28:
2840:
2583:
2224:
SLE depends on a choice of
Brownian motion on the boundary of the domain, and there are several variations depending on what sort of Brownian motion is used: for example it might start at a fixed point, or start at a uniformly distributed point on the unit circle, or might have a built in drift, and
2329:
Percolation interface: Make a lozenge of equally sized hexagons in the plane. Color its upper and left side with black, and the lower and right side with white. Then color the other hexagons “white” or “black” independently with equal probability 1/2. There is a boundary between the black and the
218:
inherent in such stochastic processes together make it possible to encode these planar curves into a one-dimensional
Brownian motion running on the boundary of the domain (the driving function in Loewner's differential equation). This way, many important questions about the planar models can be
2232:
The two domains most commonly used in
Schramm–Loewner evolution are the upper half plane and the unit disk. Although the Loewner differential equation in these two cases look different, they are equivalent up to changes of variables as the unit disk and the upper half plane are conformally
2835:{\displaystyle \mathbb {P} ={\frac {1}{2}}+{\frac {\Gamma ({\frac {4}{\kappa }})}{{\sqrt {\pi }}\,\Gamma ({\frac {8-\kappa }{2\kappa }})}}{\frac {x_{0}}{y_{0}}}\,_{2}F_{1}\left({\frac {1}{2}},{\frac {4}{\kappa }},{\frac {3}{2}},-\left({\frac {x_{0}}{y_{0}}}\right)^{2}\right)}
1328:
1170:
1463:
3154:
1569:
1723:
213:
models that exhibit conformal invariance. The SLE curves are the scaling limits of interfaces and other non-self-intersecting random curves in these models. The main idea is that the conformal invariance and a certain
2446:
89:
2572:
3218:
2151:
528:
3292:
by Lawler, Schramm and Werner. This allowed derivation of many quantitative properties of loop-erased random walk (some of which were derived earlier by
Richard Kenyon). The related random
1980:
1917:
4739:
1175:
3358:
2026:
1045:
3006:
422:
3048:
2930:
5274:
2084:
292:
945:
903:
361:
1359:
977:
569:
5098:
2863:
1612:
1037:
835:
468:
312:
1852:
1832:
1592:
809:
2054:
1780:
1753:
760:
733:
706:
671:
624:
3055:
5701:
4694:
4649:
861:
5231:
5211:
1808:
1468:
1351:
1017:
997:
780:
644:
593:
442:
381:
332:
259:
5615:
3728:
2233:
equivalent. However a conformal equivalence between them does not preserve the
Brownian motion on their boundaries used to drive Schramm–Loewner evolution.
1620:
5532:
5542:
5216:
4665:
2220:, which is related to curves joining a point on the boundary of a domain to a point in the interior (often curves joining 1 and 0 in the unit disk).
2209:, which is related to curves connecting two points on the boundary of a domain (usually the upper half plane, with the points being 0 and infinity).
5226:
5584:
3775:
Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2004). "Conformal invariance of planar loop-erased random walks and uniform spanning trees".
5299:
5481:
3592:
224:
5771:
5761:
5284:
4467:
4416:
4286:
4091:
4054:
5671:
5635:
2172:. In other words, Schramm–Loewner evolution is a probability measure on planar curves, given as the image of Wiener measure under this map.
2313:
198:
5588:
5939:
5676:
4786:
4687:
4636:
5741:
5319:
5289:
2378:
151:, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy
5592:
5576:
3726:
Garban, Christophe; Pete, Gábor; Schramm, Oded (2013). "Pivotal, cluster and interface measures for critical planar percolation".
5786:
5491:
4711:
5691:
5656:
5625:
5620:
5259:
5056:
4973:
3815:
5630:
4958:
5254:
5061:
5716:
5596:
5965:
5944:
5721:
5557:
5456:
5441:
4853:
4769:
4680:
4329:
4083:
3988:
3970:
236:
5731:
5367:
34:
5726:
2513:
5329:
3485:
Smirnov, Stanislav (2001). "Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits".
