31:
39:
574:. 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
89:) 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.
1887:
380:
514:
237:, 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
2017:
445:
613:
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
560:
2552:
108:
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
2376:
2979:
1972:
2509:
2489:
2893:
2810:
2820:
2494:
2504:
2862:
1905:
210:-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
2577:
2759:
3049:
3039:
2562:
1182:
1163:
2949:
2913:
183:
above which excluded volume is negligible. A SAW that does not satisfy the excluded volume condition was recently studied to model explicit
2866:
3217:
2954:
2064:
1965:
1351:
1384:
3263:
3019:
2597:
2567:
968:
728:
323:
2870:
2854:
818:
3258:
3064:
2769:
1989:
686:
234:
2969:
2934:
2903:
2898:
2537:
2334:
2251:
2908:
2236:
1269:
sequence A007764 (Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid)
3253:
2532:
2339:
1911:
1053:
Tishby, I.; Biham, O.; Katzav, E. (2016). "The distribution of path lengths of self avoiding walks on ErdĆsâRĂ©nyi networks".
464:
2258:
622:
2994:
2874:
1192:
Madras, N.; Sokal, A. D. (1988). "The pivot algorithm â A highly efficient Monte-Carlo method for the self-avoiding walk".
3222:
2999:
2835:
2734:
2719:
2131:
2047:
1958:
1767:
1724:
578:
networks, the distribution of path lengths of such dynamically grown SAWs can be calculated analytically, and follows the
3009:
2645:
3004:
1845:
2607:
2191:
2136:
2052:
1927:
1411:
575:
2939:
2929:
2572:
2542:
233:
One of the phenomena associated with self-avoiding walks and statistical physics models in general is the notion of
3248:
2944:
2109:
2007:
2655:
2231:
2012:
3024:
2825:
2739:
2724:
2114:
1577:
602:-step self-avoiding walks in the full plane. It is currently unknown whether the limit of the uniform measure as
2858:
2744:
2166:
2246:
2221:
1433:
680:
587:
199:
2964:
2547:
2082:
617:, that is, the limit as the length of the walk goes to infinity and the mesh of the lattice goes to zero. The
413:
1315:
Generic python implementation to simulate SAWs and expanding FiberWalks on a square lattices in n-dimensions.
3159:
3149:
2840:
2622:
2361:
2226:
2037:
1870:
2444:
3101:
3029:
2288:
1919:
1478:
1344:
3124:
3106:
3086:
3081:
2800:
2632:
2612:
2459:
2402:
2241:
2151:
1704:
1396:
579:
533:
222:
93:
2592:
3199:
3154:
3144:
2885:
2830:
2805:
2774:
2754:
2514:
2499:
2366:
1865:
1860:
1650:
1582:
1238:
1201:
1072:
1011:
853:"The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes"
771:
695:
195:
3194:
3034:
2959:
2764:
2524:
2434:
2324:
1623:
1600:
1535:
1483:
1468:
1401:
583:
238:
78:
3164:
3129:
3044:
3014:
2784:
2779:
2602:
2439:
2104:
1981:
1850:
1830:
1789:
1552:
1217:
1119:
1088:
1062:
1035:
1001:
974:
946:
915:
897:
761:
656:
180:
2845:
1107:
3243:
3184:
2989:
2640:
2397:
2314:
2283:
2176:
2156:
2146:
2002:
1997:
1893:
1855:
1779:
1687:
1592:
1498:
1473:
1463:
1406:
1389:
1379:
1374:
1337:
1292:
1178:
1159:
1027:
964:
885:
799:
724:
314:
158:
2850:
2587:
225:. There is currently no known formula, although there are rigorous methods of approximation.
3204:
3091:
2974:
2344:
2319:
2268:
2196:
2119:
2072:
1810:
1677:
1660:
1488:
1246:
1209:
1129:
1080:
1019:
956:
907:
864:
833:
789:
779:
668:
662:
455:
3169:
3069:
3054:
2815:
2749:
2427:
2371:
2354:
2099:
1825:
1762:
1423:
109:
66:
43:
2984:
2216:
1520:
1084:
590:
distribution to a node can be obtained by solving a set of coupled recurrence equations.
17:
1242:
1205:
1076:
1015:
888:(1 May 2012). "The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)".
