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The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension. (English) Zbl 1178.58010

P. Lévy [Problèmes concrèts d’analyse fonctionelle. Paris: Gauthier-Villars (1951; Zbl 0043.32302)], elaborating on R. Gáteaux [Bull. Soc. Math. Fr. 47, 47–70, 70–96 (1919; JFM 47.0382.01) and ibid. 50, 1–37 (1922; JFM 48.0470.01)], tried to develop measures on infinite dimensional spaces and integration of functionals and arrived at additive but not \(\sigma\)-additive measures [cf. the reviewer’s 1961 Cantab thesis, Some measure theoretic problems related to Brownian motion]. For instance, the Riemann integral is not \(\sigma\)-additive. Lévy considered the asymptotic behaviour of the uniform measures \(\rho_{n}\) on high-dimensional Euclidean spaces to the effect that \(\rho_{n}\) is asymptotically concentrated on the sphere of radius proportional to \(\sqrt{n}\) as \(n \to \infty\) (cf. Gelfand’s conjectures [F. A. Berezin, I. M. Gelfand, M. I. Graev and M. A. Naĭmark, Usp. Mat. Nauk 11, No. 6(72), 13–40 (1956; Zbl 0074.10302)] which led to the reviewer’s thesis [loc.cit.]). Lévy’s sphere determines averages for functionals (as non-differentiable functionals vary rapidly in the neighbourhood of the surface) but it does not lead to a Lebesgue theory of integration.
The author interprets this procedure as that the compact Lie group \(\text{SO}(n,\mathbb R)\) acting on homogeneous spaces \(S^{n-1}\), spheres of radii proportional to \(\sqrt{n}\), to asymptotically approach the standard infinite-dimensional Gaussian measure, being a ‘weak limit’ of the uniform measures on \(\text{SO}(n)\). The author puts this, in statistical mechanics terms, as that the number \(n\) of (identical) particles in a fixed energy system becomes very large the (micro) canonical ensembles are asymptotically equivalent to the grand canonical ensemble.
The infinite dimensional Gaussian measure can be looked at from Gelfand’s generalised random variable viewpoint as integration of functions of test functions; the Gelfand triple (or rigged Hilbert space) is \({\mathcal S}\subset H \subset {\mathcal S}'\) where the dual space \({\mathcal S}'\) of Schwartz distributions is a Gaussian white noise on test functions in \({\mathcal S}\). This leads to a ‘cylindrical measure’ on Hilbert space; it is not \(\sigma\)-additive and has so a multitude of different null sets.
N. Wiener’s differential \(n\)-space [J. Math. Phys. 58, 131–174 (1923)] is made up of the differentials \(B_{t_{i}} - B_{t_{i-1}}\) of Brownian paths \(B_{t}\) over a finite interval of the line partitioned into \(n\) uniform pieces; Wiener’s differential space is made up of the almost sure limits as \(n \to \infty\). Wiener modified this differential space to become \(\sigma\)-additive by expanding the space of paths, considering the paths to be measurable if they are Hölder continuous with parameter \({1 \over 4}\). (Kolmogorov’s extension theorem was too abstract for computational purposes). For infinite-dimensional spaces, like Hilbert and Banach spaces, it is not possible to construct a Lebesgue integration theory; instead one drops some Lebesgue-like condition, not necessarily \(\sigma\)-additivity.
The author’s innovative idea is to use quasi-stochastic processes in order to deal with measures with support a \(\sigma\)-finite space rather than on a probability space. The Fourier transform works for periodic functions, but can be unsatisfactory, cf. the Gibbs phenomenon when the Fourier series is cut off.
The Laplace transform involves moments and is more general than Fourier in that it can handle \(\sigma\)-finite measures. The Fourier theory relies on ‘modes’ and characteristic functionals; the Laplace theory relies on moments and moment generating functionals. The author uses generating functionals because his approximating measures do not have a well-defined limit, though he would not have been able to reverse any limiting Laplace transforms to get a measure.
The author in [Proc. Steklov Inst. Math. 259, 248–272 (2007); translation from Tr. Mat. Inst. Steklova 259, 256–281 (2007; Zbl 1165.28003)] considered the sequence of non-compact Lie groups \(\text{SL}(n,\mathbb R)\) with Cartan (basically abelian, maximal torus) subgroups \(\text{SDiag}_{+}(n,\mathbb R)\) and their Haar measures \(m_n\) which are \(\sigma\)-finite, not finite; these are restrictions of Lebesgue measures on \(\mathbb R^n\). The diagonal matrices act as multipliers on a (non-trivial) hypersphere \(M_{n,r}= \{(y_{1},\dots,y_{n}):\prod_{k=1}^n {y_k}=r^n\}\). The \(m_n\) have Laplace transforms \(D_n({\mathbf f}) = \int_{M{n,r}}\exp\{-\sum f_{k} \cdot y_{k}\}\, dm_{n}(\mathbf{y})\) where \(\mathbf{f} = (f_{1},\dots,f_{n})\) and \({\mathbf y} = (y_{1},\dots,y_{n})\).