4913:
4858:
4774:
3983:
3965:
5661:
5651:
5294:
5264:
4042:
3166:
2105:
473:
5666:
4831:
4729:
3636:
2264:) intersects itself and every point is contained in a loop but the curve is not space-filling (with probability 1).
1323:{\displaystyle {\dfrac {\partial g_{t}(z)}{\partial t}}=g_{t}(z){\dfrac {\zeta (t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.}
5377:
4953:
4734:
5970:
5746:
5547:
5461:
5446:
4836:
4659:
4626:
3680:
Schramm, Oded; Steif, Jeffrey E. (2010). "Quantitative noise sensitivity and exceptional times for percolation".
1923:
1860:
5580:
5466:
4888:
2229:
controls the rate of diffusion of the
Brownian motion, and the behavior of SLE depends critically on its value.
4968:
4943:
4632:
4576:
4531:
4228:
3321:
5686:
5269:
4804:
1986:
1165:{\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}=-zf_{t}^{\prime }(z){\frac {\zeta (t)+z}{\zeta (t)-z}}}
3369:
2193:
There are two versions of SLE, using two families of curves, each depending on a non-negative real parameter
5881:
5871:
5562:
5344:
5083:
4948:
4759:
3285:
2933:
2282:
596:
175:
5166:
5823:
5751:
5010:
3831:
2942:
386:
262:
3995:
Kager, Wouter; Nienhuis, Bernard (2004), "A Guide to
Stochastic Loewner Evolution and its Applications",
5846:
5828:
5808:
5803:
5522:
5354:
5334:
5181:
5124:
4963:
4873:
3777:
3682:
3297:
3018:
2872:
678:
210:
171:
124:
5314:
5921:
5876:
5866:
5607:
5552:
5527:
5496:
5476:
5236:
5221:
5088:
4396:
4311:
4270:
4014:
3978:
3932:
3823:
3600:
3557:
3504:
3408:
3282:
for percolation. This breakthrough, in turn, allowed further analysis of many aspects of this model.
5916:
5756:
5681:
5486:
5246:
5156:
5046:
3960:
3836:
3634:
Smirnov, Stanislav; Werner, Wendelin (2001). "Critical exponents for two-dimensional percolation".
2481:
2323:
2059:
4572:
4527:
4175:
1458:{\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}={\frac {2f_{t}^{\prime }(z)}{\zeta (t)-z}}}
275:
5886:
5851:
5766:
5736:
5506:
5501:
5324:
5161:
4826:
4764:
4703:
4606:
4588:
4561:
4543:
4496:
4445:
4386:
4366:
4338:
4259:
4213:
4187:
4112:
4030:
4004:
3948:
3922:
3899:
3873:
3785:
3754:
3736:
3708:
3690:
3662:
3644:
3616:
3547:
3494:
3468:
3460:
3442:
3416:
3398:
3259:
2348:
2299:
206:
194:
152:
95:
5567:
908:
866:
337:
2357: = 8 corresponds to the path separating the uniform spanning tree from its dual tree.