775:
3174:
3139:
3059:
2665:
2412:
2329:
2298:
2293:
2273:
2263:
2206:
2201:
2181:
2161:
2126:
2094:
2077:
1840:
1784:
1772:
1743:
1699:
1682:
1665:
1618:
1562:
1547:
1515:
1453:
1295:
794:
749:
571:
203:
184:
869:
852:
112:
condition. In higher dimensions, the SAW is believed to behave much like the ordinary
3237:
3076:
2617:
2454:
2449:
2407:
2349:
2171:
2087:
2027:
1694:
1670:
1540:
1510:
1493:
1458:
1443:
1323:
1221:
1092:
618:
614:
310:
978:
960:
919:
3134:
3096:
2650:
2582:
2471:
2466:
2278:
2211:
2186:
2022:
1939:
1934:
1835:
1815:
1572:
1505:
1039:
934:
610:
74:
73:) that does not visit the same point more than once. This is a special case of the
70:
2714:
1108:"Efficient network exploration by means of resetting self-avoiding random walkers"
30:
27:
A sequence of moves on a lattice that does not visit the same point more than once
1319:
911:
784:
143:, whose physical volume prohibits multiple occupation of the same spatial point.
3179:
2698:
2693:
2688:
2678:
2481:
2422:
2417:
2381:
2141:
2032:
1900:
1820:
1530:
1525:
937:; Werner, Wendelin (2004). "On the scaling limit of planar self-avoiding walk".
674:
124:
113:
50:
1309:
1134:
1023:
221:
Calculating the number of self-avoiding walks in any given lattice is a common
3189:
2729:
2673:
2557:
1753:
1738:
1733:
1714:
1448:
716:
191:
132:
2683:
1709:
1655:
1567:
1418:
1300:
745:
582:. For arbitrary networks, the distribution of path lengths of the walk, the
1031:
803:
689: â Properties of systems that are independent of the dynamical details
454:
has only been approximated numerically, and is believed not to even be an
1610:
1006:
837:
211:
120:
62:
1314:
1229:
Fisher, M. E. (1966). "Shape of a self-avoiding walk or polymer chain".
992:
Carlos P. Herrero (2005). "Self-avoiding walks on scale-free networks".
135:
in order to model the real-life behavior of chain-like entities such as
2510:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1950:
1640:
1557:
1360:
1213:
190:
The properties of SAWs cannot be calculated analytically, so numerical
147:
140:
136:
128:
1250:
38:
1628:
951:
750:"The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion"
1124:
1067:
1271:âthe number of self-avoiding paths joining opposite corners of an
902:
766:
214:
transformations (rotations and reflections) on the walk after the
37:
29:
1265:
562:
does not; in other words, this law is believed to be universal.
1954:
1333:
407:
is only known for the hexagonal lattice, where it is equal to:
399:
depends on the particular lattice chosen for the walk so does
375:{\displaystyle \mu =\lim _{n\to \infty }c_{n}^{\frac {1}{n}}.}
42:
Self-avoiding walk on a 20x20 square lattice, simulated using
570:
Self-avoiding walks have also been studied in the context of
131:. Indeed, SAWs may have first been introduced by the chemist
665: â Path in a graph that visits each vertex exactly once
621:
of the self-avoiding walk is conjectured to be described by
1329:
1268:
2490:
Autoregressive conditional heteroskedasticity (ARCH) model
609:
induces a measure on infinite full-plane walks. However,
119:
SAWs and SAPs play a central role in the modeling of the
2018:
Independent and identically distributed random variables
509:{\displaystyle c_{n}\approx \mu ^{n}n^{\frac {11}{32}}}
2495:
Autoregressive integrated moving average (ARIMA) model
677: â Process forming a path from many random steps
536:
530:
depends on the lattice, but the power law correction
467:
416:
326:
691:
Pages displaying wikidata descriptions as a fallback
127:
behavior of thread- and loop-like molecules such as
3117:
2922:
2884:
2793:
2707:
2664:
2631:
2523:
2480:
2390:
2307:
2063:
1988:
1879:
1803:
1752:
1723:
1639:
1609:
1591:
1432:
1367:
1283:from 0 to 12. Also includes an extended list up to
1106:Colombani, G.; Bertagnolli, G.; Artime, O. (2023).
264:-step self avoiding walk can be decomposed into an
1055:Journal of Physics A: Mathematical and Theoretical
554:
508:
439:
374:
2377:Stochastic chains with memory of variable length
671: â Mathematical problem set on a chessboard
334:
659: â Physics associated with critical points
96:, a self-avoiding walk is a chain-like path in
1966:
1345:
945:(2). American Mathematical Society: 339â364.
