On dimensional reduction of functional measures
For functional measures indexed by rectangles in \({\mathbb{R}}^{\nu}\), dimensional reduction means (loosely speaking) that, when one side of the rectangle goes to zero, the conveniently scaled functional integrals converge to the functional integrals corresponding to the limit rectangle in \({\mathbb{R}}^{\nu -1}\). A simple example is the Gaussian measure of mean \(\lambda\in {\mathbb{R}}\) and covariance \((-\Delta +1)^{-1},\) where \(\Delta\) is the Laplacian on the given rectangle with periodic boundary conditions, whose characteristic functional on functions independent of one variable is simply related to the characteristic functional of the corresponding measure in lower dimension. A sufficient condition for dimensional reducibility is given for a class of functional measures absolutely continuous with respect to a Gaussian measure. The condition is shown to hold in \(P(\Phi)_{\nu}\) and ``exponential type'' models of Euclidean field theory, wherefrom limit theorems for the partition functions of the models in two-dimensional space-time follow.
exponential type models of Euclidean field theory
scientific article; zbMATH DE number 4049935
scientific article