953:
533:
5906:
5711:
5362:
5119:
5036:
5005:
4898:
4878:
4868:
4724:
4719:
4643:
4463:
4412:
4282:
4087:
4050:
3520:
3312:
3279:
3271:
3263:
3252:
3012:
572:
5572:
5309:
3149:{\displaystyle {\frac {\kappa }{2}}\partial _{ww}h(w)+{\frac {4w}{w^{2}+1}}\partial _{w}h=0.}
2848:
1597:
1353:
is the upper half plane the
Loewner equation differs from this by changes of variable and is
1022:
814:
447:
297:
189:
Besides UST and LERW, the
Schramm–Loewner evolution is conjectured or proven to describe the
5926:
5813:
5696:
5066:
5041:
4990:
4918:
4841:
4794:
4598:
4553:
4506:
4455:
4404:
4348:
4251:
4243:
4197:
4150:
4022:
3940:
3883:
3841:
3795:
3746:
3700:
3654:
3608:
3584:
3565:
3512:
3452:
1837:
1817:
1577:
788:
4520:
4477:
4426:
4362:
4209:
4101:
4064:
3895:
2032:
1758:
1731:
738:
711:
684:
649:
602:
27:
5891:
5791:
5776:
5537:
5471:
5149:
5093:
5076:
4821:
4516:
4484:
4473:
4441:
4433:
4422:
4381:(2007), "Conformally invariant scaling limits: an overview and a collection of problems",
4358:
4295:
4255:
4224:
4205:
4171:
4097:
4060:
3891:
3433:
Schramm, Oded; Sheffield, Scott (2005), "Harmonic explorer and its convergence to SLE4.",
2303:
1564:{\displaystyle {\dfrac {\partial g_{t}(z)}{\partial t}}={\dfrac {2}{g_{t}(z)-\zeta (t)}}.}
215:
183:
156:
5706:
4938:
4026:
2322: = 4 corresponds to the path of the harmonic explorer and contour lines of the
840:
220:
4400:
4299:
4129:
4018:
3936:
3827:
3604:
3561:
3508:
3412:
17:
5896:
5861:
5781:
5387:
5134:
5051:
5020:
5015:
4995:
4985:
4928:
4923:
4903:
4883:
4848:
4816:
4799:
4318:, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
4163:
4138:
4125:
4108:
2866:
2366:
1793:
1336:
1002:
982:
765:
629:
578:
427:
366:
317:
244:
4557:
3569:
3516:
5959:
5798:
5339:
5176:
5171:
5129:
5071:
4893:
4809:
4749:
4610:
4275:
4263:
3758:
3620:
3225:
269:
190:
120:
4565:
4370:
3952:
3750:
3712:
5856:
5818:
5372:
5304:
5193:
5188:
5000:
4933:
4908:
4744:
4378:
4324:
4217:
4167:
4034:
3666:
3472:
3275:
2330:
white, running from bottom left to top right. The scaling limit of the boundary is
2099:
given by the
Loewner equation as in the previous section, for the driving function
1718:{\displaystyle f_{t}(\zeta (t))=\gamma (t){\text{ or }}\zeta (t)=g_{t}(\gamma (t))}
223:. Indeed, several mathematically non-rigorous predictions made by physicists using
163:
5436:
4155:
3903:
4327:(2000), "Scaling limits of loop-erased random walks and uniform spanning trees",
4077:
5901:
5420:
5415:
5410:
5400:
5203:
5144:
5139:
5103:
4863:
4754:
3704:
3293:
3229:
202:
179:
4511:
3456:
5911:
5451:
5395:
5279:
4602:
4201:
3944:
3910:
3658:
4316:
Univalent functions, with a chapter on quadratic differentials by Gerd Jensen
3799:
3524:
5405:
4229:"Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I"
2298:
has the restriction property and is conjectured to be the scaling limit of
4068:
147:
which gives a family of random curves from two fixed boundary points, and
143:
controlling how much the curve turns. There are two main variants of SLE,
3927:
266:
4548:
4450:
4343:
4082:, Mathematical Surveys and Monographs, vol. 114, Providence, R.I.:
4009:
2484:
of the paths (with probability 1) is equal to min(2, 1 +
5232:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
4672:
4593:
4408:
4353:
4247:
3612:
3464:
3420:
3220:, which was used in the construction of the harmonic explorer, and for
119:), is a family of random planar curves that have been proven to be the
4579:(2002b), "Conformal Field Theories of Stochastic Loewner Evolutions",
3887:
31:
Schramm–Loewner evolution on the upper half plane with hue indicating
4501:
4391:
4192:
3878:
3845:
3790:
3695:
3649:
3447:
3403:
4459:
3813:
Kenyon, Richard (2000). "Long range properties of spanning trees".
