SITE PERCOLATION ON THE SQUARE LATTICE: MEASURING p_c ON A LAPTOP Science Journaling Club, Volume 1 Issue 2, Winter 2025 python 3.12.3 | numpy 2.4.2 master seed = 20251215 | worker processes = 8 accepted value for square-lattice site percolation: p_c = 0.59274605 ============================================================================== VALIDATION A. UNION-FIND AGAINST AN INDEPENDENT FLOOD FILL ============================================================================== Random configurations, occupation probability drawn uniformly from [0.30, 0.80]. Both the spanning verdict and the largest cluster size must agree exactly. A single disagreement invalidates everything downstream. L configs span mismatches S_max mismatches 4 4000 0 0 6 3000 0 0 8 2000 0 0 12 1200 0 0 16 800 0 0 24 400 0 0 total mismatches over 11400 configurations: 0 ============================================================================== VALIDATION B. THE INCREMENTAL SWEEP LANDS ON THE EXACT CROSSING SITE ============================================================================== For each sweep we rebuild the configuration from the first n_span sites and ask the flood fill whether it spans (it must), then remove the last site and ask again (it must not). This tests the sweep, the flag bookkeeping and the path compression together. L sweeps failures 8 400 0 16 300 0 24 150 0 Validations A and B together: 0 failures. ============================================================================== MONTE CARLO CAMPAIGN ============================================================================== Newman-Ziff sweeps. One sweep = one uniformly random permutation of all L*L sites, added one at a time, recording the occupancy at which the lattice first spans and the largest cluster at occupancy round(p_c * L*L). Every sweep contributes to every point of that lattice's crossing curve, so the sweep count is the sample size at every p. L N sweeps n_target wall(s) sweeps/s 16 256 40000 152 2.1 18922.0 32 1024 40000 607 9.9 4020.2 64 4096 20000 2428 21.7 921.2 128 16384 8000 9712 35.8 223.7 256 65536 5000 38846 95.5 52.3 512 262144 1200 155385 106.5 11.3 1024 1048576 320 621539 128.4 2.5 ============================================================================== MICROCANONICAL RESULT: OCCUPANCY AT FIRST SPANNING ============================================================================== n_span / N is the fraction of sites present when the lattice first crosses. Its mean is itself an estimator of p_c, with the same finite-size correction as the crossing-probability estimator but a different amplitude. L sweeps mean n_span/N s.d. std.err min/N max/N 16 40000 0.591367 0.055587 0.000278 0.3320 0.8047 32 40000 0.592451 0.035546 0.000178 0.4385 0.7500 64 20000 0.592614 0.022005 0.000156 0.4678 0.6965 128 8000 0.592524 0.013647 0.000153 0.5372 0.6537 256 5000 0.592879 0.008248 0.000117 0.5605 0.6212 512 1200 0.592684 0.004857 0.000140 0.5751 