Short-time large deviations of the spatially averaged height of a Kardar-Parisi-Zhang interface on a ring

Using the optimal fluctuation method, we evaluate the short-time probability distribution P ( H ˉ , L , t = T ) of the spatially averaged height H ˉ = ( 1 / L ) ∫ 0 L h ( x , t = T ) d x of a one-dimensional interface h ( x , t ) governed by the Kardar-Parisi-Zhang equation ∂ t h = ν ∂ x 2 h + λ 2 ∂ x h 2 + D ξ x , t on a ring of length L. The process starts from a flat interface, h ( x , t = 0 ) = 0 . Both at λ H ˉ < 0 and at sufficiently small positive λ H ˉ the optimal (that is, the least-action) path h ( x , t ) of the interface, conditioned on H ˉ , is uniform in space, and the distribution P ( H ˉ , L , T ) is Gaussian. However, at sufficiently large λ H ˉ > 0 the spatially uniform solution becomes sub-optimal and gives way to non-uniform optimal paths. We study these, and the resulting non-Gaussian distribution P ( H ˉ , L , T ) , analytically and numerically. The loss of optimality of the uniform solution occurs via a dynamical phase transition of either first or second order, depending on the rescaled system size ℓ = L / ν T , at a critical value H ˉ = H ˉ c ( ℓ ) . At large but finite ℓ the transition is of first order. Remarkably, it becomes an ‘accidental’ second-order transition in the limit of ℓ → ∞ , where a large-deviation behavior − ln P ( H ¯ , L , T ) ≃ ( L / T ) f ( H ¯ ) (in the units λ = ν = D = 1 ) is observed. At small ℓ the transition is of second order, while at ℓ = O ( 1 ) transitions of both types occur.