2361:
When SLE corresponds to some conformal field theory, the parameter
178:(LERW) probabilistic processes, and developed by him together with
4117:
3741:
3552:
3499:
26:
4049:, Walter de Gruyter GmbH & Co. KG, Berlin, pp. 261–293,
3538:
Smirnov, Stanislav (2001). "Critical percolation in the plane".
4676:
4534:(2002a), "SLEκ growth processes and conformal field theories",
4436:(2004), "Random planar curves and Schramm–Loewner evolutions",
2347:
has the locality property. This arises in the scaling limit of
2441:{\displaystyle c={\frac {(8-3\kappa )(\kappa -6)}{2\kappa }}.}
2351:
on the triangular lattice and conjecturally on other lattices.
681:. Sometimes it is more convenient to use the inverse function
4320:(Chapter 6 treats the classical theory of Loewner's equation)
3228:, which was used by Smirnov to prove conformal invariance in
4440:, Lecture Notes in Math., vol. 1840, Berlin, New York:
3864:
Beffara, Vincent (2008), "The dimension of the SLE curves",
2190:, but is instead the unbounded component of the complement.
1418:
1104:
4141:(2009), "Conformal invariance and 2D statistical physics",
1039:
is parameterized by "capacity", then Loewner's equation is
5212:
Autoregressive conditional heteroskedasticity (ARCH) model
3015:
to obtain the following partial differential equation for
2175:
In general the curve γ need not be simple, and the domain
2285:, or equivalently, branches of the uniform spanning tree.
4740:
Independent and identically distributed random variables
4487:(2005), "Conformal restriction and related questions",
2936:. This was derived by using the martingale property of
1834:
is a Brownian motion of diffusivity zero. The function
5217:
Autoregressive integrated moving average (ARIMA) model
4176:"The dimension of the planar Brownian frontier is 4/3"
3177:
3029:
193:
of various stochastic processes in the plane, such as
4043:"An introduction to the stochastic Loewner evolution"
3372:
to simulate Schramm Loewner Evolution planar curves.
3324:
3169:
3058:
3021:
2945:
2875:
2851:
2586:
2516:
2381:
2108:
2062:
2035:
1989:
1926:
1863:
1840:
1820:
1796:
1761:
1734:
1623:
1600:
1580:
1514:
1473:
1471:
1362:
1339:
1240:
1180:
1178:
1048:
1025:
1005:
985:
956:
911:
869:
843:
817:
791:
768:
741:
714:
687:
673:, then it satisfies a differential equation found by
652:
632:
605:
581:
536:
476:
450:
430:
389:
369:
340:
334:
starting on the boundary (a continuous function with
320:
300:
278:
247:
37:
3240:
5839:
5644:
5606:
5515:
5429:
5386:
5353:
5245:
5202:
5112:
5029:
4785:
4710:
84:{\displaystyle \log(\operatorname {Im} (g_{t}(z)))}
4274:
3487:Comptes Rendus de l'Académie des Sciences, Série I
3352:
3212:
3148:
3042:
3000:
2924:
2857:
2834:
2567:{\displaystyle x_{0}+iy_{0}=z_{0}\in \mathbb {H} }
2566:
2440:
2312: = 3 is the limit of interfaces for the
2145:
2078:
2048:
2020:
1974:
1911:
1846:
1826:
1802:
1774:
1747:
1717:
1606:
1586:
1563:
1457:
1345:
1322:
1164:
1031:
1011:
991:
971:
939:
897:
855:
829:
803:
774:
754:
727:
700:
665:
638:
618:
587:
563:
522:
462:
436:
416:
375:
355:
326:
306:
286:
