A Gaussian process related to the mass spectrum of the near-critical Ising model

Abstract

Let $\Phi^h(x)$ with $x=(t,y)$ denote the near-critical scaling limit of the planar Ising magnetization field. We take the limit of $\Phi^h$ 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 is the Laplace transform of a mass spectral measure $\rho$ of the relativistic quantum field theory associated to the Euclidean field $\Phi^h$. Our analysis of the small distance/time behavior of the covariance functions of $\Phi^h$ and $X(t)$ shows that $\rho$ is finite but has infinite first moment.