Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth

We obtain several exact results for universal distributions involving the maximum of the ${\mathrm{Airy}}_2$ process minus a parabola and plus a Brownian motion, with applications to the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows one to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g., $C_{\infty }={lim}_{t_2/t_1\to +\infty }{\overline{h\left(t_1\right)h\left(t_2\right)}}^c/{\overline{h{\left(t_1\right)}^2}}^c$, with $C_{\infty }\approx 0.623$, as well as conditional moments, which quantify ergodicity breaking in the time evolution; (ii) in the same limit, the distribution of the midpoint position $x\left(t_1\right)$ of a directed polymer of length $t_2$; and (iii) the height distribution in stationary KPZ with a step. These results are derived from the replica Bethe ansatz for the KPZ continuum equation, with a “decoupling assumption” in the large time limit. They agree and confirm, whenever they can be compared, with (i) our recent tail results for two-time KPZ with the work by de Nardis and Le Doussal [J. Stat. Mech. (2017) 053212], checked in experiments with the work by Takeuchi and co-workers [De Nardis et al., Phys. Rev. Lett. 118, 125701 (2017)] and (ii) a recent result of Maes and Thiery [J. Stat. Phys. 168, 937 (2017)] on midpoint position.

Figure 1

Two directed polymers in a random potential with fixed end points, starting at $(x,0)$, ending, respectively, at $(0,t_1)$ and $(0,t_2)$. Minus their free energies maps to KPZ heights $h(0,t_1)$ and $h(0,t_2)$ with droplet initial condition centered at $x$. The PDF of the midpoint $y$, and the correlations between $h(0,t_1)$ and $h(0,t_2)$ are obtained here exactly in the limit $t_1,\mathrm{\Delta }=\frac{t_2-t_1}{t_1}\to +\infty$.

Figure 2

Conditional average ${\langle {\sigma }_2-{\sigma }_1\rangle }_{{\sigma }_1>{\sigma }_{1c}}$ ($y$ axis) as a function of ${\sigma }_{1c}$ ($x$ axis), which describes the averaged scaled KPZ height difference $h$ at large $\mathrm{\Delta }=\frac{t_2-t_1}{t_1}$ [see (37)]. (i) Orange: exact result obtained here. (ii) Blue: prediction obtained in [28, 29], which becomes exact for large positive ${\sigma }_{1c}$ [indistinguishable from (i) for ${\sigma }_{1c}>0]$. The limit ${\sigma }_{1c}\to -\infty$ (unconditioned mean), exact from (31) and to which our result (i) converges, is the orange horizontal line at $-{\kappa }_1^{\mathrm{GUE}}=1.77109$.

Figure 3

Conditional covariance ratio $C_{\infty }\left({\sigma }_{1c}\right)$, Eq. (39). Orange: asymptotic prediction for large positive ${\sigma }_{1c}$ from [28, 29]. Blue: exact result from (24), converging to (27) (orange horizontal line) for ${\sigma }_{1c}=-\infty$.