253:
123:of a variety of two-dimensional lattice models in
83:
5099:Stochastic chains with memory of variable length
4383:International Congress of Mathematicians. Vol. I
3251:that the boundary of planar Brownian motion has
2056:is the upper half-plane with the line from 0 to
3389:Schramm, Oded (2001a), "Percolation formula.",
3278:, this led to the determination of many of the
3213:{\displaystyle 1-{\tfrac {1}{\pi }}\arg(z_{0})}
2455: < 1 corresponds to two values of
2146:{\displaystyle \zeta (t)={\sqrt {\kappa }}B(t)}
523:{\displaystyle D_{t}=D\smallsetminus \gamma ()}
170:) as a conjectured scaling limit of the planar
2095:Schramm–Loewner evolution is the random curve
4688:
4438:Lectures on probability theory and statistics
4385:, Eur. Math. Soc., Zürich, pp. 513–543,
2471:
2468:
8:
4661:Conformally Invariant Scaling Limits and SLE
4079:Conformally invariant processes in the plane
3368:Computer programs (Matlab) are presented in
2302:. A version of it is the outer boundary of
1975:{\displaystyle g_{t}(z)={\sqrt {z^{2}+4t}}}
1912:{\displaystyle f_{t}(z)={\sqrt {z^{2}-4t}}}
1854:is thus identically zero almost surely and
1810:be the upper half plane and consider an SLE
5227:Autoregressive–moving-average (ARMA) model
4695:
4681:
4673:
4648:: CS1 maint: location missing publisher (
4301:Introduction to Schramm–Loewner evolutions
3913:(2005), "SLE for theoretical physicists",
3248:
626:is a suitable normalized isomorphism from
4592:
4547:
4510:
4500:
4449:
4390:
4374:Schramm's original paper, introducing SLE
4352:
4342:
4191:
4154:
4116:
4008:
3926:
3877:
3835:
3789:
3740:
3694:
3648:
3551:
3540:Comptes Rendus de l'Académie des Sciences
3498:
3446:
3402:
3353:{\displaystyle d=1+{\frac {\kappa }{8}}.}
3337:
3323:
3201:
3176:
3168:
3131:
3112:
3097:
3073:
3059:
3057:
3028:
3020:
2978:
2968:
2967:
2944:
2889:
2879:
2874:
2850:
2821:
2809:
2799:
2793:
2772:
2759:
2746:
2735:
2725:
2724:
2715:
2705:
2699:
2670:
2663:
2656:
2641:
2632:
2619:
2607:
2598:
2588:
2587:
2585:
2560:
2559:
2550:
2537:
2521:
2515:
2492:Left passage probability formulas for SLE
2388:
2380:
2124:
2107:
2069:
2061:
2040:
2034:
2011:
1988:
1955:
1949:
1931:
1925:
1892:
1886:
1868:
1862:
1839:
1819:
1795:
1766:
1760:
1739:
1733:
1691:
1667:
1628:
1622:
1599:
1579:
1524:
1513:
1483:
1472:
1470:
1417:
1412:
1402:
1373:
1363:
1361:
1338:
1298:
1262:
1239:
1224:
1190:
1179:
1177:
1118:
1103:
1098:
1059:
1049:
1047:
1024:
1004:
984:
955:
916:
910:
874:
868:
842:
816:
790:
767:
746:
740:
719:
713:
692:
686:
657:
651:
631:
610:
604:
580:
535:
481:
475:
449:
429:
388:
368:
339:
319:
299:
280:
279:
277:
246:
60:
36:
2575:
2275:) is space-filling (with probability 1).
2164:) is Brownian motion on the boundary of
2021:{\displaystyle \gamma (t)=2i{\sqrt {t}}}
135:, it gives a family of random curves in
3381:
2477:
2463:between 0 and 4, and a "dual" value 16/
674:
167:
5533:Doob's martingale convergence theorems
4641:
4581:Communications in Mathematical Physics
3770:
3768:
3585:"Scaling relations for 2D-percolation"
227:have been proven using this strategy.
