Simulation: ASEP and the KPZ Fixed Point

The plot shows one realization after subtracting the ensemble-mean height profile. With a fixed observation horizon T, time is scaled by T, space by T^2/3, and height fluctuations by T^1/3.

What this simulation shows

ASEP

The asymmetric simple exclusion process is a continuous-time particle system on a one-dimensional lattice. Each particle has an independent exponential clock of rate 1. When its clock rings, it attempts to jump right with probability p or left with probability 1 − p; the attempted jump occurs only if the neighboring destination site is unoccupied. The simulation uses random sequential Monte Carlo updates as a discrete approximation to this clock dynamics.

Step initial condition

On the infinite lattice, step initial data means η_i(0) = 1 for i ≤ 0 and η_i(0) = 0 for i > 0: every site on the left is occupied and every site on the right is empty. This produces a single particle–hole interface, whose evolution is the standard curved, or droplet, KPZ geometry. The finite periodic simulation approximates this by filling one contiguous half of the ring with particles and leaving the other half empty. The second interface created by periodicity is kept far from the displayed central one over the short time window shown.

Height function

For occupation variable η_i, the associated interface is defined by h(i+1) − h(i) = 1 − 2η_i. Empty sites therefore make upward steps and occupied sites make downward steps. The plotted height is centered by subtracting its ensemble mean.

Connection with the KPZ fixed point

ASEP belongs to the KPZ universality class. For this rough visualization, with model-dependent constants set to 1, the displayed field is H_T(x,s) = [H(T^2/3x, Ts) − E H(T^2/3x, Ts)] / T^1/3. Thus the scaling is 1:2:3: time is measured on scale T, space on scale T^2/3, and fluctuations on scale T^1/3. Here, the expectation is estimated by averaging several independent copies. The exponents agree with TASEP; the precise multiplicative constants depend on the ASEP bias and local density. This remains a finite numerical approximation, not an exact KPZ-fixed-point sample.

Controls

Copies sets the number of independent runs used to estimate the mean profile.
Right probability sets the ASEP bias. A value of 1 gives TASEP; a value close to 1/2 is nearly symmetric.
Speed sets how many Monte Carlo sweeps occur between visual updates.
Pause / Run freezes or resumes the evolution.
Restart returns every copy to the step initial condition.

Color scale: blue indicates negative fluctuations, white is near zero, and orange to red indicates positive fluctuations.

Selected references

T. Liggett, Interacting Particle Systems, Springer, 1985.
C. Tracy and H. Widom, Asymptotics in ASEP with step initial condition, Communications in Mathematical Physics 290 (2009), 129–154.
I. Corwin, The Kardar–Parisi–Zhang equation and universality class, Random Matrices: Theory and Applications 1 (2012).
K. Matetski, J. Quastel, and D. Remenik, The KPZ fixed point, Acta Mathematica 227 (2021), 115–203.
A. Aggarwal, I. Corwin, and M. Hegde, KPZ fixed point convergence of the ASEP and stochastic six-vertex models, 2024.