Abstract
We investigate the short-time regime of the KPZ equation in 1+11+1 dimensions and develop a unifying method to obtain the height distribution in this regime, valid whenever an exact solution exists in the form of a Fredholm Pfaffian or determinant. These include the droplet and stationary initial conditions in full space, previously obtained by a different method. The novel results concern the droplet initial condition in a half space for several Neumann boundary conditions: hard wall, symmetric, and critical. In all cases, the height probability distribution takes the large deviation form P(H,t) \sim \exp( - \Phi(H)/\sqrt{t})P(H,t)∼exp(−Φ(H)/t) for small time. We obtain the rate function \Phi(H)Φ(H) analytically for the above cases. It has a Gaussian form in the center with asymmetric tails, |H|^{5/2}|H|5/2 on the negative side, and H^{3/2}H3/2 on the positive side. The amplitude of the left tail for the half-space is found to be half the one of the full space. As in the full space case, we find that these left tails remain valid at all times. In addition, we present here (i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary condition and (ii) two Fredholm determinant representations for the solutions of the hard wall and the symmetric boundary respectively.
Highlights
Introduction to the cumulant methodThroughout this section we assume that the KPZ equation has been solved and yields for the moment generating function of the partition function the following Fredholm Pfaffian representation for z ≥ 0 − zα eH t (14)for some α > 0, χ > 0 and properly shifted height field H, where the set {ai}i∈ forms a Pfaffian point process [1 + zet1/3ai ]χ Pf J − σt,z K (15)with the 2 × 2 kernels K and J, the generalized Fermi factor σt,z being defined as J(r, r ) =01 −1 0 1r=r, K= K11 K21 K12 K22
In the case of the hard wall, we have obtained a new useful representation of the exact solution at all times in terms of a Fredholm Pfaffian with a matrix valued kernel which we show is equivalent to the solution of [33] expressed in terms a Fredholm determinant with a scalar valued kernel
We have developed a mathematical framework enabling to derive the short time properties of the large deviations of the distribution of the height of the KPZ solutions
Summary
Many recent works study the continuum KPZ equation in one dimension [1,2,3,4,5,6,7] which describes the stochastic growth of an interface parameterized by a height field h(x, t) at point x and time t as. A number of results have been obtained for the short time regime t 1 They all agree that the probability density function (PDF), P(H, t), of the properly shifted height at one space point x = 0, denoted H, takes the large deviation form log P(H, t). In this paper we use these exact solutions to establish (2) for the half-space problem and to calculate the large deviation rate function Φ(H) at short time for the above three cases. It exhibits a crossover between a cubic tail Φ−(z) z3/12 (matching the Tracy Widom distribution [38]) and a 5/2 tail exponent Φ−(z) z5/2 The latter can be readily obtained [36] using the method of truncation to the first cumulant described above, and is identical to the tail behavior at short time (a signature that the left tail remains identical at all times [41]). We establish that in all cases log P(H, t)
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