0.6093 1024 320 0.592797 0.003247 0.000181 0.5851 0.6035 ============================================================================== CANONICAL CROSSING CURVES R_L(p) ============================================================================== Exact binomial convolution of the microcanonical curve. The window of n used for the convolution must carry essentially all of the binomial mass at every p evaluated; the 'mass' column is the worst case seen for that L. L window width worst binomial mass in window R(pmin) R(pmax) 16 257 1.000000000000 0.00000 1.00000 32 886 1.000000000000 0.00000 1.00000 64 2057 0.999999999998 0.00000 1.00000 128 4550 1.000000000000 0.00000 1.00000 256 10140 0.999999999963 0.00000 1.00000 512 22757 0.999999999939 0.00000 1.00000 1024 51411 1.000000000000 0.00000 1.00000 ============================================================================== VALIDATION C. BRUTE-FORCE ENUMERATION ON TINY LATTICES ============================================================================== Every configuration of a 3x3 and a 4x4 lattice is enumerated (512 and 65536 of them). That gives the crossing probability as an exact polynomial in p. The Monte Carlo pipeline, run end to end on the same lattice sizes with 200000 sweeps each, must reproduce it. L p exact R(p) Monte Carlo R(p) diff diff/s.e. 3 0.40000 0.215363584 0.215407987 4.44e-05 0.05 3 0.50000 0.384765625 0.384646836 -1.19e-04 -0.11 3 0.59275 0.566703307 0.566421794 -2.82e-04 -0.25 3 0.70000 0.769564747 0.769227055 -3.38e-04 -0.36 3 0.80000 0.912621568 0.912404057 -2.18e-04 -0.34 4 0.40000 0.155794339 0.156059324 2.65e-04 0.33 4 0.50000 0.338638306 0.338653240 1.49e-05 0.01 4 0.59275 0.555588036 0.555268042 -3.20e-04 -0.29 4 0.70000 0.795369473 0.794915411 -4.54e-04 -0.50 4 0.80000 0.941359047 0.941100990 -2.58e-04 -0.49 ============================================================================== VALIDATION D. CROSSING PROBABILITY AT THE ACCEPTED THRESHOLD ============================================================================== If the crossing probability at p_c converges to a size-independent number, that is the universal crossing probability. For a square (aspect ratio 1) Cardy's formula gives exactly 1/2 in the continuum limit. L R_L(p_c) std.err R_L(p_c) - 0.5 in s.e. 16 0.516484 0.002499 0.016484 6.60 32 0.505076 0.002500 0.005076 2.03 64 0.502637 0.003535 0.002637 0.75 128 0.509643 0.005589 0.009643 1.73 256 0.500540 0.007071 0.000540 0.08 512 0.502864 0.014434 0.002864 0.20 1024 0.479735 0.027928 -0.020265 -0.73 ============================================================================== FINITE-SIZE THRESHOLD ESTIMATES p*(L) ============================================================================== p*(L) is defined by R_L(p*) = 1/2, found by bisection on the convolved curve. The error bar is a bootstrap over sweeps: 400 resamples of the sweep set at each L, with the whole convolution and bisection redone each time. L