851:LiĆkiewicz M; Ogihara M; Toda S (July 2003).
8:
34:Self-avoiding walk on a 15Ă15 square lattice
939:Proceedings of Symposia in Pure Mathematics
2505:Autoregressiveâmoving-average (ARMA) model
1973:
1959:
1951:
1352:
1338:
1330:
272:-step self-avoiding walk, it follows that
1133:
1123:
1066:
1005:
950:
901:
868:
793:
783:
765:
723:. Cornell University Press. p. 672.
541:
535:
495:
485:
472:
466:
425:
417:
415:
358:
353:
337:
325:
317:to show that the following limit exists:
440:{\displaystyle {\sqrt {2+{\sqrt {2}}}}.}
1906:List of fractals by Hausdorff dimension
708:
252:-step self-avoiding walks. Since every
2811:Doob's martingale convergence theorems
1310:Java applet of a 2D self-avoiding walk
2563:Constant elasticity of variance (CEV)
2553:ChanâKarolyiâLongstaffâSanders (CKLS)
7:
179:. The dimension is called the upper
586:of the non-visited network and the
187:resulting from expansion of a SAW.
3050:Skorokhod's representation theorem
2831:Law of large numbers (weak/strong)
555:{\displaystyle n^{\frac {11}{32}}}
344:
25:
3020:Martingale representation theorem
1888:How Long Is the Coast of Britain?
3065:Stochastic differential equation
2955:Doob's optional stopping theorem
2950:DoobâMeyer decomposition theorem
598:Consider the uniform measure on
268:-step self-avoiding walk and an
2935:Convergence of random variables
2821:FisherâTippettâGnedenko theorem
721:Principles of Polymer Chemistry
2533:Binomial options pricing model
1912:The Fractal Geometry of Nature
1194:Journal of Statistical Physics
1154:Madras, N.; Slade, G. (1996).
1112:Journal of Physics: Complexity
1085:10.1088/1751-8113/49/28/285002
341:
218:th step to create a new walk.
1:
3000:Kolmogorov continuity theorem
2836:Law of the iterated logarithm
1175:Intersections of Random Walks
870:10.1016/S0304-3975(03)00080-X
168:it is close to 5/3 while for
3005:Kolmogorov extension theorem
2684:Generalized queueing network
2192:Interacting particle systems
912:10.4007/annals.2012.175.3.14
857:Theoretical Computer Science
785:10.1371/journal.pone.0085585
202:simulations for the uniform
2137:Continuous-time random walk
1928:Chaos: Making a New Science
1231:Journal of Chemical Physics
961:10.1090/pspum/072.2/2112127
3280:
3145:Extreme value theory (EVT)
2945:Doob decomposition theorem
2237:OrnsteinâUhlenbeck process
2008:Chinese restaurant process
1024:10.1103/PhysRevE.71.016103
298:. Therefore, the sequence
241:, defined as follows. Let
3213:
3025:Optional stopping theorem
2826:Large deviation principle
2578:HeathâJarrowâMorton (HJM)
2515:Moving-average (MA) model
2500:Autoregressive (AR) model
2325:Hidden Markov model (HMM)
2259:SchrammâLoewner evolution
623:SchrammâLoewner evolution
458:. It is conjectured that
175:the fractal dimension is
18:Self-avoiding random walk
3264:Variants of random walks
2940:Doléans-Dade exponential
2770:Progressively measurable
2568:CoxâIngersollâRoss (CIR)
1322:to generate SAWs on the
1135:10.1088/2632-072X/acff33
817:Hayes B (JulâAug 1998).
698:â All are self-avoiding.
683: â Video game genre
200:Markov chain Monte Carlo
3259:Computational chemistry
3160:Mathematical statistics
3150:Large deviations theory
2980:Infinitesimal generator
2841:Maximal ergodic theorem
2760:Piecewise-deterministic
2362:Random dynamical system
2227:Markov additive process
819:"How to Avoid Yourself"
198:is a common method for
2995:KarhunenâLoĂšve theorem
2930:CameronâMartin formula
2894:BurkholderâDavisâGundy
2289:Variance gamma process
1920:The Beauty of Fractals
1173:Lawler, G. F. (1991).
