The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension

Abstract.

We consider the sequence of the hyperspheres M _n , i.e., the homogeneous transitive spaces of the Cartan subgroup $SDiag(n,{\mathbb{R}})$ of the group $SL(n,{\mathbb{R}}), n = 1, 2, \ldots ,$ and study the normalized limit of the corresponding sequence of invariant measures m _n on those spaces. In the case of compact groups and homogeneous spaces, for example, for the classical pairs (SO(n), S ^n-1), n = 1, 2, … , the limit of the corresponding measures is the classical infinite-dimensional Gaussian measure; this is the well-known Maxwell-Poincaré lemma. Simultaneously the Gaussian measure is a unique (up to a scalar) invariant measure with respect to the action of the infinite orthogonal group O(∞). This coincidence implies the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), S ^n-1). Our main result shows that the situation for noncompact groups, for example for the case $(SDiag(n,{\mathbb{R}}),M_n)$, is completely different: the limit of the measures m _n does not exist in the literal sense, and we show that only a normalized logarithmic limit of the Laplace transforms of those measures does exist. At the same time, there exists a measure which is invariant with respect to a continuous analogue of the Cartan subgroup of the group GL(∞), the so-called infinite-dimensional Lebesgue measure (see [7]). This difference is an evidence for non-equivalence between the grand and small canonical ensembles in the noncompact case.