Population dynamics simulations of large deviations for three subclasses of the Kardar-Parisi-Zhang universality class

TL;DR

This work tackles the problem of large deviations in the 1D KPZ universality class by applying a population dynamics (cloning) method to the totally asymmetric simple exclusion process (TASEP), enabling efficient sampling of rare, time-extensive height fluctuations. By analyzing step, flat, and stationary initial conditions, the authors validate exact large-deviation predictions for the step case and reveal new insights for the flat and stationary cases, including a robust negative-deviation regime linked to wedge-like interface profiles. The approach yields the cumulant generating function lambda(k,t) and its asymptotic scalings, demonstrating the method's ability to connect typical fluctuations (Tracy–Widom and Baik–Rains regimes) to large-deviation tails across different initial conditions. The results suggest that population dynamics is a versatile numerical (and potentially experimental) tool for exploring KPZ large deviations beyond analytically tractable cases.

Abstract

Recent theoretical studies have gradually deepened our understanding of the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) universality class even in the large deviation regime, but numerical methods for studying KPZ large deviations remain limited. Here we implement a method based on the population dynamics algorithm for studying large deviations of time-integrated local currents in the totally asymmetric simple exclusion process (TASEP), which is a pragmatic model in the 1D KPZ class. Carrying out simulations for the three representative initial conditions, namely step, flat, and stationary ones, we not only confirm theoretical predictions available for the step case, but also characterize large deviations for the flat and stationary cases which have not been investigated before. We reveal in particular an unexpected robustness of the deeply negative large deviation regime with respect to different initial conditions. We attribute this robustness to the spontaneous formation of a wedge shape in interface profile. Our population dynamics approach may serve as a versatile method for studying large deviations in the KPZ class numerically and, potentially, even experimentally.

Figures (5)