5285:Constant elasticity of variance (CEV)
5275:Chan–Karolyi–Longstaff–Sanders (CKLS)
4111:(2007), "Schramm–Loewner Evolution",
3315:of a curve by the following relation
7:
3241:Lawler, Schramm & Werner (2001b)
3001:{\displaystyle h(x,y):=\mathbb {P} }
735:, which is a conformal mapping from
417:{\displaystyle \gamma ((0,\infty ))}
5772:Skorokhod's representation theorem
5553:Law of large numbers (weak/strong)
4637:University of California, Berkeley
4045:, in Kaimanovich, Vadim A. (ed.),
4027:10.1023/B:JOSS.0000028058.87266.be
3128:
3070:
3043:{\displaystyle w:={\tfrac {x}{y}}}
2925:{\displaystyle _{2}F_{1}(a,b,c,d)}
2852:
2664:
2635:
2281: = 2 corresponds to the
1500:
1476:
1390:
1366:
1207:
1183:
1076:
1052:
837:, and the boundary values at time
571:is simply connected and therefore
405:
131:and a domain in the complex plane
25:
5742:Martingale representation theorem
2510:being on the left of fixed point
2372:of the conformal field theory by
2253:) is simple (with probability 1).
979:taking values in the boundary of
5787:Stochastic differential equation
5677:Doob's optional stopping theorem
5672:Doob–Meyer decomposition theorem
4625:Lawler; Schramm; Werner (2001),
3274:. Combined with earlier work of
5657:Convergence of random variables
5543:Fisher–Tippett–Gnedenko theorem
3751:10.1090/S0894-0347-2013-00772-9
3266:was proved to be related to SLE
1019:is the unit disk and the curve
5255:Binomial options pricing model
4277:The Fractal Geometry of Nature
3307:Rohde and Schramm showed that
3207:
3194:
3091:
3085:
2995:
2980: passes to the left
2972:
2961:
2949:
2919:
2895:
2693:
2667:
2651:
2638:
2613:
2600: passes to the left
2592:
2501:The probability of chordal SLE
2421:
2409:
2406:
2391:
2260: < 8 the curve γ(
2249: < 4 the curve γ(
2140:
2134:
2118:
2112:
1999:
1993:
1943:
1937:
1880:
1874:
1712:
1709:
1703:
1697:
1681:
1675:
1664:
1658:
1649:
1646:
1640:
1634:
1551:
1545:
1536:
1530:
1495:
1489:
1443:
1437:
1429:
1423:
1385:
1379:
1310:
1304:
1288:
1282:
1274:
1268:
1252:
1246:
1236:
1230:
1202:
1196:
1150:
1144:
1130:
1124:
1115:
1109:
1071:
1065:
966:
960:
928:
922:
886:
880:
558:
555:
543:
540:
517:
514:
502:
499:
411:
408:
396:
393:
350:
344:
78:
75:
72:
66:
53:
44:
1:
5722:Kolmogorov continuity theorem
5558:Law of the iterated logarithm
4558:10.1016/S0370-2693(02)02423-1
4330:Israel Journal of Mathematics
4180:Mathematical Research Letters
4156:10.1090/S0273-0979-08-01229-9
4084:American Mathematical Society
3570:10.1016/S0764-4442(01)01991-7
3517:10.1016/S0764-4442(01)01991-7
2079:{\displaystyle 2i{\sqrt {t}}}
677:, p. 121) in his work on the
237:Loewner differential equation
219:translated into exercises in
186:in a series of joint papers.
5727:Kolmogorov extension theorem
5406:Generalized queueing network
4914:Interacting particle systems
4131:Stochastic Loewner Evolution
3977:Gutlyanskii, V.Ya. (2001) ,
3300:was shown to converge to SLE
3288:was shown to converge to SLE
1782:are extended by continuity.
