20:
28:
563:. In this context, it is customary to treat the SAW as a dynamical process, such that in every time-step a walker randomly hops between neighboring nodes of the network. The walk ends when the walker reaches a dead-end state, such that it can no longer progress to newly un-visited nodes. It was recently found that on
78:) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations.
1876:
369:
503:
226:, that is, independence of macroscopic observables from microscopic details, such as the choice of the lattice. One important quantity that appears in conjectures for universal laws is the
2006:
434:
602:
has shown that such a measure exists for self-avoiding walks in the half-plane. One important question involving self-avoiding walks is the existence and conformal invariance of the
549:
2541:
97:
with a certain number of nodes, typically a fixed step length and has the property that it doesn't cross itself or another walk. A system of SAWs satisfies the so-called
2365:
2968:
1961:
2498:
2478:
2882:
2799:
2809:
2483:
2493:
2851:
1894:
199:-step self-avoiding walks. The pivot algorithm works by taking a self-avoiding walk and randomly choosing a point on this walk, and then applying
2566:
2748:
3038:
3028:
2551:
1171:
1152:
2938:
2902:
172:
above which excluded volume is negligible. A SAW that does not satisfy the excluded volume condition was recently studied to model explicit
2855:
3206:
2943:
2053:
1954:
1340:
1373:
3252:
3008:
2586:
2556:
957:
717:
312:
2859:
2843:
807:
3247:
3053:
2758:
1978:
675:
223:
2958:
2923:
2892:
2887:
2526:
2323:
2240:
2897:
2225:
1258:
sequence A007764 (Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid)
3242:
2521:
2328:
1900:
1042:
Tishby, I.; Biham, O.; Katzav, E. (2016). "The distribution of path lengths of self avoiding walks on ErdĆsâRĂ©nyi networks".
453:
2247:
611:
2983:
2863:
1181:
Madras, N.; Sokal, A. D. (1988). "The pivot algorithm â A highly efficient Monte-Carlo method for the self-avoiding walk".
3211:
2988:
2824:
2723:
2708:
2120:
2036:
1947:
1756:
1713:
567:
networks, the distribution of path lengths of such dynamically grown SAWs can be calculated analytically, and follows the
2998:
2634:
2993:
1834:
2596:
2180:
2125:
2041:
1916:
1400:
564:
2928:
2918:
2561:
2531:
222:
One of the phenomena associated with self-avoiding walks and statistical physics models in general is the notion of
3237:
2933:
2098:
1996:
2644:
2220:
2001:
3013:
2814:
2728:
2713:
2103:
1566:
591:-step self-avoiding walks in the full plane. It is currently unknown whether the limit of the uniform measure as
2847:
2733:
2155:
2235:
2210:
1422:
669:
576:
188:
2953:
2536:
2071:
606:, that is, the limit as the length of the walk goes to infinity and the mesh of the lattice goes to zero. The
402:
1304:
Generic python implementation to simulate SAWs and expanding FiberWalks on a square lattices in n-dimensions.
3148:
3138:
2829:
2611:
2350:
2215:
2026:
1859:
2433:
3090:
3018:
2277:
1908:
1467:
1333:
3113:
3095:
3075:
3070:
2789:
2621:
2601:
2448:
2391:
2230:
2140:
1693:
1385:
568:
522:
211:
82:
2581:
3188:
3143:
3133:
2874:
2819:
2794:
2763:
2743:
2503:
2488:
2355:
1854:
1849:
1639:
1571:
1227:
1190:
1061:
1000:
842:"The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes"
760:
684:
184:
3183:
3023:
2948:
2753:
2513:
2423:
2313:
1612:
1589:
1524:
1472:
1457:
1390:
572:
227:
67:
3153:
3118:
3033:
3003:
2773:
2768:
2591:
2428:
2093:
1970:
1839:
1819:
1778:
1541:
1206:
1108:
1077:
1051:
1024:
990:
963:
935:
904:
886:
750:
645:
169:
2834:
1096:
3232:
3173:
2978:
2629:
2386:
2303:
2272:
2165:
2145:
2135:
1991:
1986:
1882:
1844:
1768:
1676:
1581:
1487:
1462:
1452:
1395:
1378:
1368:
1363:
1326:
1281:
1167:
1148:
1016:
953:
874:
788:
713:
303:
147:
2839:
2576:
214:. There is currently no known formula, although there are rigorous methods of approximation.
