diff options
| -rw-r--r-- | data/cluster-size/cluster-size_2vector3d.dat | 54 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_3potts2d.dat | 65 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_3vector3d.dat | 23 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_4potts2d.dat | 54 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_4vector3d.dat | 47 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_ising2d.dat | 65 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_ising3d.dat | 54 | ||||
| -rw-r--r-- | fig_clusters_ising2d.eps | 5759 | ||||
| -rw-r--r-- | fig_clusters_ising2d.tex | 180 | ||||
| -rw-r--r-- | figs/fig_clusters_ising2d.gplot | 137 | ||||
| -rw-r--r-- | makefile | 2 | ||||
| -rw-r--r-- | monte-carlo.bib | 34 | ||||
| -rw-r--r-- | monte-carlo.pdf | bin | 147156 -> 185499 bytes | |||
| -rw-r--r-- | monte-carlo.tex | 158 |
14 files changed, 6605 insertions, 27 deletions
diff --git a/data/cluster-size/cluster-size_2vector3d.dat b/data/cluster-size/cluster-size_2vector3d.dat new file mode 100644 index 0000000..16e71a8 --- /dev/null +++ b/data/cluster-size/cluster-size_2vector3d.dat @@ -0,0 +1,54 @@ +8 2.20167 1.e-7 63.59562043795609 2.164973893102592 +8 2.20167 1.e-6 64.87819548872191 2.212648436568576 +8 2.20167 0.00001 67.46940874035968 2.301616822709248 +8 2.20167 0.0001 67.44140423335065 2.30008894357248 +8 2.20167 0.001 71.32843137254912 2.398277793810432 +8 2.20167 0.01 67.9385416666665 2.299960087698432 +8 2.20167 0.1 105.92954363490816 3.86057588781312 +8 2.20167 1. 213.1218354430377 7.762196801047552 +8 2.20167 10. 368.2691292875991 10.105035534898688 +8 2.20167 100. 470.9830508474578 7.230626947020288 + +16 2.20167 1.e-7 286.2860304287703 8.98124531728384 +16 2.20167 1.e-6 274.4344131704709 8.678468826714113 +16 2.20167 0.00001 262.02001520141926 8.166182721142784 +16 2.20167 0.0001 233.82716329775104 7.156634142302208 +16 2.20167 0.001 234.45846572854887 7.287514589700096 +16 2.20167 0.01 391.3233532934144 13.354160729767935 +16 2.20167 0.1 822.6586126266573 29.32967370950656 +16 2.20167 1. 1872.1783216783197 65.73364751310439 +16 2.20167 10. 3098.8235294117644 78.5758652691415 +16 2.20167 100. 3722.8065573770486 60.9046991010857 + +32 2.20167 1.e-7 1048.580106979115 29.22072313312051 +32 2.20167 1.e-6 971.4170222854472 26.492627007799296 +32 2.20167 0.00001 987.9398532419093 27.91466354242355 +32 2.20167 0.0001 1083.3379648008356 31.228299422072833 +32 2.20167 0.001 1151.1507002801193 33.530449835229184 +32 2.20167 0.01 2611.275275590533 85.86586555583693 +32 2.20167 0.1 6339.86940298507 227.132753381163 +32 2.20167 1. 14823.10940170938 515.0581135403254 +32 2.20167 10. 22135.92249999999 669.8054971229799 +32 2.20167 100. 30417.365771812078 417.6834673476567 + +64 2.20167 1.e-7 4289.382018782331 113.41379059607142 +64 2.20167 1.e-6 4021.5975113958357 101.65684041993421 +64 2.20167 0.00001 4077.0909943129705 103.63447013697126 +64 2.20167 0.0001 4898.0003000525785 125.73227228305818 +64 2.20167 0.001 8836.591433942041 254.10948594506138 +64 2.20167 0.01 21022.64215218081 682.6924817920819 +64 2.20167 0.1 50597.72201630827 1842.307708405678 +64 2.20167 1. 114540.33772652397 4068.2195261694933 +64 2.20167 10. 201245.50704225348 4999.55582256939 +64 2.20167 100. 239130.41503267954 3687.9212448362005 + +128 2.20167 1.e-7 17409.28502540219 666.9019482318438 +128 2.20167 1.e-6 23689.046527859227 1093.5300645443338 +128 2.20167 0.00001 20089.293792012664 786.2899335338394 +128 2.20167 0.0001 38106.45479910696 1854.133959464583 +128 2.20167 0.001 72399.92470837763 3547.6496046148814 +128 2.20167 0.01 190290.0237010039 10030.696361468363 +128 2.20167 0.1 448240.2068230266 24526.873461304327 +128 2.20167 1. 849276.2519685049 50303.14845552824 +128 2.20167 10. 1.583249774647888e6 67273.13361722566 +128 2.20167 100. 1.7961472880000002e6 61619.78659707237 diff --git a/data/cluster-size/cluster-size_3potts2d.dat b/data/cluster-size/cluster-size_3potts2d.dat new file mode 100644 index 0000000..3b8319d --- /dev/null +++ b/data/cluster-size/cluster-size_3potts2d.dat @@ -0,0 +1,65 @@ +8 0.994973 1.e-7 28.3823264201984 0.236138387891328 +8 0.994973 1.e-6 27.805823680823615 0.233244623238976 +8 0.994973 0.00001 28.043619527251906 0.23433663419584 +8 0.994973 0.0001 27.385046581398974 0.230684992491712 +8 0.994973 0.001 27.58528428093645 0.231563393720128 +8 0.994973 0.01 28.17325524646157 0.236158926337664 +8 0.994973 0.1 34.48330078124999 0.275121977917568 +8 0.994973 1. 41.996052631579005 0.322332521883392 +8 0.994973 10. 43.341899892357375 0.343088948340544 +8 0.994973 100. 43.08706700548358 0.343120888099008 + +16 0.994973 1.e-7 89.29277492358682 0.771637147299328 +16 0.994973 1.e-6 94.07098723053645 0.807376643724288 +16 0.994973 0.00001 94.24580121145267 0.80744921509248 +16 0.994973 0.0001 91.6403508771927 0.79496752019712 +16 0.994973 0.001 91.13449052603623 0.78649727606528 +16 0.994973 0.01 101.76133669235277 0.872308584550144 +16 0.994973 0.1 136.38096649866546 1.111759974312704 +16 0.994973 1. 165.2677242888389 1.251915123006976 +16 0.994973 10. 172.59167452512435 1.388648509539584 +16 0.994973 100. 168.46318695766527 1.388370676203264 + +32 0.994973 1.e-7 308.8067591965655 2.657940580622336 +32 0.994973 1.e-6 311.22099948947454 2.685346518641664 +32 0.994973 0.00001 306.9843078125906 2.652692697684992 +32 0.994973 0.0001 299.0313012127119 2.595835539974144 +32 0.994973 0.001 330.0585442086902 2.831122416274432 +32 0.994973 0.01 409.085369443626 3.420774081614848 +32 0.994973 0.1 543.6338461538417 4.462358771792896 +32 0.994973 1. 664.04034203025 5.015368884472832 +32 0.994973 10. 681.0485759914834 5.571595495469056 +32 0.994973 100. 685.512391158741 5.571552525910016 + +64 0.994973 1.e-7 1045.320553473614 8.963296885243905 +64 0.994973 1.e-6 1014.882108700717 8.67390104844288 +64 0.994973 0.00001 1015.2398430933033 8.73498768896 +64 0.994973 0.0001 1007.5999535855084 8.654159380598784 +64 0.994973 0.001 1210.7716902581003 10.254569083936769 +64 0.994973 0.01 1635.5552682133791 13.576588168851456 +64 0.994973 0.1 2161.249827331027 17.837539231420415 +64 0.994973 1. 2670.878363475968 20.079606759276544 +64 0.994973 10. 2726.4530313124455 22.30290342632243 +64 0.994973 100. 2724.6384820239687 22.302813764513793 + +128 0.994973 1.e-7 3538.730470981927 29.814243053223937 +128 0.994973 1.e-6 3483.9675201993605 29.319961695010814 +128 0.994973 0.00001 3429.912134387319 28.99572400950477 +128 0.994973 0.0001 3630.212132598497 30.704791192567807 +128 0.994973 0.001 4715.700264108196 39.40942045064397 +128 0.994973 0.01 6600.033674307486 54.57252040849818 +128 0.994973 0.1 8546.844884810596 71.10215711311463 +128 0.994973 1. 10624.082364553462 80.31972694956441 +128 0.994973 10. 10974.995574627894 89.225318947029 +128 0.994973 100. 