1156:The Self-Avoiding Walk
556:
510:
441:
376:
46:
44:sequential Monte Carlo
35:
3254:Computational physics
3125:Actuarial mathematics
3087:Uniform integrability
3082:Stratonovich integral
3010:LĂ©vyâProkhorov metric
2914:MarcinkiewiczâZygmund
2801:Central limit theorem
2403:Gaussian random field
2232:McKeanâVlasov process
2152:Dyson Brownian motion
2013:GaltonâWatson process
890:Annals of Mathematics
884:Duminil-Copin, Hugo;
748:; J.S. Weitz (2014).
580:Gompertz distribution
557:
511:
442:
403:. The exact value of
377:
248:denote the number of
223:computational problem
94:computational physics
83:self-avoiding polygon
41:
33:
3200:Time series analysis
3155:Mathematical finance
3040:Reflection principle
2367:Regenerative process
2167:FlemingâViot process
1982:Stochastic processes
1866:Lewis Fry Richardson
1861:Hamid Naderi Yeganeh
1651:Burning Ship fractal
1583:Weierstrass function
1296:"Self-Avoiding Walk"
933:Lawler, Gregory F.;
838:10.1511/1998.31.3301
696:Space-filling curves
534:
465:
450:For other lattices,
414:
324:
3195:Stochastic analysis
3035:Quadratic variation
3030:Prokhorov's theorem
2965:FeynmanâKac formula
2435:Markov random field
2083:Birthâdeath process
1624:Space-filling curve
1601:Multifractal system
1484:Space-filling curve
1469:Sierpinski triangle
1243:1966JChPh..44..616F
1206:1988JSP....50..109M
1077:2016JPhA...49B5002T
1016:2005PhRvE..71a6103H
776:2014PLoSO...985585B
584:degree distribution
390:connective constant
368:
239:connective constant
3165:Probability theory
3045:Skorokhod integral
3015:Malliavin calculus
2598:Korn-Kreer-Lenssen
2482:Time series models
2445:PitmanâYor process
1851:Aleksandr Lyapunov
1831:Desmond Paul Henry
1795:Self-avoiding walk
1790:Percolation theory
1434:Iterated function
1375:Fractal dimensions
1293:Weisstein, Eric W.
1214:10.1007/bf01022990
886:Smirnov, Stanislav
826:American Scientist
657:Critical phenomena
588:first-hitting-time
552:
506:
437:
372:
349:
348:
194:are employed. The
181:critical dimension
150:. For example, in
55:self-avoiding walk
47:
36:
3249:Discrete geometry
3231:
3230:
3185:Signal processing
2904:Doob's upcrossing
2899:Doob's martingale
2863:EngelbertâSchmidt
2806:Donsker's theorem
2740:Feller-continuous
2608:RendlemanâBartter
2398:Dirichlet process
2315:Branching process
2284:Telegraph process
2177:Geometric process
2157:Empirical process
2147:Diffusion process
2003:Branching process
1998:Bernoulli process
1948:
1947:
1894:Coastline paradox
1871:WacĆaw SierpiĆski
1856:Benoit Mandelbrot
1780:Fractal landscape
1688:Misiurewicz point
1593:Strange attractor
1474:Apollonian gasket
1464:Sierpinski carpet
1251:10.1063/1.1726734
1184:978-0-8176-3892-4
1165:978-0-8176-3891-7
549:
503:
432:
430:
366:
333:
313:and we can apply
159:fractal dimension
75:graph theoretical
16:(Redirected from
3271:
3205:Machine learning
3092:Usual hypotheses
2975:Girsanov theorem
2960:Dynkin's formula
2725:Continuous paths
2633:Actuarial models
2573:GarmanâKohlhagen
2543:BlackâKarasinski
2538:BlackâDermanâToy
2525:Financial models
2391:Fields and other
2320:Gaussian process
2269:Sigma-martingale
2073:Additive process
1975:
1968:
1961:
1952:
1811:Michael Barnsley
1678:Lyapunov fractal
1536:SierpiĆski curve
1489:Blancmange curve
1354:
1347:
1340:
1331:
1306:
1305:
1267:
1254:
1225:
1200:(1â2): 109â186.
1188:
1169:
1140:
1139:
1137:
1127:
1103:
1097:
1096:
1070:
1050:
1044:
1043:
1009:
1007:cond-mat/0412658
989:
983:
982:
954:
930:
924:
923:
905:
896:(3): 1653â1665.