3193:
3080:
2963:
2333:
2308:
2257:
2185:
2108:
2061:
1799:
1666:
1649:
1477:
1235:
1198:
1118:
1069:
1008:
945:
896:
853:
822:
778:
768:
657:
651:
444:
3158:
3058:
3043:
2804:
2738:
2416:
2360:
2343:
2088:
1814:
1751:
1412:
98:
55:
32:
2973:
2205:
1509:
1073:
579:
distribution to a node can be obtained by solving a set of coupled recurrence equations.
1231:
1194:
1065:
1004:
877:(1 May 2012). "The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)".
764:
3163:
3128:
3048:
2654:
2401:
2318:
2287:
2282:
2262:
2252:
2195:
2190:
2170:
2150:
2115:
2083:
2066:
1829:
1773:
1761:
1732:
1688:
1671:
1654:
1607:
1551:
1536:
1504:
1442:
1284:
783:
738:
560:
192:
173:
858:
841:
101:
condition. In higher dimensions, the SAW is believed to behave much like the ordinary
3226:
3065:
2606:
2443:
2438:
2396:
2338:
2160:
2076:
2016:
1683:
1659:
1529:
1499:
1482:
1447:
1432:
1312:
1210:
1081:
607:
603:
299:
967:
949:
908:
3123:
3085:
2639:
2571:
2460:
2455:
2267:
2200:
2175:
2011:
1928:
1923:
1824:
1804:
1561:
1494:
1028:
923:
599:
63:
62:) that does not visit the same point more than once. This is a special case of the
59:
2703:
1097:"Efficient network exploration by means of resetting self-avoiding random walkers"
19:
16:
A sequence of moves on a lattice that does not visit the same point more than once
1308:
900:
773:
132:, whose physical volume prohibits multiple occupation of the same spatial point.
3168:
2687:
2682:
2677:
2667:
2470:
2411:
2406:
2370:
2130:
2021:
1889:
1809:
1519:
1514:
926:; Werner, Wendelin (2004). "On the scaling limit of planar self-avoiding walk".
663:
113:
102:
39:
1298:
1123:
1012:
210:
Calculating the number of self-avoiding walks in any given lattice is a common
3178:
2718:
2662:
2546:
1742:
1727:
1722:
1703:
1437:
705:
180:
121:
2672:
1698:
1644:
1556:
1407:
1289:
734:
571:. For arbitrary networks, the distribution of path lengths of the walk, the
1020:
792:
678: â Properties of systems that are independent of the dynamical details
443:
has only been approximated numerically, and is believed not to even be an
1599:
995:
826:
200:
109:
51:
1303:
1218:
Fisher, M. E. (1966). "Shape of a self-avoiding walk or polymer chain".
981:
Carlos P. Herrero (2005). "Self-avoiding walks on scale-free networks".
124:
in order to model the real-life behavior of chain-like entities such as
2499:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1939:
1629:
1546:
1349:
1202:
179:
The properties of SAWs cannot be calculated analytically, so numerical
136:
129:
125:
117:
1239:
27:
1617:
940:
739:"The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion"
1113:
1056:
1260:âthe number of self-avoiding paths joining opposite corners of an
891:
755:
203:
transformations (rotations and reflections) on the walk after the
26:
18:
1254:
551:
does not; in other words, this law is believed to be universal.
1943:
1322:
396:
is only known for the hexagonal lattice, where it is equal to:
388:
depends on the particular lattice chosen for the walk so does
364:{\displaystyle \mu =\lim _{n\to \infty }c_{n}^{\frac {1}{n}}.}
31:
Self-avoiding walk on a 20x20 square lattice, simulated using
559:
Self-avoiding walks have also been studied in the context of
120:. Indeed, SAWs may have first been introduced by the chemist
654: â Path in a graph that visits each vertex exactly once
610:
of the self-avoiding walk is conjectured to be described by
1318:
1257:
2479:
Autoregressive conditional heteroskedasticity (ARCH) model
598:
induces a measure on infinite full-plane walks. However,
108:
SAWs and SAPs play a central role in the modeling of the
2007:
Independent and identically distributed random variables
498:{\displaystyle c_{n}\approx \mu ^{n}n^{\frac {11}{32}}}
2484:
Autoregressive integrated moving average (ARIMA) model
666: â Process forming a path from many random steps
525:
519:
depends on the lattice, but the power law correction
456:
405:
315:
680:
Pages displaying wikidata descriptions as a fallback
116:
behavior of thread- and loop-like molecules such as
3106:
2911:
2873:
2782:
2696:
2653:
2620:
2512:
2469:
2379:
2296:
2052:
1977:
1868:
1792:
1741:
1712:
1628:
1598:
1580:
1421:
1356:
1272:from 0 to 12. Also includes an extended list up to
1095:Colombani, G.; Bertagnolli, G.; Artime, O. (2023).