10864.279569892507 89.224446664704 + +256 0.994973 1.e-7 10948.04919614151 286.9151418115031 +256 0.994973 1.e-6 10144.75468889555 266.11003730886654 +256 0.994973 0.00001 11373.025692359 302.0328836857856 +256 0.994973 0.0001 14407.5266470835 378.75374401919385 +256 0.994973 0.001 19704.837992013694 512.2242250181837 +256 0.994973 0.01 25847.14064801176 687.3836664888361 +256 0.994973 0.1 36241.574112735005 917.0604850607555 +256 0.994973 1. 41424.834951456294 1019.2338743602709 +256 0.994973 10. 42021.48963730574 1132.0645859317515 +256 0.994973 100. 42350.64099216714 1132.9247419503083 diff --git a/data/cluster-size/cluster-size_3vector3d.dat b/data/cluster-size/cluster-size_3vector3d.dat new file mode 100644 index 0000000..2467b0e --- /dev/null +++ b/data/cluster-size/cluster-size_3vector3d.dat @@ -0,0 +1,23 @@ +8 1.44325 0.00001 55.5139971969664 0.554903213121536 +8 1.44325 0.0000199526 56.027379494199806 0.559909155830272 +8 1.44325 0.0000398107 56.192946736956415 0.56169179990272 +8 1.44325 0.0000794328 54.5194957803648 0.544726918441472 +8 1.44325 0.000158489 54.597054353332226 0.545821297943552 +8 1.44325 0.000316228 55.4369717283584 0.5542436232704 +8 1.44325 0.000630957 55.07394505297152 0.55047552796416 +8 1.44325 0.00125893 55.21150950161306 0.551854558490112 +8 1.44325 0.00251189 53.847208515437565 0.53806267020288 +8 1.44325 0.00501187 56.394869546151426 0.563462774585344 +8 1.44325 0.01 58.60907046476697 0.58596544898048 +8 1.44325 0.0199526 60.544246945532414 0.605157448036352 +8 1.44325 0.0398107 69.25171956575693 0.69234327766272 +8 1.44325 0.0794328 82.73071667530547 0.826681271223296 +8 1.44325 0.158489 100.86688560481588 1.007812401429504 +8 1.44325 0.316228 126.55678979886899 1.26487036726528 +8 1.44325 0.630957 161.53304886211532 1.615101040502784 +8 1.44325 1.25893 196.9797927992617 1.969288721453568 +8 1.44325 2.51189 243.51881051175528 2.433779871054848 +8 1.44325 5.01187 284.1764088204406 2.841037843352576 +8 1.44325 10. 331.5890990542551 3.315775558490112 +8 1.44325 19.9526 373.07874564459826 3.727468875063808 +8 1.44325 39.8107 407.2365374937083 4.07083755410944
\ No newline at end of file diff --git a/data/cluster-size/cluster-size_4potts2d.dat b/data/cluster-size/cluster-size_4potts2d.dat new file mode 100644 index 0000000..4c36fd5 --- /dev/null +++ b/data/cluster-size/cluster-size_4potts2d.dat @@ -0,0 +1,54 @@ +8 0.910239 1.e-7 31.25128528907744 0.246253494231872 +8 0.910239 1.e-6 30.78990703851277 0.244874662488896 +8 0.910239 0.00001 32.308278361051585 0.25217886251904 +8 0.910239 0.0001 31.30189867722483 0.246826664226944 +8 0.910239 0.001 32.18340571006183 0.251616343170432 +8 0.910239 0.01 32.33067211288518 0.251779013153728 +8 0.910239 0.1 38.88447177685043 0.280855693400128 +8 0.910239 1. 47.08413939313638 0.313063502786432 +8 0.910239 10. 47.468546958062845 0.336828361986496 +8 0.910239 100. 48.136758538046976 0.334695743642496 + +16 0.910239 1.e-7 109.95747918001254 0.887640019863296 +16 0.910239 1.e-6 109.3095789725975 0.893187923640576 +16 0.910239 0.00001 108.40977651425408 0.88709965624064 +16 0.910239 0.0001 106.5304030441874 0.872532765732608 +16 0.910239 0.001 104.64525502791116 0.866529850325248 +16 0.910239 0.01 121.32530973451341 0.954942818574848 +16 0.910239 0.1 156.60729154945435 1.142645834240256 +16 0.910239 1. 187.6613283674729 1.225325583072256 +16 0.910239 10. 189.83988139362458 1.36108803505792 +16 0.910239 100. 195.52122785583435 1.340490724936448 + +32 0.910239 1.e-7 381.88485016647985 3.121002356892672 +32 0.910239 1.e-6 351.6929717976576 2.948493945322496 +32 0.910239 0.00001 363.32742894910666 2.9950692129536 +32 0.910239 0.0001 351.5059971928033 2.93354506405888 +32 0.910239 0.001 374.4909486865367 3.059559813685248 +32 0.910239 0.01 467.12224194775655 3.664210409629696 +32 0.910239 0.1 613.5265147304018 4.56898404578816 +32 0.910239 1. 745.8312184304507 4.920557932276736 +32 0.910239 10. 761.7391692719933 5.450411114161152 +32 0.910239 100. 772.769370185771 5.411916259948544 + +64 0.910239 1.e-7 1227.513492153086 10.14512468025344 +64 0.910239 1.e-6 1150.0090557644266 9.624002896257023 +64 0.910239 0.00001 1279.3070283600364 10.607897199747072 +64 0.910239 0.0001 1214.0060714086728 10.08866700275712 +64 0.910239 0.001 1337.2163799963198 11.074506984640513 +64 0.910239 0.01 1887.081062899421 14.627000031576063 +64 0.910239 0.1 2472.9367996414157 18.29732088777933 +64 0.910239 1. 3002.5375837261045 19.63658623008768 +64 0.910239 10. 3070.5594541910427 21.72675771574272 +64 0.910239 100. 3070.4065126050405 21.73601502843699 + +128 0.910239 1.e-7 3899.50608578483 32.27619359540839 +128 0.910239 1.e-6 3777.1591532732086 31.121395790168062 +128 0.910239 0.00001 4074.4429495593367 33.50122426553139 +128 0.910239 0.0001 4724.828190248927 38.30069089855078 +128 0.910239 0.001 5624.484330850222 44.74178843721728 +128 0.910239 0.01 7617.330488423236 58.28087792766157 +128 0.910239 0.1 9930.301271234133 73.19535506664653 +128 0.910239 1. 11948.685548562318 78.73371439859302 +128 0.910239 10. 12201.149739776967 87.1172777734144 +128 0.910239 100. 12356.893854748623 86.68401818232422 diff --git a/data/cluster-size/cluster-size_4vector3d.dat b/data/cluster-size/cluster-size_4vector3d.dat new file mode 100644 index 0000000..b32f6f7 --- /dev/null +++ b/data/cluster-size/cluster-size_4vector3d.dat @@ -0,0 +1,47 @@ +8 1.e-7 6.468390804597696 0.2299249828032 +8 1.e-6 7.151050788091072 0.253524024656064 +8 0.00001 6.766711506622528 0.22870107084 +8 0.0001 6.85281749369216 0.245290188692032 +8 0.001 6.295586527293888 0.21877425067008 +8 0.01 6.58619297537344 0.230938504937856 +8 0.1 10.090617359413184 0.381796466094656 +8 1. 21.405057251908417 0.806654372150592 +8 10. 39.626443418013885 1.2837603509824 +8 100. 56.50080128205133 1.048646025543232 +16 1.e-7 12.388315738963456 0.390627946109184 +16 1.e-6 14.011522801302783 0.469495665395968 +16 0.00001 14.374053873552896 0.466301897031424 +16 0.0001 13.89824012092416 0.443478689434624 +16 0.001 13.58135211562624 0.449625053788672 +16 0.01 16.74646226415104 0.55525348888192 +16 0.1 35.82378944504883 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80.25132214784409 +64 100. 3489.254634938839 70.54946976722125 +128 1.e-7 132.48186227634994 6.020475893088256 +128 1.e-6 128.82328062859673 5.960419268018176 +128 0.00001 165.34305434979532 9.413809870815232 +128 0.0001 233.11510363380123 12.90065990230016 +128 0.001 418.1740127284593 20.95188210860032 +128 0.01 1112.215227447132 65.80112118779086 +128 0.1 2474.9717092803053 147.1845094595789