sweeps p*(L) boot s.e. L^(-1/nu) p*(L) - p_c 16 40000 0.5901536 0.0002921 0.125000 -0.0025925 32 40000 0.5922597 0.0002009 0.074325 -0.0004864 64 20000 0.5925953 0.0001707 0.044194 -0.0001507 128 8000 0.5924174 0.0001661 0.026278 -0.0003286 256 5000 0.5927350 0.0001301 0.015625 -0.0000111 512 1200 0.5927104 0.0001711 0.009291 -0.0000357 1024 320 0.5929042 0.0002065 0.005524 0.0001582 ============================================================================== FINITE-SIZE SCALING EXTRAPOLATION TO INFINITE LATTICE ============================================================================== Model: p*(L) = p_c + a * L^(-1/nu), with 1/nu = 3/4 held fixed at the exact two-dimensional value. Weighted least squares, weights 1/s.e.^2. lattices used p_c estimate slope a chi2 dof L >= 16 (7 points) 0.5930250 -0.016807 18.04 5 L >= 32 (6 points) 0.5928171 -0.007456 2.41 4 L >= 64 (5 points) 0.5928147 -0.007317 2.41 3 Bootstrapping the extrapolation: each of the 400 replicates resamples the sweeps at every L independently, re-solves p*(L) everywhere, and refits. lattices used p_c estimate boot s.e. 95% interval L >= 16 0.5930250 0.0000955 [0.5928244, 0.5931933] L >= 32 0.5928171 0.0001110 [0.5925836, 0.5930203] L >= 64 0.5928147 0.0001341 [0.5925659, 0.5930704] ============================================================================== THE HEADLINE NUMBER ============================================================================== Club estimate (lattices L = 32, 64, 128, 256, 512, 1024, 1/nu = 3/4 fixed): p_c (club) = 0.5928171 +/- 0.0001110 (1 s.e., bootstrap) p_c (accepted) = 0.59274605 difference = +0.0000710 difference = +0.64 of the club's own error bars relative precision of the club estimate: 1.87e-04 decimal places agreed with the accepted value: 4 For comparison, the same arithmetic applied to the microcanonical estimator (mean occupancy at first spanning) rather than the R = 1/2 criterion: p_c (club, mean n_span/N route) = 0.5928214 difference +0.0000754 slope a = -0.005143, chi2/dof = 3.0/4 ============================================================================== LETTING THE EXPONENT FLOAT ============================================================================== Same data, same fit, but now 1/nu is scanned instead of fixed. The chi-square is minimised on a grid; the reported nu is 1/(best exponent). best 1/nu = 0.300 (exact value 0.750) implied nu = 3.333 (exact value 1.3333) p_c at that exponent = 0.5931315 chi2 at minimum = 2.065 The chi-square surface is shallow in this direction. A laptop-sized data set constrains p_c far better than it constrains nu, because p_c is the intercept and nu only reshapes the approach to it. 1/nu p_c from that exponent chi2 0.60 0.5928692 2.269 0.65 0.5928491 2.313 0.70 0.5928319 2.360 0.75 0.5928171 2.408 0.80 0.5928041 2.456 0.85 0.5927928 2.506 0.90 0.5927827 2.556 1.00 0.5927658 2.655 ============================================================================== VALIDATION