947:. The equation depends on a
287:{\displaystyle \mathbb {C} }
108:stochastic Loewner evolution
4859:Continuous-time random walk
4076:Lawler, Gregory F. (2005),
4041:Lawler, Gregory F. (2004),
3984:Encyclopedia of Mathematics
3966:Encyclopedia of Mathematics
3705:10.4007/annals.2010.171.619
3247:to prove the conjecture of
2472:Bauer & Bernard (2002b)
2469:Bauer & Bernard (2002a)
2271: ≥ 8 the curve γ(
5987:
5867:Extreme value theory (EVT)
5667:Doob decomposition theorem
4959:Ornstein–Uhlenbeck process
4730:Chinese restaurant process
4512:10.1214/154957805100000113
3457:10.1214/009117905000000477
2300:self-avoiding random walks
1814:, so the driving function
940:{\displaystyle g_{0}(z)=z}
898:{\displaystyle f_{0}(z)=z}
356:{\displaystyle \gamma (0)}
234:
5935:
5747:Optional stopping theorem
5548:Large deviation principle
5300:Heath–Jarrow–Morton (HJM)
5237:Moving-average (MA) model
5222:Autoregressive (AR) model
5047:Hidden Markov model (HMM)
4981:Schramm–Loewner evolution
4603:10.1007/s00220-003-0881-x
4202:10.4310/mrl.2001.v8.n4.a1
4047:Random walks and geometry
3945:10.1016/j.aop.2005.04.001
3866:The Annals of Probability
3659:10.4310/mrl.2001.v8.n6.a4
2182:is not the complement of
2091:Schramm–Loewner evolution
972:{\displaystyle \zeta (t)}
564:{\displaystyle \gamma ()}
100:Schramm–Loewner evolution
5662:Doléans-Dade exponential
5492:Progressively measurable
5290:Cox–Ingersoll–Ross (CIR)
4654:( video of MSRI lecture)
4633:Lawrence Hall of Science
3959:Goluzina, E.G. (2001) ,
18:Schramm–Loewner equation
5882:Mathematical statistics
5872:Large deviations theory
5702:Infinitesimal generator
5563:Maximal ergodic theorem
5482:Piecewise-deterministic
5084:Random dynamical system
4949:Markov additive process
3286:Loop-erased random walk
2934:hypergeometric function
2858:{\displaystyle \Gamma }
2283:loop-erased random walk
1607:{\displaystyle \gamma }
1032:{\displaystyle \gamma }
830:{\displaystyle t\geq 0}
785:In Loewner's equation,
597:Riemann mapping theorem
463:{\displaystyle t\geq 0}
307:{\displaystyle \gamma }
176:loop-erased random walk
5717:Karhunen–Loève theorem
5652:Cameron–Martin formula
5616:Burkholder–Davis–Gundy
5011:Variance gamma process
4658:Schramm, Oded (2001),
4143:Bull. Amer. Math. Soc.
3800:10.1214/aop/1079021469
3583:Kesten, Harry (1987).
3370:this GitHub repository
3354:
3214:
3150:
3044:
3002:
2926:
2859:
2836:
2568:
2442:
2168:, scaled by some real
2147:
2080:
2050:
2022:
1976:
1913:
1848:
1847:{\displaystyle \zeta }
1828:
1827:{\displaystyle \zeta }
1804:
1776:
1749:
1719:
1608:
1588:
1587:{\displaystyle \zeta }
1565:
1459:
1347:
1324:
1166:
1033:
1013:
993:
973:
941:
899:
857:
831:
805:
804:{\displaystyle z\in D}
776:
756:
729:
702:
667:
640:
620:
589:
573:conformally isomorphic
565:
524:
464:
438:
418:
377:
357:
328:
308:
288:
255:
225:conformal field theory
91:
85:
5847:Actuarial mathematics
5809:Uniform integrability
5804:Stratonovich integral
5732:Lévy–Prokhorov metric
5636:Marcinkiewicz–Zygmund
5523:Central limit theorem
5125:Gaussian random field
4954:McKean–Vlasov process
4874:Dyson Brownian motion
4735:Galton–Watson process
4669:(Slides from a talk.)
