Using the weak-noise theory, we evaluate the probability distribution P(H,t) of large deviations of height H of the evolving surface height h(x,t) in the Kardar-Parisi-Zhang equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height H at time t. We argue that the tails of P behave, at arbitrary time t>0, and in a proper moving frame, as -lnP∼|H|^{5/2} and ∼|H|^{3/2}. The 3/2 tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.

Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a "target": a spherical absorber of radius R. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability P(T) that no gas particle hits the target until a long but finite time T. We also find the most likely gas density history conditional on the nonhitting. The results depend on the dimension of space d and on the rescaled parameter ℓ=R/√[D(0)T], where D(0) is the gas diffusivity. For small ℓ and d>2, P(T) is determined by an exact stationary solution of the MFT equations that we find. For large ℓ, and for any ℓ in one dimension, the relevant MFT solutions are nonstationary. In this case, lnP(T) scales differently with relevant parameters, and it also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of noninteracting random walkers. Finally, we extend the formalism to a whole class of diffusive gases of interacting particles.

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Suppose that a d-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density n_{0}. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability P that no particles are absorbed during a long time T. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time T. As a result, P decays exponentially with T for a whole class of interacting diffusive gases in any dimension. For d=1 the stationary gas density profile and P can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that -lnP≃D_{0}TL^{d-2}s(n_{0}), where D_{0} is the gas diffusivity, and L is the linear size of the system. We calculate the rescaled action s(n_{0}) for d=1, for rectangular domains in d=2, and for spherical domains. Near close packing of the SSEP s(n_{0}) can be found analytically for domains of any shape and in any dimension.

The position of an invasion front, propagating into an unstable state, fluctuates because of the shot noise coming from the discreteness of reacting particles and stochastic character of the reactions and diffusion. A recent macroscopic theory [Meerson and Sasorov, Phys. Rev. E 84, 030101(R) (2011)] yields the probability of observing, during a long time, an unusually slow front. The theory is formulated as an effective Hamiltonian mechanics which operates with the density field and the conjugate "momentum" field. Further, the theory assumes that the most probable density field history of an unusually slow front represents, up to small corrections, a traveling front solution of the Hamilton equations. Here we verify this assumption by solving the Hamilton equations numerically for models belonging to the directed percolation universality class.