881:
875:
874:
872:
848:
842:
841:
823:
814:
808:
807:
797:
787:
769:
741:
735:
734:
713:
692:
663:Hamiltonian path
647:
645:
643:
642:
639:
636:
608:
601:
561:
559:
558:
553:
551:
550:
542:
529:
525:
515:
513:
512:
507:
505:
504:
496:
490:
489:
477:
476:
456:algebraic number
453:
446:
444:
443:
438:
433:
431:
426:
418:
406:
402:
398:
387:
381:
379:
378:
373:
367:
359:
357:
347:
308:
297:
271:
267:
263:
251:
247:
217:
209:
185:surface geometry
178:
174:
167:
156:
107:
101:
21:
3279:
3278:
3274:
3273:
3272:
3270:
3269:
3268:
3234:
3233:
3232:
3227:
3209:
3170:Queueing theory
3113:
3055:Skorokhod space
2918:
2909:KunitaâWatanabe
2880:
2846:Sanov's theorem
2816:Ergodic theorem
2789:
2785:Time-reversible
2703:
2666:Queueing models
2660:
2656:SparreâAnderson
2646:CramĂ©râLundberg
2627:
2613:SABR volatility
2519:
2476:
2428:Boolean network
2386:
2372:Renewal process
2303:
2252:Non-homogeneous
2242:Poisson process
2132:Contact process
2095:Brownian motion
2065:Continuous time
2059:
2053:Maximal entropy
1984:
1979:
1949:
1944:
1875:
1826:Felix Hausdorff
1799:
1763:Brownian motion
1748:
1719:
1642:
1635:
1605:
1587:
1578:Pythagoras tree
1435:
1428:
1424:Self-similarity
1368:Characteristics
1363:
1358:
1320:Norris software
1291:
1290:
1262:
1257:
1228:
1191:
1185:
1172:
1166:
1153:
1149:
1147:Further reading
1144:
1143:
1105:
1104:
1100:
1052:
1051:
1047:
991:
990:
986:
971:
932:
931:
927:
883:
882:
878:
863:(1â3): 129â56.
850:
849:
845:
821:
816:
815:
811:
743:
742:
738:
731:
715:
714:
710:
705:
690:
653:
640:
637:
634:
633:
631:
626:
625:with parameter
603:
599:
596:
568:
537:
532:
531:
527:
520:
491:
481:
468:
463:
462:
451:
412:
411:
404:
400:
397:
393:
385:
322:
321:
305:
299:
295:
291:
285:
273:
269:
265:
253:
249:
246:
242:
231:
215:
207:
196:pivot algorithm
176:
169:
162:
151:
110:excluded volume
103:
97:
28:
23:
22:
15:
12:
11:
5:
3277:
3275:
3267:
3266:
3261:
3256:
3251:
3246:
3236:
3235:
3229:
3228:
3226:
3225:
3220:
3218:List of topics
3214:
3211:
3210:
3208:
3207:
3202:
3197:
3192:
3187:
3182:
3177:
3175:Renewal theory
3172:
3167:
3162:
3157:
3152:
3147:
3142:
3140:Ergodic theory
3137:
3132:
3130:Control theory
3127:
3121:
3119:
3115:
3114:
3112:
3111:
3110:
3109:
3104:
3094:
3089:
3084:
3079:
3074:
3073:
3072:
3062:
3060:Snell envelope
3057:
3052:
3047:
3042:
3037:
3032:
3027:
3022:
3017:
3012:
3007:
3002:
2997:
2992:
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2942:
2937:
2932:
2926:
2924:
2920:
2919:
2917:
2916:
2911:
2906:
2901:
2896:
2890:
2888:
2882:
2881:
2879:
2878:
2859:BorelâCantelli
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2797:
2795:
2794:Limit theorems
2791:
2790:
