  In a scalar theory which we use as a simplified model for the Higgs sector,
we adopt the semiclassical formalism of Son for computations of $n$-particle
production cross-sections in the high-multiplicity $n\to \infty$ weak-coupling
$\lambda \to 0$ regime with the value of $\lambda n$ held fixed and large. The
approach relies on the use of singular classical solutions to a certain
boundary value problem. In the past this formalism has been successfully used
and verified in computations of perturbative multi-particle processes at
tree-level, and also at the next-to-leading order level in the small $\lambda
n$ expansion near the multi-particle mass threshold. We apply this singular
solutions formalism in the regime of ultra-high multiplicities where $\lambda n
\gg 1$, and compute the leading positive $\sim n\,\sqrt{\lambda n}$
contribution to the exponent of the multi-particle rate in this large $\lambda
n$ limit. The computation is carried out near the multi-particle mass threshold
where the multiplicity $n$ approaches its maximal value allowed by kinematics.
This calculation relies on the idea of Gorsky and Voloshin to use a thin wall
approximation for the singular solutions that resemble critical bubbles. This
approximation is justified in precisely the high-multiplicity $\sqrt{\lambda n}
\to \infty$ regime of interest. Based on our results we show that the scalar
theory with a spontaneous symmetry breaking used here as a simplified model for
the Higgs sector, is very likely to realise the high-energy Higgsplosion
phenomenon.