\ No newline at end of file diff --git a/data/cluster-size/cluster-size_ising2d.dat b/data/cluster-size/cluster-size_ising2d.dat new file mode 100644 index 0000000..8d790cc --- /dev/null +++ b/data/cluster-size/cluster-size_ising2d.dat @@ -0,0 +1,65 @@ +8 2.26918531421 1.e-7 41.76061921842541 0.291820060441472 +8 2.26918531421 1.e-6 41.77551405395194 0.291912174276736 +8 2.26918531421 0.00001 40.99311243484749 0.29380210922656 +8 2.26918531421 0.0001 41.4345943911159 0.29368346942368 +8 2.26918531421 0.001 41.57769376181453 0.292287093854272 +8 2.26918531421 0.01 41.4138449962375 0.294463837488192 +8 2.26918531421 0.1 47.3735872750103 0.288410474637952 +8 2.26918531421 1. 60.359498031495875 0.159130800346688 +8 2.26918531421 10. 63.998666222073986 0.00066655521088 +8 2.26918531421 100. 64. 0. + +16 2.26918531421 1.e-7 137.932397547595 1.106213393458432 +16 2.26918531421 1.e-6 142.46250000000077 1.125068889773824 +16 2.26918531421 0.00001 140.3548493331131 1.115093465484288 +16 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Landscape {Pat4} {Pat5} ifelse setpattern} bind def +/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def +/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def +} def +% +% +%End of PostScript Level 2 code +% +/PatternBgnd { + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse +} def +% +% Substitute for Level 2 pattern fill codes with +% grayscale if Level 2 support is not selected. +% +/Level1PatternFill { +/Pattern1 {0.250 Density} bind def +/Pattern2 {0.500 Density} bind def +/Pattern3 {0.750 Density} bind def +/Pattern4 {0.125 Density} bind def +/Pattern5 {0.375 Density} bind def +/Pattern6 {0.625 Density} bind def +/Pattern7 {0.875 Density} bind def +} def +% +% Now test for support of Level 2 code +% +Level1 {Level1PatternFill} {Level2PatternFill} ifelse +% +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +% +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (fig_clusters_ising2d.tex) + /Subject (gnuplot plot) + /Creator (gnuplot 5.2 patchlevel 2) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Thu Apr 19 16:11:44 2018) + /DOCINFO pdfmark +end +} ifelse +% +% Support for boxed text - Ethan A Merritt Sep 2016 +% +/InitTextBox { userdict /TBy2 3 -1 roll put userdict /TBx2 3 -1 roll put + userdict /TBy1 3 -1 roll put userdict /TBx1 3 -1 roll put + /Boxing true def } def +/ExtendTextBox { dup type /stringtype eq + { Boxing { gsave dup false charpath pathbbox + dup TBy2 gt {userdict /TBy2 3 -1 roll put} {pop} ifelse + dup TBx2 gt {userdict /TBx2 3 -1 roll put} {pop} ifelse + dup TBy1 lt {userdict /TBy1 3 -1 roll put} {pop} ifelse + dup TBx1 lt {userdict /TBx1 3 -1 roll put} {pop} ifelse + grestore } if } + {} ifelse} def +/PopTextBox { newpath TBx1 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HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0323 .0968 .2258 .4839 1 ] def +/RedA [.3686 .8824 .5608 .9216 .5294 .7725 ] def +/GreenA [.5059 .6118 .6902 .3843 .4706 .4314 ] def +/BlueA [.7098 .1412 .1961 .2078 .702 .102 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 Gamma mul div def +/ColorSpace (RGB) def +Color InterpolatedColor or { % COLOUR vs. GRAY map + 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{ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0323 .0968 .2258 .4839 1 ] def +/RedA [.3686 .8824 .5608 .9216 .5294 .7725 ] def +/GreenA [.5059 .6118 .6902 .3843 .4706 .4314 ] def +/BlueA [.7098 .1412 .1961 .2078 .702 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+167 -12 R +0 8 V +-31 -8 R +62 0 V +-62 8 R +62 0 V +1 g 4295 2743 M +0 25 V +-31 -25 R +62 0 V +-62 25 R +62 0 V +166 -61 R +0 25 V +-31 -25 R +62 0 V +-62 25 R +62 0 V +167 29 R +0 26 V +-31 -26 R +62 0 V +-62 26 R +62 0 V +166 87 R +0 25 V +-31 -25 R +62 0 V +-62 25 R +62 0 V +167 125 R +0 25 V +-31 -25 R +62 0 V +-62 25 R +62 0 V +167 104 R +0 26 V +-31 -26 R +62 0 V +-62 26 R +62 0 V +166 136 R +0 24 V +-31 -24 R +62 0 V +-62 24 R +62 0 V +167 40 R +0 24 V +-31 -24 R +62 0 V +-62 24 R +62 0 V +166 -18 R +0 26 V +-31 -26 R +62 0 V +-62 26 R +62 0 V +167 -22 R +0 25 V +-31 -25 R +62 0 V +-62 25 R +62 0 V +0 g 3937 2771 Pnt +4135 2775 Pnt +4332 2764 Pnt +4530 2767 Pnt +4728 2777 Pnt +4925 2874 Pnt +5123 2968 Pnt +5320 2983 Pnt +5518 2980 Pnt +.0323 g 3851 2754 Pnt +4048 2779 Pnt +4246 2780 Pnt +4443 2767 Pnt +4641 2764 Pnt +4839 2817 Pnt +5036 2957 Pnt +5234 3048 Pnt +5431 3069 Pnt +5629 3057 Pnt +.0968 g 3962 2773 Pnt +4159 2777 Pnt +4357 2770 Pnt +4554 2758 Pnt +4752 2805 Pnt +4950 2907 Pnt +5147 3043 Pnt +5345 3139 Pnt +5542 3151 Pnt +5740 3154 Pnt +.2258 g 4073 2782 Pnt +4270 2768 Pnt +4468 2768 Pnt +4665 2764 Pnt +4863 2852 Pnt +5061 2995 Pnt +5258 3128 Pnt +5456 3230 Pnt +5653 3239 Pnt +5851 3239 Pnt +.4839 g 4184 2790 Pnt +4381 2783 Pnt +4579 2775 Pnt +4776 2803 Pnt +4974 2927 Pnt +5172 3088 Pnt +5369 3211 Pnt +5567 3315 Pnt +5764 3331 Pnt +5962 3326 Pnt +1 g 4295 2756 Pnt +4492 2720 Pnt +4690 2774 Pnt +4887 2887 Pnt +5085 3036 Pnt +5283 3166 Pnt +5480 3327 Pnt +5678 3391 Pnt +5875 3398 Pnt +6073 3402 Pnt +LTw +% End plot #1 +% Begin plot #2 +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +0 g 3742 2781 M +195 -10 V +198 4 V +197 -11 V +198 3 V +198 10 V +197 97 V +198 94 V +197 15 V +198 -3 V +.0323 g 3851 2754 M +197 25 V +198 1 V +197 -13 V +198 -3 V +198 53 V +197 140 V +198 91 V +197 21 V +198 -12 V +0 g .0968 g 3962 2773 M +197 4 V +198 -7 V +197 -12 V +198 47 V +198 102 V +197 136 V +198 96 V +197 12 V +198 3 V +0 g .2258 g 4073 2782 M +197 -14 V +198 0 V +197 -4 V +198 88 V +198 143 V +197 133 V +198 102 V +197 9 V +198 0 V +0 g .4839 g 4184 2790 M +197 -7 V +198 -8 V +197 28 V +198 124 V +198 161 V +197 123 V +198 104 V +197 16 V +198 -5 V +0 g 1 g 4295 2756 M +197 -36 V +198 54 V +197 113 V +198 149 V +198 130 V +197 161 V +198 64 V +197 7 V +198 4 V +stroke +LTw +% End plot #2 +% Begin plot #3 +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +4130 2467 M +3 3 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 20 V +24 19 V +25 20 V +24 19 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 19 V +25 20 V +24 19 V +25 20 V +24 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 19 V +25 20 V +24 19 V +25 20 V +24 19 V +25 20 V +24 19 V +24 19 V +25 20 V +24 19 V +25 20 V +24 19 V +25 20 V +24 19 V +25 19 V +24 20 V +25 19 V +24 20 V +24 19 V +25 20 V +24 19 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 20 V +24 19 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 20 V +25 19 V +24 19 V +4 3 V +stroke +LTw +% End plot #3 +2.000 UL +LTb +1.000 UL +LTb +3742 3833 N +0 -1366 V +2421 0 V +0 1366 V +-2421 0 V +Z stroke +1.000 UP +1.000 UL +LTb +grestore % colour palette end +gsave % colour palette begin +/maxcolors 0 def +/HSV2RGB { exch dup 0.0 eq {pop exch pop dup dup} % achromatic gray + { /HSVs exch def /HSVv exch def 6.0 mul dup floor dup 3 1 roll sub + /HSVf exch def /HSVi exch cvi def /HSVp HSVv 1.0 HSVs sub mul def + /HSVq HSVv 1.0 HSVs HSVf mul sub mul def + /HSVt HSVv 1.0 HSVs 1.0 HSVf sub mul sub