Rk→ {1,...,m}? WhenXandYare standard Gaussians with fixed correlation ρ ∈ (-1, 1), we show that the set of probability distributions that can be noninteractively simulated fromkGaussian samples is the same for anyk≥m2. Previously, it was not even known if this number of samplesm2would be finite or not, except whenm≤ 2. Consequently, a straightforward brute-force search deciding whether or not a probability distribution on {1,...,m}2is within distance 0kcorrelated Gaussian samples has run time bounded by (5/ε)m(log(ε/2)/ log |ρ|)m2, improving a bound of Ghazi, Kamath and Raghavendra. A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on {1,...,m}2is within distance 0ksamples of a given finite discrete distribution (X,Y) in run time that does not depend onk, with constants that again improve a bound of Ghazi, Kamath and Raghavendra.

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discretenoise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low inuence partition in discrete space for every number of parts k > 3, for every value ρ ≠ of the noise and for every prescribed measures for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures concerning partitions into sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell's result, the Majority is Stablest Theorem and many other results in isoperimetric theory.

It is shown that every measurable partition {A1,..., Ak} of R3 satisfies: ∑i=1k|intAi xe-1/2|x|22dx|22≤ 9π2. Let P1,P2,P3 be the partition of R2 into 120o sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai=Pi x R for i∈ {1,2,3} and Ai=∅ for i∈ {4,...,k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 x 4 centered and spherical hypothesis matrix equals 2π/3.