  The passage through a critical point of a many-body quantum system leads to
abundant nonadiabatic excitations. Here, we explore a regime, in which the
critical point is not crossed although the system is passing slowly very close
to it. We show that the leading exponent for the excitation probability then
can be obtained by standard arguments of the Dykhne formula but the exponential
prefactor is no longer simple, and behaves as a power law on the characteristic
transition rate. We derive this prefactor for the nonlinear Landau-Zener (nLZ)
model by adjusting the Dykhne's approach. Then, we introduce an exactly
solvable model of the transition near a critical point in the Stark ladder. We
derive the number of the excitations for it without approximations, and find
qualitatively similar results for the excitation scaling.