253:-step self avoiding walk can be decomposed into an
1044:Journal of Physics A: Mathematical and Theoretical
543:
497:
428:
363:
2366:Stochastic chains with memory of variable length
660: â Mathematical problem set on a chessboard
323:
648: â Physics associated with critical points
85:, a self-avoiding walk is a chain-like path in
1955:
1334:
934:(2). American Mathematical Society: 339â364.
840:LiĆkiewicz M; Ogihara M; Toda S (July 2003).
8:
23:Self-avoiding walk on a 15Ă15 square lattice
928:Proceedings of Symposia in Pure Mathematics
2494:Autoregressiveâmoving-average (ARMA) model
1962:
1948:
1940:
1341:
1327:
1319:
261:-step self-avoiding walk, it follows that
1122:
1112:
1055:
994:
939:
890:
857:
782:
772:
754:
712:. Cornell University Press. p. 672.
530:
524:
484:
474:
461:
455:
414:
406:
404:
347:
342:
326:
314:
306:to show that the following limit exists:
429:{\displaystyle {\sqrt {2+{\sqrt {2}}}}.}
1895:List of fractals by Hausdorff dimension
697:
241:-step self-avoiding walks. Since every
2800:Doob's martingale convergence theorems
1299:Java applet of a 2D self-avoiding walk
2552:Constant elasticity of variance (CEV)
2542:ChanâKarolyiâLongstaffâSanders (CKLS)
7:
168:. The dimension is called the upper
575:of the non-visited network and the
176:resulting from expansion of a SAW.
3039:Skorokhod's representation theorem
2820:Law of large numbers (weak/strong)
544:{\displaystyle n^{\frac {11}{32}}}
333:
14:
3009:Martingale representation theorem
1877:How Long Is the Coast of Britain?
3054:Stochastic differential equation
2944:Doob's optional stopping theorem
2939:DoobâMeyer decomposition theorem
587:Consider the uniform measure on
257:-step self-avoiding walk and an
2924:Convergence of random variables
2810:FisherâTippettâGnedenko theorem
710:Principles of Polymer Chemistry
2522:Binomial options pricing model
1901:The Fractal Geometry of Nature
1183:Journal of Statistical Physics
1143:Madras, N.; Slade, G. (1996).
1101:Journal of Physics: Complexity
1074:10.1088/1751-8113/49/28/285002
330:
207:th step to create a new walk.
1:
2989:Kolmogorov continuity theorem
2825:Law of the iterated logarithm
1164:Intersections of Random Walks
859:10.1016/S0304-3975(03)00080-X
157:it is close to 5/3 while for
2994:Kolmogorov extension theorem
2673:Generalized queueing network
2181:Interacting particle systems
901:10.4007/annals.2012.175.3.14
846:Theoretical Computer Science
774:10.1371/journal.pone.0085585
191:simulations for the uniform
2126:Continuous-time random walk
1917:Chaos: Making a New Science
1220:Journal of Chemical Physics
950:10.1090/pspum/072.2/2112127
3269:
3134:Extreme value theory (EVT)
2934:Doob decomposition theorem
2226:OrnsteinâUhlenbeck process
1997:Chinese restaurant process
1013:10.1103/PhysRevE.71.016103
287:. Therefore, the sequence
230:, defined as follows. Let
3202:
3014:Optional stopping theorem
2815:Large deviation principle
2567:HeathâJarrowâMorton (HJM)
2504:Moving-average (MA) model
2489:Autoregressive (AR) model
2314:Hidden Markov model (HMM)
2248:SchrammâLoewner evolution
612:SchrammâLoewner evolution
447:. It is conjectured that
164:the fractal dimension is
3253:Variants of random walks
2929:Doléans-Dade exponential
2759:Progressively measurable
2557:CoxâIngersollâRoss (CIR)
1311:to generate SAWs on the
1124:10.1088/2632-072X/acff33
806:Hayes B (JulâAug 1998).