F. LARGEST CLUSTER AT THRESHOLD ============================================================================== S_max is measured at fixed occupancy n = round(p_c * N). Theory says S_max ~ L^D with D = 91/48 = 1.8958333 exactly. L mean S_max std.err S_max/N S_max / L^(91/48) 16 90.06 0.13 0.351786 0.469578 32 319.89 0.51 0.312393 0.448216 64 1167.03 2.74 0.284920 0.439406 128 4309.31 16.34 0.263019 0.436002 256 15812.27 77.50 0.241276 0.429905 512 58769.00 594.08 0.224186 0.429362 1024 216149.46 4420.68 0.206136 0.424353 log-log weighted fit over all seven sizes: D (club) = 1.86068 D (exact) = 1.8958333 (91/48) difference = -0.03515 chi2 / dof = 160.4 / 5 dropping L = 16 and 32, where corrections to scaling bite hardest: D (club) = 1.88231 difference -0.01352 chi2/dof 0.8/3 ============================================================================== CONVERGENCE: WHAT THE ANSWER LOOKED LIKE AFTER m SWEEPS ============================================================================== Using only the first m sweeps at every lattice size (capped at what that size actually has), the whole pipeline is re-run: convolution, bisection, refit. The error bar is a 100-replicate bootstrap of that reduced data set. m p*(256) at m p_c(m) [L>=32] boot s.e. p_c(m) - accepted 50 0.5910267 0.5939574 0.0006601 0.0012114 100 0.5910476 0.5927641 0.0004677 0.0000181 200 0.5917502 0.5929494 0.0003237 0.0002034 320 0.5921797 0.5929017 0.0002714 0.0001556 500 0.5923281 0.5930249 0.0002163 0.0002788 1000 0.5923746 0.5928929 0.0001862 0.0001469 2000 0.5926621 0.5928957 0.0001491 0.0001496 5000 0.5927350 0.5928546 0.0001132 0.0001085 20000 0.5927350 0.5927967 0.0001100 0.0000507 40000 0.5927350 0.5928037 0.0001038 0.0000576 ============================================================================== FIGURE DATA (machine readable) ============================================================================== # CURVE L then 41 pairs p,R CURVE 16 0.252746,0.00000 0.269746,0.00000 0.286746,0.00000 0.303746,0.00002 0.320746,0.00005 0.337746,0.00011 0.354746,0.00027 0.371746,0.00060 0.388746,0.00132 0.405746,0.00281 0.422746,0.00573 0.439746,0.01117 0.456746,0.02081 0.473746,0.03697 0.490746,0.06253 0.507746,0.10059 0.524746,0.15384 0.541746,0.22384 0.558746,0.31014 0.575746,0.40962 0.592746,0.51648 0.609746,0.62314 0.626746,0.72181 0.643746,0.80628 0.660746,0.87320 0.677746,0.92230 0.694746,0.95568 0.711746,0.97663 0.728746,0.98869 0.745746,0.99499 0.762746,0.99797 0.779746,0.99925 0.796746,0.99975 0.813746,0.99993 0.830746,0.99999 0.847746,1.00000 0.864746,1.00000 0.881746,1.00000 0.898746,1.00000 0.915746,1.00000 0.932746,1.00000 CURVE 32 0.354905,0.00000 0.366797,0.00000 0.378689,0.00000 0.390581,0.00000 0.402473,0.00000 0.414365,0.00000 0.426257,0.00001 0.438149,0.00005 0.450041,0.00019 0.461933,0.00057 0.473825,0.00160 0.485717,0.00399 0.497609,0.00901 0.509502,0.01881 0.521394,0.03658 0.533286,0.06653 