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tionary time series X$_{1}$, X$_{2}$, . . . , X$_{T}$ we can define its dis- crete Fourier transform to be 

with ω = 2 πm/T and m = 0 , 1 , . . . , T − 1. It is then straightforward to show [14] that for large T we have 

We now discuss the implications of the general form (1) on two key time scales, the integrated autocorrelation time and the exponential autocorrelation time. 

From ρ$_{X}$ ( t ) the integrated autocorrelation time is de- fined [15] as 

If X denotes the sample mean of X$_{1}$, X$_{2}$, . . . X$_{T}$ then the variance of X satisfies [15] 

It is (10) that accounts for the key role played by the integrated autocorrelation time in the statistical analy- sis of Markov-chain Monte Carlo time series. If instead of a correlated time series, one considers a sequence of independent random variables, then the variance of the sample mean is simply var( X ) /T . It is in this sense that τ$_{int}$$_{,X}$ determines how many time steps we need to wait between two “effectively independent” samples. 

It can now be seen immediately from (9) that, as noted in the introduction, (1) and (3) imply 

Equation (13) provides a very simple exact expression for τ$_{int}$$_{,n}$ in terms of the physical parameters of the model. It is quite rare to have such an expression for a non-trivial model. 

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## 2.1. Autocorrelation times 

## 2.1.1. Integrated Autocorrelation time 

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Typically, we expect that ρ$_{X}$ ( t ) ∼ exp( − t/τ$_{exp}$ ) as t → ∞ , which defines the exponential autocorrelation time τ$_{exp}$ . More precisely [15], one defines the exponen- tial autocorrelation time of observable X to be 

and then the exponential autocorrelation time of the sys- tem as 

where the supremum is taken over all observables X . The autocorrelation time τ$_{exp}$ measures the decay rate of the slowest mode of the system, and it therefore sets the scale for the number of initial time steps to discard from a simulation, in order to avoid bias from initial non- stationarity. All observables that are not orthogonal to this slowest mode satisfy τ$_{exp}$$_{,X}$ = τ$_{exp}$ . 

For the TASEP in continuous time, τ$_{exp}$ was computed analytically in [16, 17] using the exact Bethe Ansatz so- lution. In particular, it was found that τ$_{exp}$ is O (1) with respect to L in the high and low density phases. We would expect the same behavior to hold generally for the NaSch model. 

However, if ρ$_{n}$ ( t ) were to have strictly finite support as claimed in (1), then we would have −| t | / log ρ$_{n}$ ( t ) = 0 for all | t | > τ , implying that τ$_{exp}$$_{,n}$ ̸ = τ$_{exp}$ . This would then mean that n is orthogonal to the slowest relaxation mode, which seems implausible. We thus conclude that although (1) provides a very good approximation, ρ$_{n}$ ( t ) cannot actually have a strictly finite support. 

To obtain a more precise ansatz for ρ$_{n}$ ( t ) we therefore fix some k ∈ N satisfying k ≤ ⌊ τ ⌋ and set 

Since we know empirically that (1) is a very good approx- imation, it must be the case that k/τ ∼ 1 as τ → ∞ . Let us then write τ = k + ε , where the only assumption we make regarding ε is that ε/τ → 0 as τ → ∞ . Since the continuum limit of ρ ( x τ ) should define a continuous function of x ∈ R we choose the parameter B by de- manding that 1 − | t | /τ = Be −| t | τ$_{exp}$ when | t | = k , which yields 

## 2.1.2. Exponential Autocorrelation time 

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## 2.2. Finite-size scaling of ρ n ( t ) 

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