mul def + /HSVi HSVi 6 mod def 0 HSVi eq {HSVv HSVt HSVp} + {1 HSVi eq {HSVq HSVv HSVp}{2 HSVi eq {HSVp HSVv HSVt} + {3 HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0323 .0968 .2258 .4839 1 ] def +/RedA [.3686 .8824 .5608 .9216 .5294 .7725 ] def +/GreenA [.5059 .6118 .6902 .3843 .4706 .4314 ] def +/BlueA [.7098 .1412 .1961 .2078 .702 .102 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 Gamma mul div def +/ColorSpace (RGB) def +Color InterpolatedColor or { % COLOUR vs. GRAY map + InterpolatedColor { %% Interpolation vs. RGB-Formula + /g {stroke pm3dround /grayv exch def interpolate + SelectSpace setrgbcolor} bind def + }{ + /g {stroke pm3dround dup cF7 Constrain exch dup cF5 Constrain exch cF15 Constrain + SelectSpace setrgbcolor} bind def + } ifelse +}{ + /g {stroke pm3dround pm3dGamma exp setgray} bind def +} ifelse +1.000 UL +LTb +7286 2467 M +31 0 V +2388 0 R +-31 0 V +-2388 36 R +31 0 V +2388 0 R +-31 0 V +-2388 32 R +31 0 V +2388 0 R +-31 0 V +-2388 27 R +31 0 V +2388 0 R +-31 0 V +-2388 25 R +63 0 V +2356 0 R +-63 0 V +stroke +7286 2750 M +31 0 V +2388 0 R +-31 0 V +-2388 95 R +31 0 V +2388 0 R +-31 0 V +-2388 68 R +31 0 V +2388 0 R +-31 0 V +-2388 53 R +31 0 V +2388 0 R +-31 0 V +-2388 42 R +31 0 V +2388 0 R +-31 0 V +-2388 37 R +31 0 V +2388 0 R +-31 0 V +-2388 31 R +31 0 V +2388 0 R +-31 0 V +-2388 28 R +31 0 V +2388 0 R +-31 0 V +-2388 25 R +63 0 V +2356 0 R +-63 0 V +stroke +7286 3292 M +31 0 V +2388 0 R +-31 0 V +-2388 95 R +31 0 V +2388 0 R +-31 0 V +-2388 68 R +31 0 V +2388 0 R +-31 0 V +-2388 52 R +31 0 V +2388 0 R +-31 0 V +-2388 43 R +31 0 V +2388 0 R +-31 0 V +-2388 36 R +31 0 V +2388 0 R +-31 0 V +-2388 32 R +31 0 V +2388 0 R +-31 0 V +-2388 27 R +31 0 V +2388 0 R +-31 0 V +-2388 25 R +63 0 V +2356 0 R +-63 0 V +stroke +7286 3833 M +31 0 V +2388 0 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0 setdash +0 g 7326 2603 M +197 5 V +197 9 V +198 0 V +197 13 V +197 -12 V +198 105 V +197 164 V +198 129 V +197 58 V +.0667 g 7473 2636 M +197 -9 V +198 -11 V +197 -27 V +198 1 V +197 120 V +197 175 V +198 193 V +197 119 V +197 43 V +0 g .2 g 7621 2622 M +197 -18 V +197 4 V +198 21 V +197 14 V +197 193 V +198 209 V +197 199 V +198 95 V +197 74 V +0 g .4667 g 7768 2633 M +197 -16 V +198 4 V +197 43 V +198 138 V +197 204 V +197 207 V +198 192 V +197 132 V +197 41 V +0 g 1 g 7916 2642 M +197 72 V +197 -39 V +198 151 V +197 151 V +197 227 V +198 202 V +197 150 V +198 146 V +197 30 V +stroke +LTw +% End plot #2 +% Begin plot #3 +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +8002 2467 M +17 20 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +24 28 V +25 28 V +24 27 V +25 28 V +24 28 V +25 28 V +24 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +24 28 V +25 28 V +24 27 V +25 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +25 28 V +24 28 V +24 28 V +25 28 V +4 4 V +stroke +LTw +% End plot #3 +2.000 UL +LTb +1.000 UL +LTb +7286 3833 N +0 -1366 V +2419 0 V +0 1366 V +-2419 0 V +Z stroke +1.000 UP +1.000 UL +LTb +grestore % colour palette end +gsave % colour palette begin +/maxcolors 0 def +/HSV2RGB { exch dup 0.0 eq {pop exch pop dup dup} % achromatic gray + { /HSVs exch def /HSVv exch def 6.0 mul dup floor dup 3 1 roll sub + /HSVf exch def /HSVi exch cvi def /HSVp HSVv 1.0 HSVs sub mul def + /HSVq HSVv 1.0 HSVs HSVf mul sub mul def + /HSVt HSVv 1.0 HSVs 1.0 HSVf sub mul sub mul def + /HSVi HSVi 6 mod def 0 HSVi eq {HSVv HSVt HSVp} + {1 HSVi eq {HSVq HSVv HSVp}{2 HSVi eq {HSVp HSVv HSVt} + {3 HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0323 .0968 .2258 .4839 1 ] def +/RedA [.3686 .8824 .5608 .9216 .5294 .7725 ] def +/GreenA [.5059 .6118 .6902 .3843 .4706 .4314 ] def +/BlueA [.7098 .1412 .1961 .2078 .702 .102 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 Gamma mul div def +/ColorSpace (RGB) def +Color InterpolatedColor or { % COLOUR vs. GRAY map + InterpolatedColor { %% Interpolation vs. RGB-Formula + /g {stroke pm3dround /grayv exch def interpolate + SelectSpace setrgbcolor} bind def + }{ + /g {stroke pm3dround dup cF7 Constrain exch dup cF5 Constrain exch cF15 Constrain + 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+LTb +1.000 UL +LTb +726 2009 N +726 704 L +2420 0 V +0 1305 V +-2420 0 V +Z stroke +1.000 UP +1.000 UL +LTb +grestore % colour palette end +gsave % colour palette begin +/maxcolors 0 def +/HSV2RGB { exch dup 0.0 eq {pop exch pop dup dup} % achromatic gray + { /HSVs exch def /HSVv exch def 6.0 mul dup floor dup 3 1 roll sub + /HSVf exch def /HSVi exch cvi def /HSVp HSVv 1.0 HSVs sub mul def + /HSVq HSVv 1.0 HSVs HSVf mul sub mul def + /HSVt HSVv 1.0 HSVs 1.0 HSVf sub mul sub mul def + /HSVi HSVi 6 mod def 0 HSVi eq {HSVv HSVt HSVp} + {1 HSVi eq {HSVq HSVv HSVp}{2 HSVi eq {HSVp HSVv HSVt} + {3 HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0323 .0968 .2258 .4839 1 ] def +/RedA [.3686 .8824 .5608 .9216 .5294 .7725 ] def +/GreenA [.5059 .6118 .6902 .3843 .4706 .4314 ] def +/BlueA [.7098 .1412 .1961 .2078 .702 .102 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 Gamma mul div def +/ColorSpace (RGB) def +Color InterpolatedColor or { % COLOUR vs. GRAY map + InterpolatedColor { %% Interpolation vs. RGB-Formula + /g {stroke pm3dround /grayv exch def interpolate + SelectSpace setrgbcolor} bind def + }{ + /g {stroke pm3dround dup cF7 Constrain exch dup cF5 Constrain exch cF15 Constrain + SelectSpace setrgbcolor} bind def + } ifelse +}{ + /g {stroke pm3dround pm3dGamma exp setgray} bind def +} ifelse +1.000 UL +LTb +3742 704 M +31 0 V +2387 0 R +-31 0 V +3742 810 M +31 0 V +2387 0 R +-31 0 V +3742 902 M +31 0 V +2387 0 R +-31 0 V +3742 983 M +31 0 V +2387 0 R +-31 0 V +-2387 72 R +63 0 V +2355 0 R +-63 0 V +stroke +3742 1532 M +31 0 V +2387 0 R +-31 0 V +3742 1811 M +31 0 V +2387 0 R +-31 0 V +3742 2009 M +31 0 V +2387 0 R +-31 0 V +3817 704 M +0 31 V +0 1274 R +0 -31 V +3873 704 M +0 31 V +0 1274 R +0 -31 V +3906 704 M +0 31 V +0 1274 R +0 -31 V +3930 704 M +0 31 V +0 1274 R +0 -31 V +3949 704 M +0 31 V +0 1274 R +0 -31 V +3964 704 M +0 31 V +0 1274 R +0 -31 V +3977 704 M +0 31 V +0 1274 R +0 -31 V +3989 704 M +0 31 V +0 1274 R +0 -31 V +3999 704 M +0 63 V +0 1242 R +0 -63 V +stroke +4212 704 M +0 31 V +0 1274 R +0 -31 V +4267 704 M +0 31 V +0 1274 R +0 -31 V +4301 704 M +0 31 V +0 1274 R +0 -31 V +4325 704 M +0 31 V +0 1274 R +0 -31 V +4344 704 M +0 31 V +0 1274 R +0 -31 V +4359 704 M +0 31 V +0 1274 R +0 -31 V +4372 704 M +0 31 V +0 1274 R +0 -31 V +4383 704 M +0 31 V +0 1274 R +0 -31 V +4393 704 M +0 63 V +0 1242 R +0 -63 V +stroke 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M +0 31 V +0 1274 R +0 -31 V +5577 704 M +0 63 V +0 1242 R +0 -63 V +stroke +5790 704 M +0 31 V +0 1274 R +0 -31 V +5846 704 M +0 31 V +0 1274 R +0 -31 V +5879 704 M +0 31 V +0 1274 R +0 -31 V +5903 704 M +0 31 V +0 1274 R +0 -31 V +5922 704 M +0 31 V +0 1274 R +0 -31 V +5937 704 M +0 31 V +0 1274 R +0 -31 V +5950 704 M +0 31 V +0 1274 R +0 -31 V +5962 704 M +0 31 V +0 1274 R +0 -31 V +5972 704 M +0 63 V +0 1242 R +0 -63 V +stroke +1.000 UL +LTb +3742 2009 N +0 -1305 V +2418 0 V +0 1305 V +-2418 0 V +Z stroke +1.000 UP +1.000 UL +LTb +% Begin plot #1 +1.000 UP +2.000 UL +LTb +0 g 3749 914 M +0 11 V +-7 -11 R +38 0 V +-38 11 R +38 0 V +166 -21 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 22 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 -32 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 9 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 -7 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 116 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 122 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +166 -3 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +167 0 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +.0667 g 3860 945 M +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 -15 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 -17 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +166 -24 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +167 -24 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 91 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 165 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 115 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +167 -1 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +166 11 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +.2 g 3972 967 M +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 -68 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +166 11 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 -34 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +166 32 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 141 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 177 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +167 125 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +166 5 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 0 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +.4667 g 4083 936 M +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 -56 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +167 61 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +166 -48 R +0 12 V +-31 -12 R +62 0 V +-62 12 R +62 0 V +166 55 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 226 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 175 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 123 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +166 6 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +167 -10 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +1 g 4194 897 M +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 -33 R +0 11 V +-31 -11 R +62 0 V +stroke 4423 875 M +-62 11 R +62 0 V +166 41 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 91 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +167 109 R +0 11 V +-31 -11 R +62 0 V +-62 11 R +62 0 V +166 198 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 172 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +167 118 R +0 9 V +-31 -9 R +62 0 V +-62 9 R +62 0 V +166 5 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +166 -1 R +0 10 V +-31 -10 R +62 0 V +-62 10 R +62 0 V +0 g 3749 920 Pnt +3946 910 Pnt +4144 943 Pnt +4341 921 Pnt +4538 940 Pnt +4735 943 Pnt +4933 1070 Pnt +5130 1202 Pnt +5327 1207 Pnt +5525 1217 Pnt +.0667 g 3860 951 Pnt +4058 947 Pnt +4255 941 Pnt +4452 929 Pnt +4650 917 Pnt +4847 1019 Pnt +5044 1194 Pnt +5241 1319 Pnt +5439 1327 Pnt +5636 1347 Pnt +.2 g 3972 973 Pnt +4169 916 Pnt +4366 939 Pnt +4564 916 Pnt +4761 959 Pnt +4958 1111 Pnt +5155 1299 Pnt +5353 1433 Pnt +5550 1448 Pnt +5747 1458 Pnt +.4667 g 4083 942 Pnt +4280 897 Pnt +4478 970 Pnt +4675 934 Pnt +4872 1001 Pnt +5070 1238 Pnt +5267 1423 Pnt +5464 1557 Pnt +5661 1572 Pnt +5859 1572 Pnt +1 g 4194 902 Pnt +4392 880 Pnt +4589 933 Pnt +4786 1034 Pnt +4984 1154 Pnt +5181 1363 Pnt +5378 1545 Pnt +5576 1673 Pnt +5773 1687 Pnt +5970 1696 Pnt +LTw +% End plot #1 +% Begin plot #2 +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +0 g 3749 920 M +197 -10 V +198 33 V +197 -22 V +197 19 V +197 3 V +198 127 V +197 132 V +197 5 V +198 10 V +.0667 g 3860 951 M +198 -4 V +197 -6 V +197 -12 V +198 -12 V +197 102 V +197 175 V +197 125 V +198 8 V +197 20 V +0 g .2 g 3972 973 M +197 -57 V +197 23 V +198 -23 V +197 43 V +197 152 V +197 188 V +198 134 V +197 15 V +197 10 V +0 g .4667 g 4083 942 M +197 -45 V +198 73 V +197 -36 V +197 67 V +198 237 V +197 185 V +197 134 V +197 15 V +198 0 V +0 g 1 g 4194 902 M +198 -22 V +197 53 V +197 101 V +198 120 V +197 209 V +197 182 V +198 128 V +197 14 V +197 9 V +stroke +LTw +% End plot #2 +% Begin plot #3 +1.000 UL +LTb +LCb setrgbcolor +[] 0 setdash +4343 704 M +10 11 V +24 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +25 26 V +24 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +25 26 V +24 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +25 26 V +24 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +24 26 V +25 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +24 26 V +25 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +24 26 V +25 26 V +24 26 V +25 26 V +24 26 V +25 27 V +24 26 V +24 26 V +13 13 V +stroke +LTw +% End plot #3 +2.000 UL +LTb +1.000 UL +LTb +3742 2009 N +0 -1305 V +2418 0 V +0 1305 V +-2418 0 V +Z stroke +1.000 UP +1.000 UL +LTb +grestore % colour palette end +gsave % colour palette begin +/maxcolors 0 def +/HSV2RGB { exch dup 0.0 eq {pop exch pop dup dup} % achromatic gray + { /HSVs exch def /HSVv exch def 6.0 mul dup floor dup 3 1 roll sub + /HSVf exch def /HSVi exch cvi def /HSVp HSVv 1.0 HSVs sub mul def + /HSVq HSVv 1.0 HSVs HSVf mul sub mul def + /HSVt HSVv 1.0 HSVs 1.0 HSVf sub mul sub mul def + /HSVi HSVi 6 mod def 0 HSVi eq {HSVv HSVt HSVp} + {1 HSVi eq {HSVq HSVv HSVp}{2 HSVi eq {HSVp HSVv HSVt} + {3 HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0323 .0968 .2258 .4839 1 ] def +/RedA [.3686 .8824 .5608 .9216 .5294 .7725 ] def +/GreenA [.5059 .6118 .6902 .3843 .4706 .4314 ] def +/BlueA [.7098 .1412 .1961 .2078 .702 .102 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 Gamma mul div def +/ColorSpace (RGB) def +Color InterpolatedColor or { % COLOUR vs. GRAY map + InterpolatedColor { %% Interpolation vs. RGB-Formula + /g {stroke pm3dround /grayv exch def interpolate + SelectSpace setrgbcolor} bind def + }{ + /g {stroke pm3dround dup cF7 Constrain exch dup cF5 Constrain exch cF15 Constrain + SelectSpace setrgbcolor} bind def + } ifelse +}{ + /g {stroke pm3dround pm3dGamma exp setgray} bind def +} ifelse +1.000 UL +LTb +7286 704 M +31 0 