A Gaussian Process Related to the Mass Spectrum of the Near-Critical Ising Model

Let Φ ^h(x) with x= (t, y) denote the near-critical scaling limit of the planar Ising magnetization field. We take the limit of Φ ^h as the spatial coordinate y scales to infinity with t fixed and prove that it is a stationary Gaussian process X(t) whose covariance function K(t) is the Laplace transform of a mass spectral measure ρ of the relativistic quantum field theory associated to the Euclidean field Φ ^h. X and K should provide a useful tool for studying the mass spectrum; e.g., the small distance/time behavior of the covariance functions of Φ ^h and X(t) shows that ρ is finite but has infinite first moment.