0.545178,0.11341 0.557070,0.18142 0.568962,0.27228 0.580854,0.38295 0.592746,0.50508 0.604638,0.62711 0.616530,0.73770 0.628422,0.82872 0.640314,0.89682 0.652206,0.94293 0.664098,0.97103 0.675991,0.98641 0.687883,0.99405 0.699775,0.99756 0.711667,0.99907 0.723559,0.99966 0.735451,0.99988 0.747343,0.99996 0.759235,0.99999 0.771127,1.00000 0.783019,1.00000 0.794911,1.00000 0.806803,1.00000 0.818695,1.00000 0.830587,1.00000 CURVE 64 0.451325,0.00000 0.458396,0.00001 0.465467,0.00002 0.472538,0.00004 0.479609,0.00005 0.486680,0.00007 0.493751,0.00011 0.500822,0.00019 0.507893,0.00039 0.514964,0.00086 0.522035,0.00196 0.529106,0.00434 0.536178,0.00925 0.543249,0.01874 0.550320,0.03620 0.557391,0.06619 0.564462,0.11312 0.571533,0.18081 0.578604,0.27110 0.585675,0.38107 0.592746,0.50264 0.599817,0.62441 0.606888,0.73443 0.613959,0.82447 0.621030,0.89178 0.628101,0.93805 0.635172,0.96732 0.642244,0.98406 0.649315,0.99273 0.656386,0.99687 0.663457,0.99871 0.670528,0.99949 0.677599,0.99980 0.684670,0.99989 0.691741,0.99993 0.698812,0.99997 0.705883,0.99999 0.712954,1.00000 0.720025,1.00000 0.727096,1.00000 0.734167,1.00000 CURVE 128 0.508656,0.00000 0.512861,0.00000 0.517065,0.00000 0.521270,0.00000 0.525474,0.00000 0.529679,0.00000 0.533883,0.00003 0.538088,0.00010 0.542292,0.00030 0.546497,0.00091 0.550701,0.00241 0.554906,0.00545 0.559110,0.01101 0.563315,0.02096 0.567519,0.03872 0.571724,0.06966 0.575928,0.11813 0.580133,0.18636 0.584337,0.27677 0.588542,0.38755 0.592746,0.50964 0.596951,0.62950 0.601155,0.73522 0.605359,0.82079 0.609564,0.88600 0.613768,0.93326 0.617973,0.96424 0.622177,0.98189 0.626382,0.99092 0.630586,0.99555 0.634791,0.99787 0.638995,0.99896 0.643200,0.99953 0.647404,0.99980 0.651609,0.99990 0.655813,0.99996 0.660018,0.99999 0.664222,1.00000 0.668427,1.00000 0.672631,1.00000 0.676836,1.00000 CURVE 256 0.542746,0.00000 0.545246,0.00000 0.547746,0.00000 0.550246,0.00000 0.552746,0.00000 0.555246,0.00000 0.557746,0.00002 0.560246,0.00011 0.562746,0.00030 0.565246,0.00072 0.567746,0.00202 0.570246,0.00498 0.572746,0.01007 0.575246,0.01879 0.577746,0.03479 0.580246,0.06423 0.582746,0.11080 0.585246,0.17891 0.587746,0.27076 0.590246,0.38021 0.592746,0.50054 0.595246,0.61861 0.597746,0.72291 0.600246,0.80915 0.602746,0.87809 0.605246,0.92711 0.607746,0.95849 0.610246,0.97838 0.612746,0.98936 0.615246,0.99454 0.617746,0.99732 0.620246,0.99919 0.622746,0.99989 0.625246,0.99999 0.627746,1.00000 0.630246,1.00000 0.632746,1.00000 0.635246,1.00000 0.637746,1.00000 0.640246,1.00000 0.642746,1.00000 CURVE 512 0.563016,0.00000 0.564502,0.00000 0.565989,0.00000 0.567475,0.00000 0.568962,0.00000 0.570448,0.00000 0.571935,0.00000 0.573421,0.00004 0.574908,0.00040 0.576394,0.00112 0.577881,0.00185 0.579367,0.00328 0.580854,0.00658 0.582340,0.01837 0.583827,0.04070 0.585314,0.07212 0.586800,0.12018 0.588287,0.19038 0.589773,0.27776 0.591260,0.38283 0.592746,0.50286 0.594233,0.61414 0.595719,0.71860 0.597206,0.81785 