687:â All are self-avoiding.
672: â Video game genre
189:Markov chain Monte Carlo
3248:Computational chemistry
3149:Mathematical statistics
3139:Large deviations theory
2969:Infinitesimal generator
2830:Maximal ergodic theorem
2749:Piecewise-deterministic
2351:Random dynamical system
2216:Markov additive process
808:"How to Avoid Yourself"
187:is a common method for
2984:KarhunenâLoĂšve theorem
2919:CameronâMartin formula
2883:BurkholderâDavisâGundy
2278:Variance gamma process
1909:The Beauty of Fractals
1162:Lawler, G. F. (1991).
1145:The Self-Avoiding Walk
545:
499:
430:
365:
35:
33:sequential Monte Carlo
24:
3243:Computational physics
3114:Actuarial mathematics
3076:Uniform integrability
3071:Stratonovich integral
2999:LĂ©vyâProkhorov metric
2903:MarcinkiewiczâZygmund
2790:Central limit theorem
2392:Gaussian random field
2221:McKeanâVlasov process
2141:Dyson Brownian motion
2002:GaltonâWatson process
879:Annals of Mathematics
873:Duminil-Copin, Hugo;
737:; J.S. Weitz (2014).
569:Gompertz distribution
546:
500:
431:
392:. The exact value of
366:
237:denote the number of
212:computational problem
83:computational physics
72:self-avoiding polygon
30:
22:
3189:Time series analysis
3144:Mathematical finance
3029:Reflection principle
2356:Regenerative process
2156:FlemingâViot process
1971:Stochastic processes
1855:Lewis Fry Richardson
1850:Hamid Naderi Yeganeh
1640:Burning Ship fractal
1572:Weierstrass function
1285:"Self-Avoiding Walk"
922:Lawler, Gregory F.;
827:10.1511/1998.31.3301
685:Space-filling curves
523:
454:
439:For other lattices,
403:
313:
3184:Stochastic analysis
3024:Quadratic variation
3019:Prokhorov's theorem
2954:FeynmanâKac formula
2424:Markov random field
2072:Birthâdeath process
1613:Space-filling curve
1590:Multifractal system
1473:Space-filling curve
1458:Sierpinski triangle
1232:1966JChPh..44..616F
1195:1988JSP....50..109M
1066:2016JPhA...49B5002T
1005:2005PhRvE..71a6103H
765:2014PLoSO...985585B
573:degree distribution
379:connective constant
357:
228:connective constant
3154:Probability theory
3034:Skorokhod integral
3004:Malliavin calculus
2587:Korn-Kreer-Lenssen
2471:Time series models
2434:PitmanâYor process
1840:Aleksandr Lyapunov
1820:Desmond Paul Henry
1784:Self-avoiding walk
1779:Percolation theory
1423:Iterated function
1364:Fractal dimensions
1282:Weisstein, Eric W.
1203:10.1007/bf01022990
875:Smirnov, Stanislav
815:American Scientist
646:Critical phenomena
577:first-hitting-time
541:
495:
426:
361:
338:
337:
183:are employed. The
170:critical dimension
139:. For example, in
44:self-avoiding walk
36:
25:
3238:Discrete geometry
3220:
3219:
3174:Signal processing
2893:Doob's upcrossing
2888:Doob's martingale
2852:EngelbertâSchmidt
2795:Donsker's theorem
2729:Feller-continuous
2597:RendlemanâBartter
2387:Dirichlet process
2304:Branching process
2273:Telegraph process
2166:Geometric process
2146:Empirical process
2136:Diffusion process
1992:Branching process
1987:Bernoulli process
1937:
1936:
1883:Coastline paradox
1860:WacĆaw SierpiĆski
1845:Benoit Mandelbrot
1769:Fractal landscape
1677:Misiurewicz point
1582:Strange attractor
1463:Apollonian gasket
1453:Sierpinski carpet
1240:10.1063/1.1726734
1173:978-0-8176-3892-4
1154:978-0-8176-3891-7
538:
492:
421:
419:
355:
322:
302:and we can apply
148:fractal dimension
64:graph theoretical
3260:
3194:Machine learning
3081:Usual hypotheses
2964:Girsanov theorem
2949:Dynkin's formula
2714:Continuous paths
2622:Actuarial models
2562:GarmanâKohlhagen
2532:BlackâKarasinski
2527:BlackâDermanâToy
2514:Financial models
2380:Fields and other
2309:Gaussian process
2258:Sigma-martingale
2062:Additive process
1964:
1957:
1950:
1941:
1800:Michael Barnsley
1667:Lyapunov fractal
1525:SierpiĆski curve
1478:Blancmange curve
1343:
1336:
1329:
1320:
1295:
1294:
1256:
1243:
1214:
1189:(1â2): 109â186.
