The intrinsically nonperturbative “vacuum persistence probability” P for e + e− production in the overlap region of a pair of high-intensity lasers is estimated in the context of three models, each of which adapt and simplify the exact Fradkin representation for the logarithm of the fermion determinant in the fields of the crossed lasers. In each case, one finds for P an expression resembling Schwinger's 1951 result for the probability of pair production in a constant electric field, proportional to an exponential factor which contains an essential singularity and hence does not admit a perturbative expansion about zero coupling. Qualitative estimates of these models suggest that realistic yields for this form of e + e− production must await lasers of intensity 1029 W/m2. The possibility of producing a quark–antiquark pair in this way is noted, in particular, with temporary, but large separations of the qq.