Having carefully investigated and compared temporal percolation statis-tics for several finite timesTi, parameter valuespj, and realizationsωk, we now turn to a larger ensemble of 400 realizations of the random walk on the half-line for several finite system sizes Ti up to Tmax = 104 and several parameter values in the critical region.
The averaged histograms of the return periods resemble power laws (Fig-ure12.9) up to the full simulation time.
The averaged finite-time temporal percolation statistics allow to visually localize the transition as the points of intersection of the percolation prob-ability Πij or the percolation strengthPij; whereas the peak in the average return time Mij,1 is not very pronounced (yet) and exhibits large variance (Figure 12.10). On the contrary, the combined-average finite return period M¯ ij,1 is numerically stable, as expected. However, algorithmic finite-size
12.3 full ensemble statistics 121
0.47 0.48 0.49 0.50 0.51 0.52 0.53
p
Figure 12.5:Finite-time statistics of a single realization ωk of the random walk on the half-line for several finite system sizes Ti and parameter val-uespj in the critical region. PanelA shows whether there is a span-ning cluster or not (Πijk, where “Yes” means Πijk = 1 and “No”
means Πijk = 0). Panel B shows the relative size of the largest re-turn period (including the truncated last rere-turn period)Pijk. PanelC shows the relative number of returns per time stepLijk/Ti. For com-parison, Panel D shows the relative size of the largest finite return period (excluding the truncated and possibly infinite last return pe-riod)Pijk0 = T1
imaxl6Lijkτijk,l. Finally, PanelEshows the mean finite return periodMijk,1/Lijk. Lines are a guide for the eyes.
122 connecting the dots
0 5000 10000
n 0
100 200 300 400
Xn
p=0.49
ω1 ω2 ω3 ω4 ω5 ω6
0 5000 10000
n p=0.5
0 5000 10000
n p=0.51
Figure 12.6: A small ensemble of6realizationsωkof the random walk on the half-line at several values of the parameterpin the critical region. Upper panels: Finite-time trajectories. Lower panels: Vertical bars colored as the trajectory of the respective realizationωkhighlight returns to the origin.
0 2000 4000 6000 8000 10000
n 0
50 100 150 200
hXni
p=0.51 p=0.5032 p=0.5 p=0.4968 p=0.49
Figure 12.7: Average positionhXni= K1P
kXn(ωk)of a small ensemble ofK= 6 realizations of the random walk on the half-line at several values of the parameter p in the critical region (see Figure 12.6 for the individual trajectories)
12.3 full ensemble statistics 123
0.40 0.45 0.50 0.55 0.60 p
0.40 0.45 0.50 0.55 0.60 p
Figure 12.8:Finite-time statistics for each runωk of a small ensemble ofK=6 re-alizations of the random walk on the half-line at several values of the parameterpj in the critical region (see Figure12.6 for the individual trajectories) for finite timeTi=105. PanelAshows whether there is a spanning cluster or not (Πijk). PanelBshows the relative size of the largest return period (including the truncated last return period)Pijk. Panel C shows the relative number of returns per time stepLijk/Ti. For comparison, PanelDshows the relative size of the largest finite re-turn period (excluding the truncated and possibly infinite last rere-turn period) Pijk0 = T1
i maxl6Lijkτijk,l. Finally, Panel E shows the mean finite return periodMijk,1/Lijkfor each realizationωk. Note the mag-nification in thexscale as compared to the upper panels. Lines are a guide for the eyes.
124 connecting the dots
101 102 103
return periods 10−1
100 101 102
averagefrequency
p=0.4900 p=0.4964 p=0.5000 p=0.5036 p=0.5100
Figure 12.9: Average histograms of the return periods of finite-time trajectories of an ensemble of 400 realizations of the random walk on the half-line over a simulation time of T = 104 for multiple parameter values p in the critical region. Each data point is the average of the number of return times in the respective bin over all realizations. Both axes, the bin edges and the frequencies, are log-scaled. The data points are centered at the geometric mean of the bin edges. Missing errorbars are smaller than the marker at the respective data point. The lines are a guide for the eyes.
12.3 full ensemble statistics 125
Figure 12.10:Average finite-time temporal percolation statistics of an ensemble of 400realizations of the random walk on the half-line for several finite system sizesTi and parameter valuespjin the critical region. Panel Ashows the percolation probabilityΠij. PanelBshows the percola-tion strengthPij(the average relative size of the largest return period, including the truncated last return period). PanelCshows the prob-ability of no return ¯Lij. For comparison, PanelD shows the average relative size of the largest finite return periodPij0 (excluding the trun-cated and possibly infinite last return period). Finally, PanelEshows the average first raw moment, or the average finite return period, Mij,1. For comparison, PanelFshows the combined-average finite re-turn period, ¯Mij,1. Missing errorbars are smaller than the marker at the respective data point. Lines are a guide for the eyes.
126 connecting the dots
Table 12.13:Results of the algorithmic finite-time analysis of temporal percolation statistics of a random walk on the half-line in 400 runs up to finite simulation timeTmax = 106. The analysis yields the critical pointpc, the critical exponentνof the temporal coherence scale and the critical exponentβ=ζof the percolation strengthPand the critical exponent γ=ζof the average return time ¯τ, and their respective errors. For the scaling procedure, finite-time data atTi=105.0,105.5,106.0were used.
P τ¯
pc 0.4999 0.4999 dpc 0.0002 0.0013 ν 2.0407 2.0651 dν 0.1785 0.2943 ζ −0.0001 1.0023 dζ 0.0050 0.2337
scaling analysis needs larger system sizes to be reliable, and in particular, it needs more pronounced peak in the susceptibility (average return time).