1177:
1158:
1129:
1128:
1126:
1116:
1092:
1086:
1085:
1059:
1039:
1033:
1032:
998:
996:cond-mat/0412658
978:
972:
971:
943:
919:
913:
912:
894:
885:(3): 1653â1665.
870:
864:
863:
861:
837:
831:
830:
812:
803:
797:
796:
786:
776:
758:
730:
724:
723:
702:
681:
652:Hamiltonian path
636:
634:
632:
631:
628:
625:
597:
590:
550:
548:
547:
542:
540:
539:
531:
518:
514:
504:
502:
501:
496:
494:
493:
485:
479:
478:
466:
465:
445:algebraic number
442:
435:
433:
432:
427:
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420:
415:
407:
395:
391:
387:
376:
370:
368:
367:
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356:
348:
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297:
286:
260:
256:
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198:
174:surface geometry
167:
163:
156:
145:
96:
90:
3268:
3267:
3263:
3262:
3261:
3259:
3258:
3257:
3223:
3222:
3221:
3216:
3198:
3159:Queueing theory
3102:
3044:Skorokhod space
2907:
2898:KunitaâWatanabe
2869:
2835:Sanov's theorem
2805:Ergodic theorem
2778:
2774:Time-reversible
2692:
2655:Queueing models
2649:
2645:SparreâAnderson
2635:CramĂ©râLundberg
2616:
2602:SABR volatility
2508:
2465:
2417:Boolean network
2375:
2361:Renewal process
2292:
2241:Non-homogeneous
2231:Poisson process
2121:Contact process
2084:Brownian motion
2054:Continuous time
2048:
2042:Maximal entropy
1973:
1968:
1938:
1933:
1864:
1815:Felix Hausdorff
1788:
1752:Brownian motion
1737:
1708:
1631:
1624:
1594:
1576:
1567:Pythagoras tree
1424:
1417:
1413:Self-similarity
1357:Characteristics
1352:
1347:
1309:Norris software
1280:
1279:
1251:
1246:
1217:
1180:
1174:
1161:
1155:
1142:
1138:
1136:Further reading
1133:
1132:
1094:
1093:
1089:
1041:
1040:
1036:
980:
979:
975:
960:
921:
920:
916:
872:
871:
867:
852:(1â3): 129â56.
839:
838:
834:
810:
805:
804:
800:
732:
731:
727:
720:
704:
703:
699:
694:
679:
642:
629:
626:
623:
622:
620:
615:
614:with parameter
592:
588:
585:
557:
526:
521:
520:
516:
509:
480:
470:
457:
452:
451:
440:
401:
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311:
310:
294:
288:
284:
280:
274:
262:
258:
254:
242:
238:
235:
231:
220:
204:
196:
185:pivot algorithm
165:
158:
151:
140:
99:excluded volume
92:
86:
17:
12:
11:
5:
3266:
3264:
3256:
3255:
3250:
3245:
3240:
3235:
3225:
3224:
3218:
3217:
3215:
3214:
3209:
3207:List of topics
3203:
3200:
3199:
3197:
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3164:Renewal theory
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3129:Ergodic theory
3126:
3121:
3119:Control theory
3116:
3110:
3108:
3104:
3103:
3101:
3100:
3099:
3098:
3093:
3083:
3078:
3073:
3068:
3063:
3062:
3061:
3051:
3049:Snell envelope
3046:
3041:
3036:
3031:
3026:
3021:
3016:
3011:
3006:
3001:
2996:
2991:
2986:
2981:
2976:
2971:
2966:
2961:
2956:
2951:
2946:
2941:
2936:
2931:
2926:
2921:
2915:
2913:
2909:
2908:
2906:
2905:
2900:
2895:
2890:
2885:
2879:
2877:
2871:
2870:
2868:
2867:
2848:BorelâCantelli
2837:
2832:
2827:
2822:
2817:
2812:
2807:
2802:
2797:
