Critical fluctuations of sums of weakly dependent random vectors

Abstract

LetSn be sums of iid random vectors taking values in a Banach space andF be a smooth function. We study the fluctuations ofSn under the transformed measurePn given byd Pn/d P=exp (nF(Sn/n))/Zn. If degeneracy occurs then the projection ofSn onto the degenerate subspace, properly centered and scaled, converges to a non-Gaussian probability measure with the degenerate subspace as its support. The projection ofSn onto the non-degenerate subspace, scaled with the usual order\(\sqrt {n,} \) converges to a Gaussian probability measure with the non-degenerate subspace as its support. The two projective limits are in general dependent. We apply this theory to the critical mean field Heisenberg model and prove a central limit type theorem for the empirical measure of this model.