High-Precision Simulations of the Parity Conserving Directed Percolation Universality Class in 1+1 Dimensions

TL;DR

This study conducts high-statistics simulations of the parity-conserving directed percolation (pcDP) universality class in 1+1 dimensions to resolve long-standing questions about critical exponents and scaling. By analyzing both even and odd particle-number sectors across multiple observables, the authors provide strong evidence that the order-parameter exponent is exactly $\beta=1$ and the fractal dimension is $D_f=1/2$, while the correlation-time exponent $\nu_t$ is consistent with an irrational dynamical exponent $z\approx1.74$, linked by $\nu_t=2z\beta$ and $\nu=2\beta$. They extract precise estimates for $p_c$ and demonstrate that the system behaves like a branching process of sub-clusters at large scales, offering a plausible route to deep analytic understanding of pcDP beyond traditional DP universality. Overall, the work reveals a striking mix of near-rational exponents and an irrational $z$, with implications for the RG flow and possible solvable limits in absorbing-state transitions outside DP. $p_c \approx 0.450977(2)$, $\beta \approx 1$, $\nu \approx 2$, $\nu_t \approx 3.481(4)$, $D_f=1/2$, and $z=\nu_t/2 \approx 1.74$ are among the key consolidated findings, underscoring the unusual and rich critical behavior of pcDP in 1+1D.

Abstract

Next to the directed percolation (DP) universality class, parity conserving directed percolation (pcDP; also called parity conserving branching annihilating random walks, pcBARW) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP it is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent $β$ (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcPD clusters. In particular, we find that all exponents which were close to rationals are even closer, and $β= 1.000$ with the error in the next digit.

Figures (19)

• Figure 1: (Color online) Log-linear plots of $N_2(t)$ against $t$, for five values of $p$ close to $p_c$. $N_2(t)$ is the average number of active sites in runs starting with two active sites at positions $x=\pm 1$.
• Figure 2: (Color online) The same data as in the previous figure, but plotted with three correction terms with exponents -1, -2, and -3. Obviously these terms are sufficient to take into account all visible corrections to scaling.
• Figure 3: (Color online) Log-linear plot of $t^{0.2872}P_2(t)$ versus $t$, where $0.2872$ is the value of $\delta_2$ estimated in Park13. The dotted curve is an eyeball fit with a leading correction exponent $\beta_1=0.5$.
• Figure 4: (Color online) Log-linear plot of $t^{0.28718}N_1(t)$ versus $t$, with three correction terms added so that the central curve becomes flat for all $t$.
• Figure 5: (Color online) Log-linear plot of $t^{2/z}R^2_1(t)$ versus $t$, with four correction terms added so that the central curve becomes flat for all $t$.