2792:
2786:
2784:
2783:Limit theorems
2780:
2779:
2777:
2776:
2771:
2766:
2761:
2756:
2751:
2746:
2741:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2700:
2698:
2694:
2693:
2691:
2690:
2685:
2680:
2675:
2670:
2665:
2659:
2657:
2651:
2650:
2648:
2647:
2642:
2637:
2632:
2626:
2624:
2618:
2617:
2615:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2559:
2554:
2549:
2544:
2539:
2534:
2529:
2524:
2518:
2516:
2510:
2509:
2507:
2506:
2501:
2496:
2491:
2486:
2481:
2475:
2473:
2467:
2466:
2464:
2463:
2458:
2453:
2452:
2451:
2446:
2436:
2431:
2426:
2421:
2420:
2419:
2414:
2404:
2402:Hopfield model
2399:
2394:
2389:
2383:
2381:
2377:
2376:
2374:
2373:
2368:
2363:
2358:
2353:
2348:
2347:
2346:
2341:
2336:
2331:
2321:
2319:Markov process
2316:
2311:
2306:
2300:
2298:
2294:
2293:
2291:
2290:
2288:Wiener sausage
2285:
2283:Wiener process
2280:
2275:
2270:
2265:
2263:Stable process
2260:
2255:
2253:Semimartingale
2250:
2245:
2244:
2243:
2238:
2228:
2223:
2218:
2213:
2208:
2203:
2198:
2196:Jump diffusion
2193:
2188:
2183:
2178:
2173:
2171:Hawkes process
2168:
2163:
2158:
2153:
2151:Feller process
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2116:Cauchy process
2113:
2112:
2111:
2106:
2101:
2096:
2091:
2081:
2080:
2079:
2069:
2067:Bessel process
2064:
2058:
2056:
2050:
2049:
2047:
2046:
2045:
2044:
2039:
2034:
2029:
2019:
2014:
2009:
2004:
1999:
1994:
1989:
1983:
1981:
1975:
1974:
1969:
1967:
1966:
1959:
1952:
1944:
1935:
1934:
1932:
1931:
1926:
1921:
1913:
1905:
1897:
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1887:
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1872:
1870:
1866:
1865:
1863:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1830:Helge von Koch
1827:
1822:
1817:
1812:
1807:
1802:
1796:
1794:
1790:
1789:
1787:
1786:
1781:
1776:
1771:
1766:
1765:
1764:
1762:Brownian motor
1759:
1748:
1746:
1739:
1738:
1736:
1735:
1733:Pickover stalk
1730:
1725:
1719:
1717:
1710:
1709:
1707:
1706:
1701:
1696:
1691:
1689:Newton fractal
1686:
1681:
1680:
1679:
1672:Mandelbrot set
1669:
1664:
1663:
1662:
1657:
1655:Newton fractal
1652:
1642:
1636:
1634:
1626:
1625:
1623:
1622:
1621:
1620:
1610:
1608:Fractal canopy
1604:
1602:
1596:
1595:
1593:
1592:
1586:
1584:
1578:
1577:
1575:
1574:
1569:
1564:
1559:
1554:
1552:Vicsek fractal
1549:
1544:
1539:
1534:
1533:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1491:
1490:
1480:
1470:
1468:Fibonacci word
1465:
1460:
1455:
1450:
1445:
1443:Koch snowflake
1440:
1435:
1429:
1427:
1419:
1418:
1416:
1415:
1410:
1405:
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1403:
1398:
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1371:
1360:
1358:
1354:
1353:
1348:
1346:
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1331:
1323:
1317:
1316:
1306:
1301:
1296:
1277:
1250:
1249:External links
1247:
1245:
1244:
1226:(2): 616â622.
1215:
1178:
1172:
1166:. BirkhÀuser.
1159:
1153:
1147:. BirkhÀuser.
1139:
1137:
1134:
1131:
1130:
1087:
1050:(28): 285002.
