Exact Solution for the Stationary Kardar-Parisi-Zhang Equation

We obtain the first exact solution for the stationary one-dimensional Kardar-Parisi-Zhang equation. A formula for the distribution of the height is given in terms of a Fredholm determinant, which is valid for any finite time $t$. The expression is explicit and compact enough so that it can be evaluated numerically. Furthermore, by extending the same scheme, we find an exact formula for the stationary two-point correlation function.

Figure 1

The two-sided BM initial condition and the two-point function.

Figure 2

Stationary height distribution for the KPZ equation for ${\gamma }_t=1$. The dashed curve is $F_0$. The dots indicate the Monte Carlo simulation for WASEP. We set the difference between left and right hopping rates 0.15 with their sum normalized to be unity, 3951 Monte Carlo steps (corresponding to ${\gamma }_t=1$), and 10 000 samples.

Figure 3

Stationary two-point correlation function $g_t^{\prime \prime }(y)$ for the KPZ equation for ${\gamma }_t=1$. The dashed curve represents the one in the scaling limit $g^{\prime \prime }(y)$. The dots indicates the Monte Carlo result of the WASEP, where parameters are the same as those in Fig. 2.