In the same way as for Lévy, letting \(n \to \infty\), the measures are concentrated on a hypersphere as \(n \to \infty\) and will not converge to a \(\sigma\)-additive measure. The author envisages a one-parameter family of unique measures hoping they would be (like) Lebesgue measures.
The author is concerned with a continuum infinite-dimensional analogue of \(\text{SL}(n,\mathbb{R})\) as a group of bounded linear operators operators on a space \(D_{+}\) of atomic measures (provided with the topology of \({\mathcal S}'\)) invariant over an infinite dimensional measurable manifold \((X, \mu)\), which would be the continuum analogue of the hypersphere.
He does not specify his reason for choosing these \(D_{+}\) but the hypothesis is understandable; indeed the reviewer showed in his thesis [loc. cit.] that Lévy’s sphere can be given an atomic measure structure. The author’s orbits \(\xi\) in \(D_{+}\) are \(\{ a_{k} \delta_{x_{k}}\}\) where \(x_{k} \in X, a_{k} \in {\mathbb R}_{+}\), \(k \in \mathbb{N}\).
His continuum analogue of the ‘Cartan subgroup’ consists of a group of multipliers \( \xi \mapsto \alpha(\cdot)\xi(\cdot)\) where the \(\alpha\) run over tame functionals, i.e., the \(\alpha\) are non-negative measurable functions with \(\int\log\alpha(x)\,d\mu(x) < \infty\). (He adds a further restriction that \(\int\log \alpha (x)\, d\mu(x) = 0\) in order to have the analogue of determinants having value 1.)
However, it is unlikely that this continuum analogue is a limit of the measures \(m_{n}\). Indeed, the author uses hard analysis, with an unjustified short cut and some confusing arguments (like many diverse \(\lambda\)’s), but probably repairable, to suggest, see below, that as \(n \to \infty\) the Laplace transforms of the \(m_{n}\) cannot globally converge weakly as \(n \to \infty\). The author makes no mention as to what the group is for which the Cartan subgroup is a subgroup. This opens a can of worms (see the Appendix below).
The idea behind continuum products of Hilbert spaces (leading to exponential or Fock spaces) of measure spaces and for current groups [cf. I. M. Gelfand, M. I. Graev and A. M. Vershik, Compos. Math. 35, 299–334 (1977; Zbl 0368.53034)], can be reduced to its essence, a continuum product of complex numbers. A continuum product of values of a non-vanishing function \(f\) on a set \(X\) with measure \(\mu\) may is taken as \(\exp\int{\{ \log f(t)\, d\mu(t)} \}\), when it exists, corresponding to the discrete product \(\exp\{ \sum_{i} f(t_{i})\} = \prod_{i} \exp\{ f(t_{i})\}\).
The author rephrases the expression for the \(D_{n}(\mathbf{f})\). He denotes \( \prod_{k =1}^{n}f_{k}{}^{1/n}\), a geometric average of the coordinates of \(\mathbf{f}\), by \(\rho_{n}({\mathbf f})\). Starting with the original expression for \(D_n(\mathbf{f})\), he transforms \(y_{k}\) to \(\rho_{n}(\mathbf{f}){z_{k}} \over {f_{k}}\) and the Laplace transforms simplify to become
\[ \int_{M{n,r}}{\exp\{ - \rho_{n}(\mathbf{f})}\sum z_{k}\}\, dm_{n}(\mathbf{z}). \]
Noting that \(dm_{n}(\mathbf{x}) = \prod{dx_{k}}\) and writing \(z_{k}\) as \(e^{x_{k}}e^{n \log r}\) leads to the expression
\[ \int_{H_{n}}{\exp \{ - \rho_{n}(\mathbf{f})} r^{n} \sum e^{x}{}_{k}\}\, d\mathbf{x} \]
where \(H_{n}\) denotes the hyperplane \(\{ (x_1,\dots,x_n) : \sum_k x_k = 0\}\).
