The Behavior of Laplace Transform of the Invariant Measure on the Hypersphere of High Dimension

TL;DR

This paper investigates the asymptotic behavior of invariant measures on high-dimensional hyperspheres related to noncompact groups, revealing fundamental differences from the compact case and implications for statistical ensemble equivalence.

Contribution

It demonstrates that for noncompact groups, the invariant measures do not have a literal limit, but their Laplace transforms' logarithmic limits exist, contrasting with the compact case.

Findings

Limit of measures does not exist in the noncompact case.

Normalized logarithmic Laplace transform converges in the noncompact case.

Difference indicates non-equivalence of grand and small ensembles for noncompact groups.

Abstract

We consider the sequence of the hyperspheres $M_{n,r}$ i.e. the homogeneous transitive spaces - of the Cartan subgroup $SDiag(n,\Bbb R)$ of the group $SL(n,\Bbb R), n=1 ...$, and studied the normalized limit of the corresponding sequence of the invariant measures $m_n$ on those spaces. In the case of compact groups and homogeneous spaces, as example - for classical pairs $(SO(n), S^{n-1}), n=1 ...$ - the limit of corresponding measures is the classical infinite dimensional gaussian measure - this is well-known Maxwell-Poincare lemma. Simultaneously that Gaussian measure is a unique (up to scalar) invariant measure with respect to the action of infinite orthogonal group $O(\infty)$. This coincidences means 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 situation for noncompact groups, for example for the case $(SDiag(n,\Bbb R),M_{n,r})$ (the definitions see below), is completely different: the limit of measures $m_n $ does not exist in literal sense, and we show that only normalized logarithmic limit of the Laplace transform of those measures does exist. In the same time there exists the measure which is invariant measure with respect to continuous analogue of Cartan subgroup of the group $GL(\infty)$ - this is so called infinite dimensional Lebesgue (see \cite{V}). This difference is an evidence of the non-equivalence between the grand and small canonical ensembles for the noncompact case.