2788:
2787:
2782:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2711:
2709:
2705:
2704:
2702:
2701:
2696:
2691:
2686:
2681:
2676:
2670:
2668:
2662:
2661:
2659:
2658:
2653:
2648:
2643:
2637:
2635:
2629:
2628:
2626:
2625:
2620:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2550:
2545:
2540:
2535:
2529:
2527:
2521:
2520:
2518:
2517:
2512:
2507:
2502:
2497:
2492:
2486:
2484:
2478:
2477:
2475:
2474:
2469:
2464:
2463:
2462:
2457:
2447:
2442:
2437:
2432:
2431:
2430:
2425:
2415:
2413:Hopfield model
2410:
2405:
2400:
2394:
2392:
2388:
2387:
2385:
2384:
2379:
2374:
2369:
2364:
2359:
2358:
2357:
2352:
2347:
2342:
2332:
2330:Markov process
2327:
2322:
2317:
2311:
2309:
2305:
2304:
2302:
2301:
2299:Wiener sausage
2296:
2294:Wiener process
2291:
2286:
2281:
2276:
2274:Stable process
2271:
2266:
2264:Semimartingale
2261:
2256:
2255:
2254:
2249:
2239:
2234:
2229:
2224:
2219:
2214:
2209:
2207:Jump diffusion
2204:
2199:
2194:
2189:
2184:
2182:Hawkes process
2179:
2174:
2169:
2164:
2162:Feller process
2159:
2154:
2149:
2144:
2139:
2134:
2129:
2127:Cauchy process
2124:
2123:
2122:
2117:
2112:
2107:
2102:
2092:
2091:
2090:
2080:
2078:Bessel process
2075:
2069:
2067:
2061:
2060:
2058:
2057:
2056:
2055:
2050:
2045:
2040:
2030:
2025:
2020:
2015:
2010:
2005:
2000:
1994:
1992:
1986:
1985:
1980:
1978:
1977:
1970:
1963:
1955:
1946:
1945:
1943:
1942:
1937:
1932:
1924:
1916:
1908:
1903:
1898:
1897:
1896:
1883:
1881:
1877:
1876:
1874:
1873:
1868:
1863:
1858:
1853:
1848:
1843:
1841:Helge von Koch
1838:
1833:
1828:
1823:
1818:
1813:
1807:
1805:
1801:
1800:
1798:
1797:
1792:
1787:
1782:
1777:
1776:
1775:
1773:Brownian motor
1770:
1759:
1757:
1750:
1749:
1747:
1746:
1744:Pickover stalk
1741:
1736:
1730:
1728:
1721:
1720:
1718:
1717:
1712:
1707:
1702:
1700:Newton fractal
1697:
1692:
1691:
1690:
1683:Mandelbrot set
1680:
1675:
1674:
1673:
1668:
1666:Newton fractal
1663:
1653:
1647:
1645:
1637:
1636:
1634:
1633:
1632:
1631:
1621:
1619:Fractal canopy
1615:
1613:
1607:
1606:
1604:
1603:
1597:
1595:
1589:
1588:
1586:
1585:
1580:
1575:
1570:
1565:
1563:Vicsek fractal
1560:
1555:
1550:
1545:
1544:
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1502:
1501:
1491:
1481:
1479:Fibonacci word
1476:
1471:
1466:
1461:
1456:
1454:Koch snowflake
1451:
1446:
1440:
1438:
1430:
1429:
1427:
1426:
1421:
1416:
1415:
1414:
1409:
1404:
1399:
1394:
1393:
1392:
1382:
1371:
1369:
1365:
1364:
1359:
1357:
1356:
1349:
1342:
1334:
1328:
1327:
1317:
1312:
1307:
1288:
1261:
1260:External links
1258:
1256:
1255:
1237:(2): 616â622.
1226:
1189:
1183:
1177:. BirkhÀuser.
1170:
1164:
1158:. BirkhÀuser.
1150:
1148:
1145:
1142:
1141:
1098:
1061:(28): 285002.