V +2388 0 R +-31 0 V +7286 739 M +31 0 V +2388 0 R +-31 0 V +7286 769 M +31 0 V +2388 0 R +-31 0 V +7286 795 M +31 0 V +2388 0 R +-31 0 V +7286 819 M +63 0 V +2356 0 R +-63 0 V +stroke +7286 974 M +31 0 V +2388 0 R +-31 0 V +-2388 92 R +31 0 V +2388 0 R +-31 0 V +-2388 64 R +31 0 V +2388 0 R +-31 0 V +-2388 50 R +31 0 V +2388 0 R +-31 0 V +-2388 41 R +31 0 V +2388 0 R +-31 0 V +-2388 35 R +31 0 V +2388 0 R +-31 0 V +-2388 30 R +31 0 V +2388 0 R +-31 0 V +-2388 26 R +31 0 V +2388 0 R +-31 0 V +-2388 24 R +63 0 V +2356 0 R +-63 0 V +stroke +7286 1492 M +31 0 V +2388 0 R +-31 0 V +-2388 91 R +31 0 V +2388 0 R +-31 0 V 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b/figs/fig_clusters_ising2d.gplot new file mode 100644 index 0000000..cfbc7c9 --- /dev/null +++ b/figs/fig_clusters_ising2d.gplot @@ -0,0 +1,137 @@ + +set terminal epslatex size 6.834, 2.81576 +set output "fig_clusters_ising2d.tex" + +cc1 = "#5e81b5" +cc2 = "#e19c24" +cc3 = "#8fb032" +cc4 = "#eb6235" +cc5 = "#8778b3" +cc6 = "#c56e1a" + +set palette model RGB defined ( 8 cc1, 16 cc2, 32 cc3, 64 cc4, 128 cc5, 256 cc6 ) + +unset colorbox +set nokey +set format x '$10^{%T}$' +set logscale xy +set ylabel '$\avg{s_{\text{\sc 1c}}}L^{-\gamma/\nu}$' +set xlabel '$hL^{\beta\delta/\nu}$' + +set xrange [0.000005:12000000] + + +set multiplot layout 2,3 + +set size 0.36, 0.5 + +dataIsing2D = "data/cluster-size/cluster-size_ising2d.dat" + +set yrange [0.8:8] + +unset xlabel + +set label "2D Ising" at graph 0.1, 0.75 + +betaIsing2D = 1.0 / 8.0 +nuIsing2D = 1.0 +deltaIsing2D = 15.0 +gammaIsing2D = 7.0 / 4.0 + +plot dataIsing2D using ($3/$2 * exp(log($1) * betaIsing2D * deltaIsing2D / nuIsing2D)):($4 * exp(-log($1) * gammaIsing2D / nuIsing2D)):($5 * exp(-log($1) * gammaIsing2D/nuIsing2D)):1 with yerrorbars pt 0 lw 2 palette, dataIsing2D using ($3/$2 * exp(log($1) * betaIsing2D * deltaIsing2D / nuIsing2D)):($4 * exp(-log($1) * gammaIsing2D / nuIsing2D)):1 with lines palette, 1.5 * exp(log(x) * 2 / deltaIsing2D) with lines lc black + +unset label + +data3Potts2D = "data/cluster-size/cluster-size_3potts2d.dat" + +set xrange [0.000005:9000000] +set yrange [0.4:7] + +set label "2D 3-State Potts" at graph 0.1, 0.75 + +unset xlabel +unset ylabel + +beta3Potts2D = 1.0 / 9.0 +nu3Potts2D = 5.0 / 6.0 +delta3Potts2D = 14.0 +gamma3Potts2D = 13.0 / 9.0 + +plot data3Potts2D using ($3/$2 * exp(log($1) * beta3Potts2D * delta3Potts2D / nu3Potts2D)):($4 * exp(-log($1) * gamma3Potts2D / nu3Potts2D)):($5 * exp(-log($1) * gamma3Potts2D/nu3Potts2D)):1 with yerrorbars pt 0 lw 2 palette, data3Potts2D using ($3/$2 * exp(log($1) * beta3Potts2D * delta3Potts2D / nu3Potts2D)):($4 * exp(-log($1) * gamma3Potts2D / nu3Potts2D)):1 with lines palette, 1.2 * exp(log(x) * 2 / delta3Potts2D) with lines lc black + +unset label +set size 0.36, 0.5 + +data2Vector3D = "data/cluster-size/cluster-size_2vector3d.dat" + +set xrange [0.000005:9000000] +set yrange [0.6:200] + +set label "3D O(2)" at graph 0.1, 0.75 + +beta2Vector3D = 0.3470 +nu2Vector3D = 0.6703 +delta2Vector3D = 4.79539 +gamma2Vector3D = 1.3169 + +plot data2Vector3D using ($3/$2 * exp(log($1) * beta2Vector3D * delta2Vector3D / nu2Vector3D)):($4 * exp(-log($1) * gamma2Vector3D / nu2Vector3D)):($5 * exp(-log($1) * gamma2Vector3D/nu2Vector3D)):1 with yerrorbars pt 0 lw 2 palette, data2Vector3D using ($3/$2 * exp(log($1) * beta2Vector3D * delta2Vector3D / nu2Vector3D)):($4 * exp(-log($1) * gamma2Vector3D / nu2Vector3D)):1 with lines palette, 3 * exp(log(x) * 2 / delta2Vector3D) with lines lc black + +unset label + +set origin -0.05,0.0 +set size 0.41, 0.55 + +dataIsing3D = "data/cluster-size/cluster-size_ising3d.dat" + +set xrange [0.000005:9000000] +set yrange [0.9:400] + +set label "3D Ising" at graph 0.1, 0.75 + +set ylabel '$\avg{s_{\text{\sc 1c}}}L^{-\gamma/\nu}$' +set xlabel '$hL^{\beta\delta/\nu}$' + +betaIsing3D = 0.326419 +nuIsing3D = 0.629971 +deltaIsing3D = 4.78984 +gammaIsing3D = 1.237075 + +plot dataIsing3D using ($3/$2 * exp(log($1) * betaIsing3D * deltaIsing3D / nuIsing3D)):($4 * exp(-log($1) * gammaIsing3D / nuIsing3D)):($5 * exp(-log($1) * gammaIsing3D/nuIsing3D)):1 with yerrorbars pt 0 lw 2 palette, dataIsing3D using ($3/$2 * exp(log($1) * betaIsing3D * deltaIsing3D / nuIsing3D)):($4 * exp(-log($1) * gammaIsing3D / nuIsing3D)):1 with lines palette, 5 * exp(log(x) * 2 / deltaIsing3D) with lines lc black + + +unset label +unset ylabel +set size 0.333, 0.55 + +data4Potts2D = "data/cluster-size/cluster-size_4potts2d.dat" + +set xrange [0.000005:9000000] +set yrange [0.6:4] + +set label "2D 4-State Potts" at graph 0.1, 0.75 + +beta4Potts2D = 1.0 / 12.0 +nu4Potts2D = 2.0 / 3.0 +delta4Potts2D = 15.0 +gamma4Potts2D = 7.0 / 6.0 + +plot data4Potts2D using ($3/$2 * exp(log($1) * beta4Potts2D * delta4Potts2D / nu4Potts2D)):($4 * exp(-log($1) * gamma4Potts2D / nu4Potts2D)):($5 * exp(-log($1) * gamma4Potts2D/nu4Potts2D)):1 with yerrorbars pt 0 lw 2 palette, data4Potts2D using ($3/$2 * exp(log($1) * beta4Potts2D * delta4Potts2D / nu4Potts2D)):($4 * exp(-log($1) * gamma4Potts2D / nu4Potts2D)):1 with lines palette, 1.2 * exp(log(x) * 2 / delta4Potts2D) with lines lc black + +unset label +set size 0.36, 0.55 + +data3Vector3D = "data/cluster-size/cluster-size_3vector3d.dat" + +set xrange [0.000005:9000000] +set yrange [0.6:200] + +set label "3D O(3)" at graph 0.1, 0.75 + +beta3Vector3D = 0.3662 +nu3Vector3D = 0.7073 +delta3Vector3D = 4.79465 +gamma3Vector3D = 1.3895 + +plot data3Vector3D using ($3/$2 * exp(log($1) * beta3Vector3D * delta3Vector3D / nu3Vector3D)):($4 * exp(-log($1) * gamma3Vector3D / nu3Vector3D)):($5 * exp(-log($1) * gamma3Vector3D/nu3Vector3D)):1 with yerrorbars pt 0 lw 2 palette, data3Vector3D using ($3/$2 * exp(log($1) * beta3Vector3D * delta3Vector3D / nu3Vector3D)):($4 * exp(-log($1) * gamma3Vector3D / nu3Vector3D)):1 with lines palette, 2 * exp(log(x) * 2 / delta3Vector3D) with lines lc black + +unset multiplot @@ -1,6 +1,6 @@ DOC=monte-carlo -FIGS=fig_correlation_collapse-hL fig_correlation fig_metastable fig_correlation-temp +FIGS=fig_correlation_collapse-hL fig_correlation fig_metastable fig_correlation-temp fig_clusters_ising2d DATA=correlation correlation_temp metastable_raw metastable_func all: ${DOC}.pdf diff --git a/monte-carlo.bib b/monte-carlo.bib index f4afc74..789b06d 100644 --- a/monte-carlo.bib +++ b/monte-carlo.bib @@ -213,6 +213,40 @@ publisher={APS} } + +@article{dimitrovic_finite-size_1991, + title = {Finite-size effects, goldstone bosons and critical exponents in the d = 3 {Heisenberg} model}, + volume = {350}, + issn = {0550-3213}, + url = {http://www.sciencedirect.com/science/article/pii/055032139190167V}, + doi = {10.1016/0550-3213(91)90167-V}, + abstract = {The d = 3 classical O(3) Heisenberg model is studied numerically in the broken phase close to the critical point. The finite-size behaviour of the magnetisation and the correlation functions are shown to be in excellent agreement with the theoretical predictions obtained by chiral perturbation theory. The finite-size effects are governed by two constants, which are defined at infinite volume and zero magnetic field: the Goldstone boson decay constant F (or helicity modulus T = F2) and the magnetisation Σ. The data determine the scaling behaviour of Σ and F leading to the prediction 0.6930(2) for the critical coupling on simple cubic lattices and v′ = 0.73(4) and β = 0.36(2) for the correlation length and magnetisation critical indices, respectively.