1034:
973:
958:
914:
865:
832:
798:
725:
718:
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695:
693:
690:
689:
688:
682:
673:
667:
661:
655:
649:
641:
638:
584:
581:
561:network theory
556:
553:
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534:
529:
506:
505:
491:
488:
483:
477:
473:
469:
464:
460:
437:
436:
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418:
413:
410:
384:
377:is called the
372:
371:
360:
354:
351:
345:
341:
335:
332:
329:
325:
321:
318:
304:Fekete's lemma
292:
282:
278:
266:
233:
219:
216:
114:knot-theoretic
54:of moves on a
15:
13:
10:
9:
6:
4:
3:
2:
3265:
3254:
3251:
3249:
3246:
3244:
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3239:
3236:
3234:
3231:
3230:
3228:
3213:
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3208:
3205:
3204:
3201:
3195:
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3187:
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3180:
3177:
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3162:
3160:
3157:
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3147:
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3137:
3135:
3132:
3130:
3127:
3125:
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3120:
3117:
3115:
3112:
3111:
3109:
3105:
3097:
3094:
3092:
3089:
3088:
3087:
3084:
3082:
3079:
3077:
3074:
3072:
3069:
3067:
3066:Stopping time
3064:
3060:
3057:
3056:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
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3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
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3000:
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2917:
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2910:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2880:
2878:
2876:
2872:
2865:
2861:
2857:
2856:HewittâSavage
2853:
2849:
2845:
2841:
2840:Zeroâone laws
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
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2801:
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2796:
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2791:
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2770:
2767:
2765:
2762:
2760:
2757:
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2712:
2710:
2707:
2705:
2702:
2701:
2699:
2695:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2669:
2666:
2664:
2661:
2660:
2658:
2656:
2652:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2627:
2625:
2623:
2619:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
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2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2537:BlackâScholes
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2519:
2517:
2515:
2511:
2505:
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2476:
2474:
2472:
2468:
2462:
2459:
2457:
2454:
2450:
2447:
2445:
2442:
2441:
2440:
2439:Point process
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2418:
2415:
2413:
2410:
2409:
2408:
2405:
2403:
2400:
2398:
2397:Gibbs measure
2395:
2393:
2390:
2388:
2385:
2384:
2382:
2378:
2372:
2369:
2367:
2364:
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2279:
2276:
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2269:
2266:
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2261:
2259:
2256:
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2251:
2249:
2246:
2242:
2239:
2237:
2234:
2233:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2186:ItĂŽ diffusion
2184:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2161:Gamma process
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2110:
2107:
2105:
2102:
2100:
2097:
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2090:
2087:
2086:
2085:
2082:
2078:
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2073:
2070:
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2065:
2063:
2060:
2059:
2057:
2055:
2051:
2043:
2040:
2038:
2035:
2033:
2032:Self-avoiding
2030:
2028:
2025:
2024:
2023:
2020:
2018:
2017:Moran process
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1984:
1982:
1980:
1979:Discrete time
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1972:
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1960:
1958:
1953:
1951:
1946:
1945:
1942:
1930:
1927:
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1757:Brownian tree
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1684:Multibrot set
1682:
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1668:
1665:
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1660:Douady rabbit
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1530:Z-order curve
1528:
1526:
1523:
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1511:
1508:
1506:
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1500:Hilbert curve
1498:
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1485:
1484:
1483:De Rham curve
1481:
1479:
1476:
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1471:
1469:
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1448:Menger sponge
1446:
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1433:Barnsley fern
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1314:
1313:Diamond cubic
1310:
1307:
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1300:
1297:
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1006:
1002:
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992:
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984:
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969:
965:
961:
959:0-8218-3638-2
955:
951:
947:
942:
937:
933:
929:
925:
924:Schramm, Oded
918:
915:
910:
906:
902:
898:
893:
888:
884:
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869:
866:
860:
855:
851:
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843:
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833:
828:
824:
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816:
809:
802:
799:
794:
790:
785:
780:
775:
770:
766:
762:
757:
752:
749:(1): e85585.