0.598692,0.89038 0.600179,0.93957 0.601665,0.97013 0.603152,0.98530 0.604638,0.99340 0.606125,0.99757 0.607611,0.99845 0.609098,0.99923 0.610584,0.99991 0.612071,1.00000 0.613557,1.00000 0.615044,1.00000 0.616530,1.00000 0.618017,1.00000 0.619503,1.00000 0.620990,1.00000 0.622476,1.00000 CURVE 1024 0.575068,0.00000 0.575952,0.00000 0.576836,0.00000 0.577720,0.00000 0.578604,0.00000 0.579488,0.00000 0.580372,0.00000 0.581256,0.00000 0.582139,0.00000 0.583023,0.00000 0.583907,0.00002 0.584791,0.00133 0.585675,0.01190 0.586559,0.02876 0.587443,0.05464 0.588327,0.09409 0.589211,0.15904 0.590094,0.21991 0.590978,0.28902 0.591862,0.37588 0.592746,0.47973 0.593630,0.59011 0.594514,0.69246 0.595398,0.78536 0.596282,0.85400 0.597165,0.91160 0.598049,0.94938 0.598933,0.97368 0.599817,0.98679 0.600701,0.99421 0.601585,0.99676 0.602469,0.99692 0.603353,0.99798 0.604237,0.99978 0.605120,1.00000 0.606004,1.00000 0.606888,1.00000 0.607772,1.00000 0.608656,1.00000 0.609540,1.00000 0.610424,1.00000 # SIZE L sweeps pstar se meanNspanFrac se R_at_pc se meanSmax se SIZE 16 40000 0.5901536 0.0002921 0.5913669 0.0002779 0.516484 0.002499 90.057 0.134 SIZE 32 40000 0.5922597 0.0002009 0.5924514 0.0001777 0.505076 0.002500 319.890 0.509 SIZE 64 20000 0.5925953 0.0001707 0.5926144 0.0001556 0.502637 0.003535 1167.030 2.741 SIZE 128 8000 0.5924174 0.0001661 0.5925241 0.0001526 0.509643 0.005589 4309.307 16.335 SIZE 256 5000 0.5927350 0.0001301 0.5928794 0.0001166 0.500540 0.007071 15812.266 77.504 SIZE 512 1200 0.5927104 0.0001711 0.5926844 0.0001402 0.502864 0.014434 58769.000 594.078 SIZE 1024 320 0.5929042 0.0002065 0.5927966 0.0001815 0.479735 0.027928 216149.462 4420.680 # CONV m pstar256 pc se pc-accepted CONV 50 0.5910267 0.5939574 0.0006601 0.0012114 CONV 100 0.5910476 0.5927641 0.0004677 0.0000181 CONV 200 0.5917502 0.5929494 0.0003237 0.0002034 CONV 320 0.5921797 0.5929017 0.0002714 0.0001556 CONV 500 0.5923281 0.5930249 0.0002163 0.0002788 CONV 1000 0.5923746 0.5928929 0.0001862 0.0001469 CONV 2000 0.5926621 0.5928957 0.0001491 0.0001496 CONV 5000 0.5927350 0.5928546 0.0001132 0.0001085 CONV 20000 0.5927350 0.5927967 0.0001100 0.0000507 CONV 40000 0.5927350 0.5928037 0.0001038 0.0000576 # FIT pc se slope chi2 dof Lmin FIT 0.5928171 0.0001110 -0.007456 2.408 4 32 # BOOTHIST 30 bins lo,hi,count over the bootstrap p_c replicates BOOTHIST 0.5924708,0.5924926,1 0.5924926,0.5925144,0 0.5925144,0.5925362,0 0.5925362,0.5925580,1 0.5925580,0.5925798,3 0.5925798,0.5926016,9 0.5926016,0.5926234,7 0.5926234,0.5926452,6 0.5926452,0.5926670,11 0.5926670,0.5926888,15 0.5926888,0.5927106,13 0.5927106,0.5927324,25 0.5927324,0.5927542,25 0.5927542,0.5927760,29 0.5927760,0.5927978,20 0.5927978,0.5928196,40 0.5928196,0.5928414,31 0.5928414,0.5928632,32 0.5928632,0.5928850,25 0.5928850,0.5929067,21 0.5929067,0.5929285,23 0.5929285,0.5929503,20 0.5929503,0.5929721,14 0.5929721,0.5929939,8 0.5929939,0.5930157,7 0.5930157,0.5930375,6 0.5930375,0.5930593,3 