Title: Bernoulli Correlations and Cut Polytopes,
Abstract: Given $n$ symmetric Bernoulli variables, what can be said about their
correlation matrix viewed as a vector? We show that the set of those vectors
$R(\mathcal{B}_n)$ is a polytope and identify its vertices. Those extreme
points correspond to correlation vectors associated to the discrete uniform
distributions on diagonals of the cube $[0,1]^n$. We also show that the
polytope is affinely isomorphic to a well-known cut polytope ${\rm CUT}(n)$
which is defined as a convex hull of the cut vectors in a complete graph with
vertex set $\{1,\ldots,n\}$. The isomorphism is obtained explicitly as
$R(\mathcal{B}_n)= {\mathbf{1}}-2~{\rm CUT}(n)$. As a corollary of this work,
it is straightforward using linear programming to determine if a particular
correlation matrix is realizable or not. Furthermore, a sampling method for
multivariate symmetric Bernoullis with given correlation is obtained. In some
cases the method can also be used for general, not exclusively Bernoulli,
marginals.