748:
744:
740:
736:
729:
726:
721:
719:9780801401343
715:
711:
707:
701:
698:
691:
686:
683:
677:
674:
671:
668:
665:
662:
659:
658:Knight's tour
656:
653:
650:
647:
644:
643:
639:
637:
618:
613:
609:
608:scaling limit
605:
604:scaling limit
601:
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45:
41:
34:
29:
21:
3124:Econometrics
3086:Wiener space
2974:ItĂŽ integral
2875:Inequalities
2764:Self-similar
2734:GaussâMarkov
2724:Exchangeable
2704:CĂ dlĂ g paths
2640:Risk process
2592:LIBOR market
2461:Random graph
2456:Random field
2268:Superprocess
2206:LĂ©vy process
2201:Jump process
2176:Hunt process
2031:
2012:Markov chain
1929:Chaos theory
1924:Kaleidoscope
1915:
1907:
1899:
1825:Gaston Julia
1805:Georg Cantor
1783:
1630:Escape-time
1562:Gosper curve
1510:LĂ©vy C curve
1495:Dragon curve
1374:Box-counting
1288:
1273:
1269:
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983:Phys. Rev. E
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878:
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733:A. Bucksch;
728:
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676:Universality
616:
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593:
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558:
510:
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438:
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87:
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71:
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60:lattice path
47:
43:
37:
3169:Ruin theory
3107:Disciplines
2979:ItĂŽ's lemma
2754:Predictable
2429:Percolation
2412:Potts model
2407:Ising model
2371:White noise
2329:Differences
2191:ItĂŽ process
2131:Cox process
2027:Loop-erased
2022:Random walk
1920:(1987 book)
1912:(1986 book)
1904:(1982 book)
1890:Fractal art
1810:Bill Gosper
1774:LĂ©vy flight
1520:Peano curve
1515:Moore curve
1401:Topological
1386:Correlation
989:(3): 1728.
664:Random walk
565:ErdĆsâRĂ©nyi
555:On networks
300:subadditive
201:symmetrical
181:simulations
110:topological
103:random walk
40:mathematics
3227:Categories
3179:Statistics
2959:Filtration
2860:Kolmogorov
2844:Blumenthal
2769:Stationary
2709:Continuous
2697:Properties
2582:HullâWhite
2324:Martingale
2211:Local time
2099:Fractional
2077:pure birth
1728:Orbit trap
1723:Buddhabrot
1716:techniques
1704:Mandelbulb
1505:Koch curve
1438:Cantor set
1268:grid, for
1114:2310.03203
1057:1603.06613
821:(4): 314.
692:References
122:Paul Flory
3091:Classical
2104:Geometric
2094:Excursion
1835:Paul LĂ©vy
1714:Rendering
1699:Mandelbox
1645:Julia set
1557:Hexaflake
1488:Minkowski
1408:Recursion
1391:Hausdorff
1290:MathWorld
1211:123272694
1082:119182848
892:1007.0575
756:1304.3521
472:μ
468:≈
334:∞
331:→
317:μ
135:SAWs are
3233:Polygons
3212:Category
3096:Abstract
2630:BĂŒhlmann
2236:Compound
1745:fractals
1632:fractals
1600:L-system
1542:T-square
1350:Fractals
1021:15697654
968:16710180
909:59164280
793:24465607
743:PLOS ONE
708:(1953).
706:P. Flory
640:See also
515:, where
381:, since
137:fractals
130:polymers
126:solvents
118:proteins
52:sequence
2719:Ergodic
2607:VaĆĄĂÄek
2449:Poisson
2109:Meander
1694:Tricorn
1547:n-flake
1396:Packing
1379:Higuchi
1369:Assouad
1264:×
1228:Bibcode
1191:Bibcode
1062:Bibcode
1029:2707668
1001:Bibcode
784:3899046
761:Bibcode
735:G. Turk
633:
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193:measure
56:lattice
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3059:Tanaka
2744:Mixing
2739:Markov
2612:Wilkie
2577:HoâLee
2572:Heston
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2089:Bridge
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1793:People
1743:Random
1650:Filled
1618:H tree
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1425:system
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2688:M/M/c
2683:M/M/1
2678:M/G/1
2668:Fluid
2334:Local
1869:Other
1276:= 21.
1207:S2CID
1109:arXiv
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1078:S2CID
1052:arXiv
1025:S2CID
991:arXiv
964:S2CID
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887:arXiv
811:(PDF)
751:arXiv
670:Snake
289:{log
2864:LĂ©vy
2663:Bulk
2547:Chen
2339:Sub-
2297:Both
1255:OEIS
1168:ISBN
1149:ISBN
1017:PMID
954:ISBN
789:PMID
714:ISBN
146:the
128:and
112:and
70:. A
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2444:Cox
1236:doi
1199:doi
1119:doi
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