1045:
984:
969:
925:
876:
843:
809:
736:
729:
707:
706:
704:
701:
700:
699:
693:
684:
678:
672:
666:
660:
652:
649:
595:
592:
572:network theory
567:
564:
548:
545:
540:
517:
516:
502:
499:
494:
488:
484:
480:
475:
471:
448:
447:
436:
429:
424:
421:
395:
388:is called the
383:
382:
371:
365:
362:
356:
352:
346:
343:
340:
336:
332:
329:
315:Fekete's lemma
303:
293:
289:
277:
244:
230:
227:
125:knot-theoretic
65:of moves on a
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3276:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3241:
3239:
3224:
3221:
3219:
3216:
3215:
3212:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3122:
3120:
3116:
3108:
3105:
3103:
3100:
3099:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3077:Stopping time
3075:
3071:
3068:
3067:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2927:
2925:
2921:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2891:
2889:
2887:
2883:
2876:
2872:
2868:
2867:HewittâSavage
2864:
2860:
2856:
2852:
2851:Zeroâone laws
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2798:
2796:
2792:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2712:
2710:
2706:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2671:
2669:
2667:
2663:
2657:
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2638:
2636:
2634:
2630:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2548:BlackâScholes
2546:
2544:
2541:
2539:
2536:
2534:
2531:
2530:
2528:
2526:
2522:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2488:
2487:
2485:
2483:
2479:
2473:
2470:
2468:
2465:
2461:
2458:
2456:
2453:
2452:
2451:
2450:Point process
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2429:
2426:
2424:
2421:
2420:
2419:
2416:
2414:
2411:
2409:
2408:Gibbs measure
2406:
2404:
2401:
2399:
2396:
2395:
2393:
2389:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2337:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2312:
2310:
2306:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2253:
2250:
2248:
2245:
2244:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2197:ItĂŽ diffusion
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2172:Gamma process
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2133:
2130:
2128:
2125:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2097:
2096:
2093:
2089:
2086:
2085:
2084:
2081:
2079:
2076:
2074:
2071:
2070:
2068:
2066:
2062:
2054:
2051:
2049:
2046:
2044:
2043:Self-avoiding
2041:
2039:
2036:
2035:
2034:
2031:
2029:
2028:Moran process
2026:
2024:
2021:
2019:
2016:
2014:
2011:
2009:
2006:
2004:
2001:
1999:
1996:
1995:
1993:
1991:
1990:Discrete time
1987:
1983:
1976:
1971:
1969:
1964:
1962:
1957:
1956:
1953:
1941:
1938:
1936:
1933:
1930:
1929:
1925:
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1917:
1914:
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1827:
1824:
1822:
1819:
1817:
1814:
1812:
1809:
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1802:
1796:
1793:
1791:
1788:
1786:
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1781:
1778:
1774:
1771:
1769:
1768:Brownian tree
1766:
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1760:
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1726:
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1716:
1713:
1711:
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1703:
1701:
1698:
1696:
1695:Multibrot set
1693:
1689:
1686:
1685:
1684:
1681:
1679:
1676:
1672:
1671:Douady rabbit
1669:
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1569:
1566:
1564:
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1559:
1556:
1554:
1551:
1549:
1546:
1542:
1541:Z-order curve
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1511:Hilbert curve
1509:
1507:
1504:
1500:
1497:
1496:
1495:
1494:De Rham curve
1492:
1490:
1487:
1486:
1485:
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1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1459:Menger sponge
1457:
1455:
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1450:
1447:
1445:
1444:Barnsley fern
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1366:
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1355:
1350:
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1325:
1324:Diamond cubic
1321:
1318:
1316:
1313:
1311:
1308:
1303:
1302:
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1025:
1021:
1017:
1013:
1008:
1003:
999:
995:
988:
985:
980:
976:
972:
970:0-8218-3638-2
966:
962:
958:
953:
948:
944:
940:
936:
935:Schramm, Oded
929:
926:
921:
917:
913:
909:
904:
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880:
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871:
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862:
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827:
820:
813:
810:
805:
801:
796:
791:
786:
781:
777:
773:
768:
763:
760:(1): e85585.
759:
755:
751:
747:
740:
737:
732:
730:9780801401343
726:
722:
718:
712:
709:
702:
697:
694:
688:
685:
682:
679:
676:
673:
670:
669:Knight's tour
667:
664:
661:
658:
655:
654:
650:
648:
629:
624:
620:
619:scaling limit
616:
615:scaling limit
612:
606:
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577:
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68:
64:
60:
56:
52:
45:
40:
32:
19:
3135:Econometrics
3097:Wiener space
2985:ItĂŽ integral
2886:Inequalities
2775:Self-similar
2745:GaussâMarkov
2735:Exchangeable
2715:CĂ dlĂ g paths
2651:Risk process
2603:LIBOR market
2472:Random graph
2467:Random field
2279:Superprocess
2217:LĂ©vy process
2212:Jump process
2187:Hunt process
2042:
2023:Markov chain
1940:Chaos theory
1935:Kaleidoscope
1926:
1918:
1910:
1836:Gaston Julia
1816:Georg Cantor
1794:
1641:Escape-time
1573:Gosper curve
1521:LĂ©vy C curve
1506:Dragon curve
1385:Box-counting
1299:
1284:
1280:
1276:
1272:
1234:
1230:
1197:
1193:
1174:
1155:
1115:
1111:
1101:
1058:
1054:
1048:
997:
994:Phys. Rev. E
993:
987:
952:math/0204277
942:
938:
928:
893:
889:
879:
860:
856:
846:
829:
825:
812:
757:
753:
744:A. Bucksch;
739:
720:
711:
687:Universality
627:
611:Harry Kesten
604:
597:
569:
521:
518:
449:
389:
384:
301:
287:
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278:
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235:universality
232:
229:Universality
220:
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161:is 4/3, for
152:
145:
118:
104:
98:
91:
86:
82:
77:notion of a
71:lattice path
58:
54:
48:
3180:Ruin theory
3118:Disciplines
2990:ItĂŽ's lemma
2765:Predictable
2440:Percolation
2423:Potts model
2418:Ising model
2382:White noise
2340:Differences
2202:ItĂŽ process
2142:Cox process
2038:Loop-erased
2033:Random walk
1931:(1987 book)
1923:(1986 book)
1915:(1982 book)
1901:Fractal art
1821:Bill Gosper
1785:LĂ©vy flight
1531:Peano curve
1526:Moore curve
1412:Topological
1397:Correlation
1000:(3): 1728.