}, + number = {3}, + urldate = {2018-04-05}, + journal = {Nuclear Physics B}, + author = {Dimitrović, I. and Hasenfratz, P. and Nager, J. and Niedermayer, F.}, + month = feb, + year = {1991}, + pages = {893--905}, + file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/QD7P2N8S/Dimitrović et al. - 1991 - Finite-size effects, goldstone bosons and critical.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/H5G2N432/055032139190167V.html:text/html} +} + +@article{stauffer_scaling_1979, + title = {Scaling theory of percolation clusters}, + volume = {54}, + issn = {0370-1573}, + url = {http://www.sciencedirect.com/science/article/pii/0370157379900607}, + doi = {10.1016/0370-1573(79)90060-7}, + number = {1}, + urldate = {2018-04-20}, + journal = {Physics Reports}, + author = {Stauffer, D.}, + month = jul, + year = {1979}, + pages = {1--74}, + file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/KTNA6MWQ/Stauffer - 1979 - Scaling theory of percolation clusters.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/FYC4TRJZ/0370157379900607.html:text/html} +} + @article{wolff1989comparison, title={Comparison between cluster Monte Carlo algorithms in the Ising model}, author={Wolff, Ulli}, diff --git a/monte-carlo.pdf b/monte-carlo.pdf Binary files differindex 7b65de4..c8a15de 100644 --- a/monte-carlo.pdf +++ b/monte-carlo.pdf diff --git a/monte-carlo.tex b/monte-carlo.tex index 23ce92b..96e5f91 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -96,6 +96,14 @@ \date\today \begin{abstract} + We introduce a generalization of the `ghost spin' representation of spin + systems that restores full symmetry group invariance in an + arbitrary external field via the introduction of a `ghost transformation.' + This offers a natural way to extend celebrated spin-cluster + Monte Carlo algorithms to systems in arbitrary fields by running the + ordinary cluster-flipping process on the new representation. For several + canonical systems, we show that this extension with field preserves the scaling of + dynamics so celebrated without field. \end{abstract} \maketitle @@ -123,16 +131,17 @@ external fields based on applying the ghost site representation spin systems that returns global rotation invariance to spin Hamiltonians at the cost of an extra degree of freedom, but these results only allow the application of a narrow category of fields -\cite{alexandrowicz1989swendsen,destri1992swendsen,lauwers1989critical,wang1989clusters}. We show, by a redefinition of the spin--spin coupling in a -generic class of such systems, systems with arbitrary external fields applied -can be treated using cluster methods. The scaling of correlation +\cite{alexandrowicz1989swendsen,destri1992swendsen,lauwers1989critical,wang1989clusters}. +We show that the scaling of correlation time near the critical point of several models suggests that this approach is a natural one, e.g., that it extends the celebrated scaling of dynamics in -these algorithms at zero field to various non-symmetric perturbations. +these algorithms at zero field to various non-symmetric perturbations. We also show, by a redefinition of the spin--spin coupling in a +generic class of such systems, systems with arbitrary external fields applied +can be treated using cluster methods. Let $G=(V,E)$ be a graph, where the set of vertices $V=\{1,\ldots,N\}$ enumerates the sites of a lattice and the set of edges $E$ contains pairs of -neighboring sites. Let $R$ be a group and $X$ an $R$-set, with the action +neighboring sites. Let $R$ be a group acting on a set $X$, with the action of group elements $r\in R$ on elements $s\in X$ denoted $r\cdot s$. $X$ is the set of states accessible by a spin, and $R$ is the \emph{symmetry group} of $X$. The set $X$ must admit a measure $\mu$ that is invariant under the action of $R$, e.g., for any @@ -274,13 +283,13 @@ coupling $Z$ depends only on the relative orientation of the spins---global reorientations by acting by some rotation do not affect the Hamiltonian. The external field $B$ breaks this symmetry. However, this can be resolved. Define a new graph $\tilde G=(\tilde V,\tilde E)$, where $\tilde V=\{0,1,\ldots,N\}$ -and +adds a new `ghost' site $0$ which is connected by \[ - \tilde E=E\cup\big\{\{0,i\}\mid i\in V\big\}. + \tilde E=E\cup\big\{\{0,i\}\mid i\in V\big\} \] -We have introduced a new site to the lattice that neighbors every other site. -Instead of assigning this site a spin whose value comes from the set $X$, we -will assign it values $s_0\in R$, symmetry group elements, so that the new +to all other sites. +Instead of assigning this ghost site a spin whose value comes from the set $X$, we +will assign it values in the symmetry group $s_0\in R$, so that the new configuration space of the model is $R\times X^N$. We introduce a Hamiltonian $\tilde\H:R\times X^N\to\R$ defined by \[ @@ -375,7 +384,9 @@ To summarize, spin systems in a field may be treated in the following way. lattice, substituting $\tilde Z$ as defined in \eqref{eq:new.z} for $Z$. \end{enumerate} Ensemble averages of observables $A$ can then be estimated by sampling the -value of $\tilde A$ on the new system. +value of $\tilde A$ on the new system. In contrast with the simpler ghost spin +representation, this form of the Hamiltonian mya be considered the ``ghost +transformation'' representation. \section{Examples} @@ -388,7 +399,9 @@ exactly the same as the spins themselves. The only nontrivial element is of order two. Because the symmetry group and the spins are described by the same elements, performing the algorithm on the Ising model in a field is very accurately described by simply adding an extra spin coupled to all others and -running the ordinary algorithm. +running the ordinary algorithm. The ghost spin version of the algorithm has +been applied by several researchers previously +\cite{wang1989clusters,ray1990metastability,destri1992swendsen,lauwers1989critical} \subsection{The $n$-component Model} @@ -401,7 +414,9 @@ through the origin. Since the former generate the entire group, the set of reflections alone suffices to provide ergodicity. Computation of the