We consider a clean two-dimensional interacting electron gas subject to a random perpendicular magnetic field, h({bf r}). The field is nonquantizing, in the sense, that {cal N}_h-a typical flux into the area lambda_{text{tiny F}}^2 in the units of the flux quantum (lambda_{text{tiny F}} is the de Broglie wavelength) is small, {cal N}_hll 1. If the spacial scale, xi, of change of h({bf r}) is much larger than lambda_{text{tiny F}}, the electrons move along semiclassical trajectories. We demonstrate that a weak field-induced curving of the trajectories affects the interaction-induced electron lifetime in a singular fashion: it gives rise to the correction to the lifetime with a very sharp energy dependence. The correction persists within the interval omega sim omega_0= E_{text{tiny F}}{cal N}_h^{2/3} much smaller than the Fermi energy, E_{text{tiny F}}. It emerges in the third order in the interaction strength; the underlying physics is that a small phase volume sim (omega/E_{text{tiny F}})^{1/2} for scattering processes, involving {em two} electron-hole pairs, is suppressed by curving. Even more surprising effect that we find is that {em disorder-averaged} interaction correction to the density of states, delta u(omega), exhibits {em oscillatory} behavior, periodic in bigl(omega/omega_0bigr)^{3/2}. In our calculations of interaction corrections random field is incorporated via the phases of the Green functions in the coordinate space. We discuss the relevance of the new low-energy scale for realizations of a smooth random field in composite fermions and in disordered phase of spin-fermion model of ferromagnetic quantum criticality.
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