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

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.

💡 Research Summary

This paper addresses the challenging problem of probing large‑deviation statistics in the one‑dimensional Kardar‑Parisi‑Zhang (KPZ) universality class. While typical fluctuations of KPZ interfaces are now well understood—being governed by Tracy‑Widom distributions that depend on the initial condition—large deviations, where the height fluctuation scales as O(t) rather than the usual t^{1/3}, remain largely unexplored both analytically and numerically. The authors adopt the totally asymmetric simple exclusion process (TASEP) as a concrete microscopic model belonging to the KPZ class, and they apply a population‑dynamics (cloning) algorithm to sample rare events of the time‑integrated current at a single site (x=0).

The method works as follows. A fixed number N_cl of independent “clones” are initialized with the same configuration. At each discrete time step Δt=1 each clone evolves according to the TASEP hopping rule (particles hop rightward with unit rate if the target site is empty). The change Δh_i of the rescaled height variable δh = h − ρ(1−ρ)t (with density ρ=½) is recorded for each clone. A biased weight e^{kΔh_i} is assigned, where k is a tunable bias parameter. The partition function Z_t = Σ_i e^{kΔh_i} is computed, and each clone is replicated a number n_i = ⌊ e^{kΔh_i} Z_t^{-1} N_cl + η ⌋ (η∈