Symmetry, Moment Problems, and Statistical Mechanical Systems on Complete Graphs

The Ising model on a complete graph with $n$ vertices is known as the Curie-Weiss model.
The corresponding Gibbs state is a simple probability measure on $\{0,1\}^n$ that is
invariant under permutations of the vertices. Such a measure is said to be exchangeable. De Finetti's Theorem states that if $n$ were infinite, an exchangeable measure would be a mixture of
homogeneous product measures. For finite $n$, this is the case if and only if a certain finite sequence can be expressed as the first $n$ moments of a probability distribution on $[0,1]$. This provides a connection to the classical Hausdorff moment problem. Using this connection, we will show that the Curie-Weiss model is a mixture of homogeneous product measures if and only if the model is ferromagnetic. In the antiferromagnetic case, we will ask whether the measure can
be extended to $\{0,1\}^l$ for $l>n$. This question leads to a new moment problem, which we will
solve and apply to get a rather precise asymptotic result for the Curie-Weiss model. This talk is based on joint work with J. Steif and B. Toth.