= Large-deviation decay rates of a Gaussian scalar packet
{title2=$\gamma_\mu=\inf_{s\ge0}[\mathcal I(s)+\mu s]$}
After <diffusion arrest of Gaussian packet compression>, define the logarithmic area increase $A_t=\tfrac12\log[\det J(t)/\det J(0)]$. It is nonnegative. Assume $A_t/t$ has a <large deviation principle> with convex <rate function> $\mathcal I$, minimized at $\lambda>0$. The exact amplitude relation $f(t)=f(0)e^{-A_t}$ and the <Laplace principle for probability measures> give the displayed moment decay rate. The rate function refers to area, so rare width fluctuations are included rather than discarded without justification. For differentiable strictly convex $\mathcal I$, the interior minimizer satisfies $\mathcal I'(s_\mu)=-\mu$. If $\mu_*=-\mathcal I'(0)$ is finite, then $\gamma_\mu=\mathcal I(0)$ for $\mu\ge\mu_*$: rare nearly unstretched packets dominate high moments. Below the threshold, $d(\gamma_\mu/\mu)/d\mu=-\mathcal I(s_\mu)/\mu^2<0$. These statements concern a positive Gaussian packet with flow-only averaging; a homogeneous signed random initial scalar requires a different averaging kernel. Arbitrary random $C(t)$ alone does not imply this statistical model.
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