  In rigidly supersymmetric quantum theories, the Nicolai map allows one to
turn on a coupling constant (from zero to a finite value) by keeping the (free)
functional integration measure but subjecting the fields to a particular
nonlocal and nonlinear transformation. A recursive perturbative construction of
the Nicolai-transformed field configuration expresses it as a power series in
the coupling, with its coefficient function at order $n$ being a sum of
particular tree diagrams. For a quantum-mechanical example, the size of these
tree diagrams (under a certain functional norm) is estimated by the $(n{+}1)$st
power of the field size, and their number grows like
$n^{-3/2}\times4.967^{\;n}$. Such an asymptotic behavior translates to a finite
convergence radius for the formal perturbative expansion of the Nicolai map,
which establishes its non-perturbative existence. The known factorial growth of
the number of Feynman diagrams for quantum correlators is reproduced by the
combinatorics of free-field Wick contractions as usual. We expect our results
to extend to higher dimensions, including super Yang-Mills theory.
