Short-time large deviations of the spatially averaged height of a KPZ interface on a ring

Using the optimal fluctuation method, we evaluate the short-time probability distribution
P(H, L, t ¯ = T) of the spatially averaged height H¯ = (1/L)
R L
0
h(x, t = T) dx of a one-dimensional
interface h(x, t) governed by the Kardar–Parisi–Zhang equation
∂th = ν∂2
xh +
λ
2
(∂xh)
2 +
√
Dξ(x, t)
on a ring of length L. The process starts from a flat interface, h(x, t = 0) = 0. Both at λH <¯ 0, and
at sufficiently small positive λH¯ the optimal (that is, the least-action) path h(x, t) of the interface,
conditioned on H¯ , is uniform in space, and the distribution P(H, L, T ¯ ) is Gaussian. However, at
sufficiently large λH >¯ 0 the spatially uniform solution becomes sub-optimal and gives way to nonuniform optimal paths. We study them, and the resulting non-Gaussian distribution P(H, 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/√
νT,
at a critical value H¯ = H¯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.