coupling of ordinary spins with the external field and expectation values requires a matrix inversion, but since the matrices in question are orthogonal this is -quickly accomplished by a transpose. +quickly accomplished by a transpose. The ghost-spin version of the algorithm +has been used to apply a simple vector field by previous researchers +\cite{dimitrovic_finite-size_1991}. \subsection{The Potts \& Clock Models} @@ -461,27 +476,118 @@ one-dimensional space---are equally well described. \section{Dynamic scaling} -We measured the autocorrelation time using methods here -\cite{geyer1992practical}. +No algorithm worthwhile if it doesn't run efficiently. Our algorithm, being an +extension of the Wolff algorithm, should be considered successful if it +likewise extends its efficiency in the systems that algorithm succeeds. The +Wolff algorithm succeeds at -\begin{figure} - \centering - \input{fig_correlation_collapse-hL} - \caption{The correlation time $\tau$ as a function of the renormalization - invarient $hL^{-\beta\delta/\nu}$ for the $N=L\times L$ square lattice - Ising model for $L=8$, $16$, $32$, $64$, $128$, and $256$. $z=0.3$ - } -\end{figure} + +Cluster algorithms were celebrated for their small dynamic exponents $z$, +which with the correlation time $\tau$ scales like $L^z$, where $L=N^{-D}$. In +the vicinity of the critical point, the renormalization group predicts scaling +behavior for the correlation time of the form +\[ + \tau=t^{-z\nu}\mathcal T(ht^{-\beta\delta},Lt^\nu) + =h^{-z\nu/\beta\delta}\mathcal T'(ht^{-\beta\delta},Lt^\nu). +\] +If a given dynamics for a system at zero field results in scaling like +$t^{-z\nu}$, one should expect its natural extension in the presence of a +field to scale like $h^{-z\nu/\beta\delta}$. We measured the autocorrelation +time for the 2D square-lattice model at a variety of system sizes, +temperatures, and fields using methods here +\cite{geyer1992practical}. The resulting scaling behavior, plotted in +Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with the +zero-field scaling behavior. \begin{figure} \centering + \input{fig_correlation_collapse-hL} \input{fig_correlation-temp} - \caption{The correlation time $\tau$ as a function of the renormalization - invarient $ht^{-\beta\delta}$ for the $N=128\times 128$ square lattice - Ising model. $z=0.3$ + \caption{Collapses of the correlation time $\tau$ of the 2D square lattice + Ising model (top) along the critical + isotherm at various systems sizes $N=L\times L$ for $L=8$, $16$, $32$, $64$, $128$, and $256$ as a function of the renormalization + invariant $hL^{\beta\delta/\nu}$ and (bottom) in the low-temperature phase + at $L=128$ for various temperatures as a function of the invariant + $ht^{-\beta\delta}$. } + \label{fig:correlation_time-collapse} \end{figure} +Since the formation and flipping of clusters is the hallmark of the Wolff +dynamics, another way to ensure that the dynamics with field scale like those without is +to analyze the distribution of cluster sizes. The success of the algorithm at +zero field is related to the way that clusters formed undergo a percolation +transition at models' critical point. +According to the scaling theory of percolation \cite{stauffer_scaling_1979}, the distribution of cluster sizes in a full decomposition of the system scales +consistently near the critical point if it has the form +\[ + P_{\text{SW}}(s)=s^{-\tau}f(ts^\sigma,th^{-1/\beta\delta},tL^{1/\nu}). +\] +The distribution of cluster sizes in the Wolff algorithm can be computed from +this using the fact that the algorithm selects clusters with probability +proportional to their size, or +\[ + \begin{aligned} + \avg{s_{\text{\sc 1c}}}&=\sum_ssP_{\text{\sc + 1c}}(s)=\sum_ss\frac sNP_{\text{SW}}(s)\\ + &=t^{-\gamma}g(th^{-1/\beta\delta},tL^{1/\nu})\\ + &=L^{\gamma/\nu}\mathcal G(ht^{-\beta\delta},hL^{\beta\delta/\nu}) + \end{aligned} +\] + +For the Ising model, an additional scaling relation can be written. Since in +that case the average cluster size is the average squared magnetization, it +can be related to the scaling functions of the magnetization and +susceptibility per site by (with $ht^{-\beta\delta}$ dependence dropped) +\[ + \begin{aligned} + \avg{s_{\text{\sc 1c}}} + &=L^{D}\avg{M^2}=\beta\avg\chi+L^{D}\avg{M}^2\\ + &=L^{\gamma/\nu}\big[(hL^{\beta\delta/\nu})^{-\gamma/\beta\delta}\beta \mathcal + Y(hL^{\beta\delta/\nu})\\ + &\hspace{7em}+(hL^{\beta\delta/\nu})^{2/\delta}\mathcal + M(hL^{\beta\delta/\nu})\big]. + \end{aligned} +\] +We therefore expect that, for the Ising model, $\avg{s_{\text{\sc 1c}}}$ +should go as $(hL^{\beta\delta})^{2/\delta}$ for large argument. We further +conjecture that this scaling behavior should hold for other models whose +critical points correspond with the percolation transition of Wolff clusters. +This behavior is supported by our numeric work along the critical isotherm for various Ising, Potts, and +$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. As can be +seen, the average cluster size collapses for each model according to the +scaling hypothesis, and the large-field behavior likewise scales as we expect +from the na\"ive Ising conjecture. + +\begin{figure*} + \input{fig_clusters_ising2d} + \caption{Collapses of rescaled average Wolff cluster size $\avg s_{\text{\sc + 1c}}L^{-\gamma/\nu}$ as + a function of field scaling variable $hL^{\beta\delta/\nu}$ for a variety + of models. Critical exponents $\gamma$, $\nu$, $\beta$, and $\delta$ are + model-dependant. Colored lines and points depict values as measured by the + extended algorithm. Solid black lines show a plot of $f(x)=x^{2/\delta}$ + for each model. + } + \label{fig:cluster_scaling} +\end{figure*} + +We have taken several disparate extensions of cluster methods to models in an +external field and generalized them to any model of a broad class. This new +algorithm has an elegant statement that involves the introduction of not a +ghost spin, but a ghost transformation. We provided evidence that extensions +deriving from this method are the natural way to extend cluster methods tithe +presence of a field, in the sense that it appears to reproduce the scaling +of the dynamics in a field that would be expected from renormalization group +predictions. + +In addition to uniting several extensions of cluster methods under a single +description, our approach allows the application of fields not possible under +prior methods. Instead of simply applying a spin-like field, this method +allows for the application of \emph{arbitrary functions} of the spins. For +instance, theoretical predictions for the effect of symmetry-breaking +perturbations on spin models can be tested numerically +\cite{jose1977renormalization}. \begin{acknowledgments} \end{acknowledgments} |