675:Random walk
576:ErdĆsâRĂ©nyi
566:On networks
311:subadditive
212:symmetrical
192:simulations
121:topological
114:random walk
51:mathematics
3238:Categories
3190:Statistics
2970:Filtration
2871:Kolmogorov
2855:Blumenthal
2780:Stationary
2720:Continuous
2708:Properties
2593:HullâWhite
2335:Martingale
2222:Local time
2110:Fractional
2088:pure birth
1739:Orbit trap
1734:Buddhabrot
1727:techniques
1715:Mandelbulb
1516:Koch curve
1449:Cantor set
1279:grid, for
1125:2310.03203
1068:1603.06613
832:(4): 314.
703:References
133:Paul Flory
3102:Classical
2115:Geometric
2105:Excursion
1846:Paul LĂ©vy
1725:Rendering
1710:Mandelbox
1656:Julia set
1568:Hexaflake
1499:Minkowski
1419:Recursion
1402:Hausdorff
1301:MathWorld
1222:123272694
1093:119182848
903:1007.0575
767:1304.3521
483:μ
479:≈
345:∞
342:→
328:μ
146:SAWs are
3244:Polygons
3223:Category
3107:Abstract
2641:BĂŒhlmann
2247:Compound
1756:fractals
1643:fractals
1611:L-system
1553:T-square
1361:Fractals
1032:15697654
979:16710180
920:59164280
804:24465607
754:PLOS ONE
719:(1953).
717:P. Flory
651:See also
526:, where
392:, since
148:fractals
141:polymers
137:solvents
129:proteins
63:sequence
2730:Ergodic
2618:VaĆĄĂÄek
2460:Poisson
2120:Meander
1705:Tricorn
1558:n-flake
1407:Packing
1390:Higuchi
1380:Assouad
1275:×
1239:Bibcode
1202:Bibcode
1073:Bibcode
1040:2707668
1012:Bibcode
795:3899046
772:Bibcode
746:G. Turk
644:
632:
204:measure
67:lattice
61:) is a
3070:Tanaka
2755:Mixing
2750:Markov
2623:Wilkie
2588:HoâLee
2583:Heston
2355:Super-
2100:Bridge
2048:Biased
1804:People
1754:Random
1661:Filled
1629:H tree
1548:String
1436:system
1220:
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1038:
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594:Limits
2923:Tools
2699:M/M/c
2694:M/M/1
2689:M/G/1
2679:Fluid
2345:Local
1880:Other
1287:= 21.
1218:S2CID
1120:arXiv
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1089:S2CID
1063:arXiv
1036:S2CID
1002:arXiv
975:S2CID
947:arXiv
916:S2CID
898:arXiv
822:(PDF)
762:arXiv
681:Snake
300:{log
2875:LĂ©vy
2674:Bulk
2558:Chen
2350:Sub-
2308:Both
1266:OEIS
1179:ISBN
1160:ISBN
1028:PMID
965:ISBN
800:PMID
725:ISBN
157:the
139:and
123:and
81:. A
79:path
53:, a
2455:Cox
1247:doi
1210:doi
1130:doi
1081:doi
1020:doi
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865:doi
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790:PMC
780:doi
607:â â
524:â â
519:as
335:lim
309:is
206:on
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487:n
474:n
470:c
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428:2
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294:m
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