• Figure 1: Model and method. (a) Concept of the population dynamics algorithm. (b) The model TASEP. Particles hop stochastically to their right neighbor sites at a constant rate if the target sites are empty. (c)-(e) Three initial conditions considered in this work. The top and bottom panels show particle configurations and the corresponding interface profiles, respectively.
• Figure 2: Examples of time evolution of interface profiles for the step initial condition, for $k=-1$ (a), $k=0$ (b), and $k=1$ (c). Simulation parameters: $N=1024, N_\mathrm{cl} = 10000$.
• Figure 3: Simulation results for the step initial condition with $N=10000$ and $N_\mathrm{cl} = 10000$. (a)(b) The absolute value of CGF $|\lambda(k,t)|$ (a) and its local exponent $\frac{\mathrm{d} \ln |\lambda(k,t)|}{\mathrm{d} \ln t}$ (b) for negative bias $k<0$. In (a), simulation data (symbols) are compared with Johansson's exact solution for negative large deviations (solid lines), Eq. (\ref{['eq:step_mu-']}), or the prediction for typical fluctuations (dashed line), Eq. (\ref{['eq:step_lambda_typical']}). (c) The scaled CGF $\mu_-(k)$ for the negative tail. Simulation data of $\lambda(k,t)/t$ with $t=10000$ (symbols) are compared with the exact solution (line), Eq. (\ref{['eq:step_mu-']}). (d)(e) The CGF $\lambda(k,t)$ (d) and its local exponent $\frac{\mathrm{d} \ln \lambda(k,t)}{\mathrm{d} \ln t}$ (e) for positive bias $k>0$. In (d), simulation data (symbols) are compared with Johansson's exact solution for positive large deviations (solid lines), Eq. (\ref{['eq:step_mu+']}), or the prediction for typical fluctuations (dashed line), Eq. (\ref{['eq:step_lambda_typical']}). (f) The scaled CGF $\mu_+(k/t)$ for the positive tail. Simulation data of $\lambda(k,t)/t^2$ (symbols) are plotted against $k/t$ and compared with the exact solution (line), Eq. (\ref{['eq:step_mu+']}).
• Figure 4: Simulation results for the flat initial condition with $N=10000$ (unless otherwise stipulated) and $N_\mathrm{cl} = 10000$. (a)(b) The absolute value of CGF $|\lambda(k,t)|$ (a) and its local exponent $\frac{\mathrm{d} \ln |\lambda(k,t)|}{\mathrm{d} \ln t}$ (b) for negative bias $k<0$. In (a), simulation data for the flat initial condition (symbols) are compared with the exact solution for the step initial condition (solid lines), Eq. (\ref{['eq:step_mu-']}), or the prediction for typical fluctuations for the flat initial condition (dashed line), Eq. (\ref{['eq:flat_lambda_typical']}). (c) The scaled CGF $\mu_-(k)$ for the negative tail. Simulation data of $\lambda(k,t)/t$ with $t=10000$ for the flat initial condition (symbols) are compared with the exact solution for the step initial condition (line), Eq. (\ref{['eq:step_mu-']}). Inset: the ratio of $\lambda(k,t)/t$ for the flat initial condition to the exact solution of $\mu^\mathrm{step}_-(k)$ for the step initial condition, with $k=-0.215$ (blue), $-1$ (green), and $-10$ (red). (d) An example of time evolution of interface profiles for $k=-1$, here with $N=1024$. (e)(f) The CGF $\lambda(k,t)$ (e) and its local exponent $\frac{\mathrm{d} \ln \lambda(k,t)}{\mathrm{d} \ln t}$ (f) for positive bias $k>0$. In (e), simulation data for the flat initial condition (symbols) are compared with the exact solution for the step initial condition (solid lines), Eq. (\ref{['eq:step_mu+']}), or the prediction for typical fluctuations for the flat initial condition (dashed line), Eq. (\ref{['eq:flat_lambda_typical']}). (g) The scaled CGF $\mu_+(k/t)$ for the positive tail. Simulation data of $\lambda(k,t)/t^2$ for the flat initial condition (symbols) are plotted against $k/t$ and compared with the exact solution for the step initial condition (line), Eq. (\ref{['eq:step_mu+']}). Inset: the ratio of $\lambda(k,t)/t^2$ for the flat initial condition to the exact solution of $\mu^\mathrm{step}_+(k/t)$ for the step initial condition. The dashed and dotted lines are guides for the eyes indicating $1/\sqrt{2}$ and $1/2$, respectively. (h) An example of time evolution of interface profiles for $k=1$, here with $N=1024$.
• Figure 5: Simulation results for the stationary initial condition with $N=10000$ and $N_\mathrm{cl} = 10000$ from $1000$ realizations (unless otherwise stipulated). (a)(b) The absolute value of CGF $|\lambda(k,t)|$ (a) and its local exponent $\frac{\mathrm{d} \ln |\lambda(k,t)|}{\mathrm{d} \ln t}$ (b) for negative bias $k<0$. In (a), simulation data for the stationary initial condition (symbols) are compared with the exact solution for the step initial condition (solid lines), Eq. (\ref{['eq:step_mu-']}), or the prediction for typical fluctuations for the stationary initial condition (dashed line), Eq. (\ref{['eq:stat_lambda_typical']}). (c) The scaled CGF $\mu_-(k)$ for the negative tail. Simulation data of $\lambda(k,t)/t$ for the stationary initial condition (symbols) are compared with the exact solution for the step initial condition (line), Eq. (\ref{['eq:step_mu-']}). The values of $t$ used are $1000, 1000, 575, 184, 75$ from the leftmost to the rightmost datapoints. Inset: the ratio of $\lambda(k,t)/t$ for the stationary initial condition to the exact solution of $\mu^\mathrm{step}_-(k)$ for the step initial condition, with $k=-0.464$ (blue), $-1$ (green), and $-4.64$ (red). (d) An example of time evolution of interface profiles for $k=-1$, here with $N=1024$. (e)(f) The CGF $\lambda(k,t)$ (e) and its local exponent $\frac{\mathrm{d} \ln \lambda(k,t)}{\mathrm{d} \ln t}$ (f) for positive bias $k>0$. In (e), simulation data for the stationary initial condition (symbols) are compared with the exact solution for the step initial condition (solid lines), Eq. (\ref{['eq:step_mu+']}), or the prediction for typical fluctuations for the stationary initial condition (dashed line), Eq. (\ref{['eq:flat_lambda_typical']}). Inset: the scaled CGF $\mu_+(k/t)$ for the positive tail. Simulation data of $\lambda(k,t)/t^2$ for the stationary initial condition (symbols) are plotted against $k/t$ and compared with the exact solution for the step initial condition (line), Eq. (\ref{['eq:step_mu+']}). Inset: the ratio of $\lambda(k,t)/t^2$ for the stationary initial condition to the exact solution of $\mu^\mathrm{step}_+(k/t)$ for the step initial condition. (h) An example of time evolution of interface profiles for $k=1$, here with $N=1024$.