  The probabilities of various elementary laser - photon - electron/positron
interactions display in selected phase space and parameter regions typical
non-perturbative dependencies such as $\propto {\cal P} \exp\{- a E_{crit}
/E\}$, where ${\cal P}$ is a pre-exponential factor, $E_{crit}$ denotes the
critical Sauter-Schwinger field strength, and $E$ characterizes the (laser)
field strength. While the Schwinger process with $a = a_S \equiv \pi$ and the
non-linear Breit-Wheeler process in the tunneling regime with $a = a_{n \ell
BW} \equiv 4 m / 3 \omega'$ (with $\omega'$ the probe photon energy and $m$ the
electron/positron mass) are famous results, we point out here that also the
non-linear Compton scattering exhibits a similar behavior when focusing on high
harmonics. Using a suitable cut-off $c > 0$, the factor $a$ becomes $a = a_{n
\ell C} \equiv \frac23 c m /(p_0 + \sqrt{p_0^2 -m^2)}$. This opens the avenue
towards a new signature of the boiling point of the vacuum even for field
strengths $E$ below $E_{crit}$ by employing a high electron beam-energy $p_0$
to counter balance the large ratio $E_{crit} / E$ by a small factor $a$ to
achieve $E / a \to E_{crit}$. In the weak-field regime, the cut-off facilitates
a threshold leading to multi-photon signatures showing up in the total cross
section at sub-threshold energies.