By prematurely going to a continuum model \(\lim_n -\rho_{n}(\mathbf{f})\) is replaced heuristically by
\(\exp \int \log\mathbf{f}(t)\,d\mu(t)\) which he denotes as \(\lambda \in\mathbb{R}\) (but rather \(\lambda(\mathbf{f})\), where \(\mathbf{f}\) is such that \(\int\log\mathbf{f}(t)\, d\mu(t) < \infty\). This \(\lambda\) is used as a parameter which varies with \(\mathbf{f}\) and determines for which \(\mathbf{f}\) the \(D_{n}(\mathbf{f})\) converge. The author works with the assumption that \(D_{n}(\mathbf{f})\) is essentially close enough to \(F_{n}(\lambda_{n})\), defined as \(\int_{H_n}\exp \{ -\lambda_n \sum_1^n e^{x_k}\}\, d\mathbf{x}\), for a meaningful consideration of convergence. The \(r^{n}\) seem to have been absorbed. He is left with the problem as to whether the \(F_{n}(\lambda_{n})\) converge for almost all the \(\mathbf{f}\) such that \(\int\log f(t)\,dm(t)\), never mind any continuum group analogue.
An integral with integrand \(e^{g(\zeta)}h(\zeta)\); \(\zeta \in \mathbb{C}\) is convenient for exponential asymptotics. The \(F_{n}\) have the useful property of being related to inverse Mellin transforms of products of \(\gamma\)-functions. Indeed, he adds another dimension to use (\({\mathbf{x}},t\)) instead of \(\mathbf{x}\). Then, \(\int_{0}^{\infty} F_{n}(e^{-t/n}e^{-ts}\,dt = \Gamma(s)^{n}\); substituting \(e^{- t/n} = \lambda_{n}\), instead of the author’s \(\lambda\), one would get an integral form \(\int_{0}^{\infty}nF_n(\lambda_n)e^{ns -1}\,d\lambda_n = \Gamma(\lambda_n)^n\).
The author extends \(F_{n}\) to the complex plane and tries for a saddle point on the positive real axis, say at \((\gamma,0)\), through which the ‘vertical’ line will be the line of steepest descent, ‘mountains’ to the left and right. Considering \(\lambda\) to be a continuum limit of the \(\rho_{n}(f)\), \(\gamma\) is expressed in terms of the parameter \(\lambda\) such that \(\lambda = \exp\{ {\Gamma'(\gamma) \over \Gamma(\gamma)}\}\), the same as \(e^{\psi(\gamma)}\) where \(\psi\) is the digamma function.
It seems that he is defining \(L(\lambda)\), or alternatively \(\overline{L}(\gamma)\), to be \(\lim_n {1 \over n}\log F_n(\rho_n(\mathbf{f})\) claiming, for large \(n\), this to be (asymptotically) \(\Gamma(\gamma) \over \lambda^{\gamma}\). He conjectures that \(L\) is related to the free energy of Gibb’s states and also is analogous to the generator of a semigroup of convolution measures subsequently involving infinite divisibility.
The author denotes by \(\lambda_{\text{cr}}\) the value of \(\lambda\) for which \(L = 1\) though the reviewer suspects the critical value should be smaller than 1. The author shows that the \(F_{n}\) converge only in a neighbourhood of \(\lambda_{\text{cr}}\), and tends to be small for small \(\lambda > 0+\) and exponentially diverges for large \(\lambda\). He cannot determine convergence as \(\lambda \to 0\) as in that case the saddle point approaches the origin where \(\Gamma\) has a pole. Thus \(F_{n}(\rho_{n})\) converges locally but not globally.
Appendix: A current group, denoted \(G^{X}\), where \(G\) is a Lie group and \(X\) is a smooth manifold with a finite measure \(\mu\), is usually the collection of bounded Borel \(G\)-valued functions on \(X\), the group operation being the pointwise multiplication. The reviewer is thinking rather of limits of fibre bundles over approximating manifolds \(X_{n}\) leading to a group of fields of operators acting on a field of Hilbert spaces [see J. Dixmier and A. Douady, Bull. Soc. Math. Fr. 91, 227–284 (1963; Zbl 0127.33102)], the base spaces \(D_{+}\) made up of point masses over the approximating manifolds.
The reviewer does not believe in infinite-dimensional Lebesgue measures and instead considers sequences \(\text{SL}(n,\mathbb{R})\) as \(n \to \infty\) and their related point processes leading to a random measure limit as \(n \to \infty\). For \(\text{SL}(2,\mathbb{R})\) the limit of gamma point processes will be the gamma random measure [see I. Gelfand, I. Graev and A. M. Vershik, in: Representations of Lie groups and Lie algebras, Proc. Summer Sch., Budapest 1971, Pt. 2, 121–179 (1985; Zbl 0595.43005)].

MSC:

58J37 Perturbations of PDEs on manifolds; asymptotics
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
44A10 Laplace transform
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