For a new class of Gaussian function space integrals depending upon n ∈ {1, 2,⋯}, the exponential rate of growth or decay as n → ∞ is determined. The result is applied to the calculation of the specific free energy in a model in statistical mechanics. The physical discussion is self-contained. The paper ends by proving upper bounds on certain probabilities. These bounds will be used in a sequel to this paper, in which asymptotic expansions and limit theorems will be proved for the Gaussian integrals considered here.