We note that for p = 0 we have | v$_{c}$ | = 1 identically throughout the high and low density regimes so that we simply have τ = L in this case. We also note that in the low-density (high-density) phase τ is independent of β ( α ). We now turn our attention to our Monte Carlo simu- lations. We simulated the parallel-update TASEP at a variety of values of α, β and p corresponding to both the low and high density phases, for system sizes L = 10 $^{3}$, 5 × 10 3 and 10 $^{4}$. Each simulation consisted of 10$^{4}$L/v$_{c}$ iterations, with the first 10 $^{3}$L/v$_{c}$ time-steps discarded to ensure negligible bias due to initial non-stationarity (initially the system was empty). Assuming the validity of (3), this implies we generated 1 . 8 × 10 $^{4}$τ$_{int}$$_{,n}$ samples of the stationary distribution in each simulation. For each simulation, we measured n at each iteration, and from the resulting time series we estimated the au- tocorrelation function ρ$_{n}$ ( t ) using the standard estima- tors [15]. Fig. 2 shows a finite-size scaling plot of ρ$_{n}$ ( t ) assuming the ansatz given by (1) with τ = L , in the p = 0 case. The agreement is clearly very good, and the sharp- ness of the cusp at t = L suggests that any corrections to the finite-support ansatz (1) are very small. 3.3. Simulations FIG. 2: Color online. Finite-size scaling plot of ρ n ( t ) for p = 0 parallel-update TASEP in the high-density and low-density phases, for L = 10 $^{3}$, 5 × 10 $^{3}$, 10 4 and a variety of α, β . Figs. 3 and 4 show finite-size scaling plots of ρ$_{n}$ ( t ) for p = 0 . 25 , 0 . 5, assuming the ansatz given by (1) and (25). There is again excellent data collapse, however we note that there is some noticeable curvature near the edge of the support, so that the sharp cusp present in the p = 0 case becomes smoothed out somewhat for p > 0. As discussed in section 2.2, this does not affect the use of (13) for setting Monte Carlo error bars, but it would be interesting from a theoretical perspective to better un- derstand how this curvature depends on the model pa- rameters p, α, β and L (as well as v$_{max}$ ; c.f. the discussion in section 4). We remark that many other quantities (in- An important generalization of the TASEP is the Nagel-Schreckenberg model [6], in which each particle (vehicle) can move up to v$_{max}$ ∈ N sites per itera- tion. Although the precise form of the phase diagram depends on v$_{max}$ , the NaSch model exhibits, in gen- eral, the same three qualitatively distinct phases as the TASEP [21]. We now briefly review the dynamical rules defining the NaSch model. Suppose at time t ∈ N a vehicle with speed v$_{t}$ ∈ { 0 , 1 , . . . , v$_{max}$ } is located on site x$_{t}$ , and has headway (number of empty sites to its FIG. 4: Color online. Finite-size scaling plot of ρ n ( t ) for p = 0 . 5 parallel-update TASEP in the high-density and low- density phases, for L = 10 $^{3}$, 5 × 10 $^{3}$, 10 4 and a variety of α, β . The choices of α, β shown correspond to four distinct values of v c providing strong evidence for the conjecture (25). 4. NAGEL-SCHRECKENBERG MODEL FIG. 3: Color online. Finite-size scaling plot of ρ n ( t ) for p = 0 . 25 parallel-update TASEP in the high-density and low- density p h ases, for L = 10 $^{3}$, 5 × 10 $^{3}$, 10 4 and a variety of α, β . The choic es of α, β shown correspond to four distinct values of v c prov iding strong evidence for the conjecture (25). cluding the fundamental diagram) have cusps at p = 0 which are smoothed out for p > 0. 5  0  0.2  0.4  0.6  0.8  1 0  0.5  1  1.5  2 n (t) |  v$_{c}$  |  t  L = 1000, p = 0.25,  = 0.10,   = 0.20 = 1000, p = 0.25,  = 0.15,   = 0.90 = 1000, p = 0.25,  = 0.40,   = 0.05 = 1000, p = 0.25,  = 0.80,   = 0.20 = 5000, p = 0.25,  = 0.10,   = 0.20 = 5000, p = 0.25,  = 0.15,   = 0.90 = 5000, p = 0.25,  = 0.40,   = 0.05 = 5000, p = 0.25,  = 0.80,   = 0.20 = 10000, p = 0.25,  = 0.10,   = 0.20 = 10000, p = 0.25,  = 0.15,   = 0.90 = 10000, p = 0.25,  = 0.40,   = 0.05 = 10000, p = 0.25,  = 0.80,   = 0.20  0  0.2  0.4  0.6  0.8  1  0  0.5  1  1.5  2 n (t) |  v$_{c}$  |  t  L = 1000, p = 0.5,  = 0.02,   = 0.10 = 1000, p = 0.5,  = 0.06,   = 0.90 = 1000, p = 0.5,  = 0.20,   = 0.08 = 1000, p = 0.5,  = 0.80,   = 0.04 = 5000, p = 0.5,  = 0.02,   = 0.10 = 5000, p = 0.5,  = 0.06,   = 0.90 = 5000, p = 0.5,  = 0.20,   = 0.08 = 5000, p = 0.5,  = 0.80,   = 0.04 = 10000, p = 0.5,  = 0.02,   = 0.10 = 10000, p = 0.5,  = 0.06,   = 0.90 = 10000, p = 0.5,  = 0.20,   = 0.08 = 10000, p = 0.5,  = 0.80,   = 0.04  0  0.2  0.4  0.6  0.8  1  0  0.5  1  1.5  2 n (t) t  L = 1000, p = 0,  = 0.25,   = 0.50 = 1000, p = 0,  = 0.50,   = 0.25 = 1000, p = 0,  = 0.50,   = 0.75 = 1000, p = 0,  = 0.75,   = 0.50 = 5000, p = 0,  = 0.25,   = 0.50 = 5000, p = 0,  = 0.50,   = 0.25 = 5000, p = 0,  = 0.50,   = 0.75 = 5000, p = 0,  = 0.75,   = 0.50 = 10000, p = 0,  = 0.25,   = 0.50 = 10000, p = 0,  = 0.50,   = 0.25 = 10000, p = 0,  = 0.50,   = 0.75 = 10000, p = 0,  = 0.75,   = 0.50