0.5930593,0.5930811,2 0.5930811,0.5931029,0 0.5931029,0.5931247,3 # NSPAN L then 40 bins lo,count of n_span/N over [0.40,0.80] NSPAN 16 0.4000,17 0.4100,36 0.4200,47 0.4300,57 0.4400,152 0.4500,125 0.4600,261 0.4700,241 0.4800,511 0.4900,486 0.5000,983 0.5100,1279 0.5200,1042 0.5300,1869 0.5400,1513 0.5500,2685 0.5600,1869 0.5700,3247 0.5800,3382 0.5900,2298 0.6000,3338 0.6100,2166 0.6200,2854 0.6300,1723 0.6400,2149 0.6500,1187 0.6600,1405 0.6700,1064 0.6800,490 0.6900,628 0.7000,272 0.7100,262 0.7200,118 0.7300,110 0.7400,33 0.7500,31 0.7600,25 0.7700,5 0.7800,8 0.7900,2 NSPAN 32 0.4000,0 0.4100,0 0.4200,0 0.4300,1 0.4400,2 0.4500,4 0.4600,6 0.4700,39 0.4800,62 0.4900,116 0.5000,278 0.5100,440 0.5200,728 0.5300,1115 0.5400,1899 0.5500,2414 0.5600,3111 0.5700,3915 0.5800,4786 0.5900,4365 0.6000,4235 0.6100,3650 0.6200,3182 0.6300,2166 0.6400,1470 0.6500,915 0.6600,592 0.6700,244 0.6800,144 0.6900,65 0.7000,38 0.7100,9 0.7200,5 0.7300,3 0.7400,0 0.7500,1 0.7600,0 0.7700,0 0.7800,0 0.7900,0 NSPAN 64 0.4000,0 0.4100,0 0.4200,0 0.4300,0 0.4400,0 0.4500,0 0.4600,1 0.4700,0 0.4800,1 0.4900,0 0.5000,4 0.5100,13 0.5200,45 0.5300,144 0.5400,330 0.5500,844 0.5600,1569 0.5700,2609 0.5800,3419 0.5900,3686 0.6000,3085 0.6100,2173 0.6200,1185 0.6300,585 0.6400,217 0.6500,64 0.6600,21 0.6700,3 0.6800,0 0.6900,2 0.7000,0 0.7100,0 0.7200,0 0.7300,0 0.7400,0 0.7500,0 0.7600,0 0.7700,0 0.7800,0 0.7900,0 NSPAN 128 0.4000,0 0.4100,0 0.4200,0 0.4300,0 0.4400,0 0.4500,0 0.4600,0 0.4700,0 0.4800,0 0.4900,0 0.5000,0 0.5100,0 0.5200,0 0.5300,1 0.5400,11 0.5500,68 0.5600,307 0.5700,990 0.5800,2027 0.5900,2310 0.6000,1478 0.6100,640 0.6200,135 0.6300,27 0.6400,5 0.6500,1 0.6600,0 0.6700,0 0.6800,0 0.6900,0 0.7000,0 0.7100,0 0.7200,0 0.7300,0 0.7400,0 0.7500,0 0.7600,0 0.7700,0 0.7800,0 0.7900,0 NSPAN 256 0.4000,0 0.4100,0 0.4200,0 0.4300,0 0.4400,0 0.4500,0 0.4600,0 0.4700,0 0.4800,0 0.4900,0 0.5000,0 0.5100,0 0.5200,0 0.5300,0 0.5400,0 0.5500,0 0.5600,19 0.5700,262 0.5800,1550 0.5900,2189 0.6000,881 0.6100,96 0.6200,3 0.6300,0 0.6400,0 0.6500,0 0.6600,0 0.6700,0 0.6800,0 0.6900,0 0.7000,0 0.7100,0 0.7200,0 0.7300,0 0.7400,0 0.7500,0 0.7600,0 0.7700,0 0.7800,0 0.7900,0 NSPAN 512 0.4000,0 0.4100,0 0.4200,0 0.4300,0 0.4400,0 0.4500,0 0.4600,0 0.4700,0 0.4800,0 0.4900,0 0.5000,0 0.5100,0 0.5200,0 0.5300,0 0.5400,0 0.5500,0 0.5600,0 0.5700,5 0.5800,343 0.5900,777 0.6000,75 0.6100,0 0.6200,0 0.6300,0 0.6400,0 0.6500,0 0.6600,0 0.6700,0 0.6800,0 0.6900,0 0.7000,0 0.7100,0 0.7200,0 0.7300,0 0.7400,0 0.7500,0 0.7600,0 0.7700,0 0.7800,0 0.7900,0 NSPAN 1024 0.4000,0 0.4100,0 0.4200,0 0.4300,0 0.4400,0 0.4500,0 0.4600,0 0.4700,0 0.4800,0 0.4900,0 0.5000,0 0.5100,0 0.5200,0 0.5300,0 0.5400,0 0.5500,0 0.5600,0 0.5700,0 0.5800,67 0.5900,250 0.6000,3 0.6100,0 0.6200,0 0.6300,0 0.6400,0 0.6500,0 0.6600,0 0.6700,0 0.6800,0 0.6900,0 0.7000,0 0.7100,0 0.7200,0 0.7300,0 0.7400,0 0.7500,0 0.7600,0 0.7700,0 0.7800,0 0.7900,0 total runtime: 526.5 s