  We review the construction of a stationary Gaussian process $X(t)$ starting
from the near-critical continuum scaling limit $\Phi^h$ of the Ising
magnetization and its relation to the mass spectrum of the relativistic quantum
field theory associated to $\Phi^h$. Then for the near-critical Ising model on
$a \mathbb{Z}^2$ with external field $a^{15/8} h$, we study the renormalized
magnetization along a vertical line (with horizontal coordinate approximately
$t$) and prove that the limit as $a\downarrow 0$ is the same Gaussian process
$X(t)$. We also explore the possible extension of this approach to dimensions
$d > 2$.