4312:Pommerenke, Christian
4071:on September 18, 2009
3435:Annals of Probability
3355:
3298:uniform spanning tree
3215:
3151:
3045:
3003:
2927:
2860:
2837:
2569:
2467:greater than 4. (see
2443:
2292: = 8/3, SLE
2256:For 4 <
2225:so on. The parameter
2148:
2081:
2051:
2049:{\displaystyle D_{t}}
2023:
1977:
1914:
1849:
1829:
1805:
1777:
1775:{\displaystyle g_{t}}
1750:
1748:{\displaystyle f_{t}}
1720:
1609:
1589:
1574:The driving function
1566:
1460:
1348:
1325:
1167:
1034:
1014:
994:
974:
942:
900:
858:
832:
806:
777:
757:
755:{\displaystyle D_{t}}
730:
728:{\displaystyle f_{t}}
703:
701:{\displaystyle g_{t}}
679:Bieberbach conjecture
668:
666:{\displaystyle D_{t}}
641:
621:
619:{\displaystyle f_{t}}
590:
566:
525:
465:
439:
419:
378:
358:
329:
314:is a simple curve in
309:
289:
256:
211:statistical mechanics
209:, and other critical
174:(UST) and the planar
172:uniform spanning tree
162:It was discovered by
125:statistical mechanics
86:
30:
5966:Stochastic processes
5922:Time series analysis
5877:Mathematical finance
5762:Reflection principle
5089:Regenerative process
4889:Fleming–Viot process
4704:Stochastic processes
4444:, pp. 107–195,
3322:
3167:
3056:
3019:
2943:
2873:
2849:
2584:
2514:
2379:
2349:critical percolation
2106:
2060:
2033:
1987:
1924:
1861:
1838:
1818:
1794:
1759:
1732:
1621:
1598:
1578:
1469:
1360:
1337:
1176:
1046:
1023:
1003:
983:
954:
909:
867:
841:
815:
789:
766:
739:
712:
685:
650:
630:
603:
579:
534:
474:
448:
428:
387:
367:
338:
318:
298:
276:
245:
231:The Loewner equation
199:critical Ising model
195:critical percolation
153:conformal invariance
127:. Given a parameter
35:
5917:Stochastic analysis
5757:Quadratic variation
5752:Prokhorov's theorem
5687:Feynman–Kac formula
5157:Markov random field
4805:Birth–death process
4489:Probability Surveys
4401:2006math......2151S
4019:2004JSP...115.1149K
3937:2005AnPhy.318...81C
3828:2000JMP....41.1338K
3729:J. Amer. Math. Soc.
3605:1987CMaPh.109..109K
3562:2001CRASM.333..239S
3509:2001CRASM.333..239S
3413:2001math......7096S
2482:Hausdorff dimension
2341: = 6, SLE
2324:Gaussian free field
1422:
1108:
856:{\displaystyle t=0}
363:on the boundary of
207:self-avoiding walks
5887:Probability theory
5767:Skorokhod integral
5737:Malliavin calculus
5320:Korn-Kreer-Lenssen
5204:Time series models
5167:Pitman–Yor process
4354:10.1007/BF02803524
4271:Mandelbrot, Benoît
4248:10.1007/BF01448091
4164:Lawler, Gregory F.
4139:Lawler, Gregory F.
4126:Lawler, Gregory F.
4109:Lawler, Gregory F.
4003:(5/6): 1149–1229,
3613:10.1007/BF01205674
3350:
3311:is related to the
3280:critical exponents
3264:triangular lattice
3210:
3186:
3146:
3040:
3038:
2998:
2922:
2855:
2832:
2564:
2438:
2365:is related to the
2245:For 0 ≤
2237:Special values of
2143:
2076:
2046:
2018:
1972:
1909:
1844:
1824:
1800:
1772:
1745:
1715:
1604:
1584:
1561:
1556:
1508:
1455:
1408:
1343:
1320:
1315:
1215:
1162:
1094:
1029:
1009:
989:
969:
937:
895:
853:
827:
801:
772:
752:
725:
698:
663:
636:
616:
585:
561:
520:
460:
434:
414:
373:
353:
324:
304:
284:
251:
203:double-dimer model
96:probability theory
92:
81:
5953:
5952:
5907:Signal processing
5626:Doob's upcrossing
5621:Doob's martingale
5585:Engelbert–Schmidt
5528:Donsker's theorem
5462:Feller-continuous
5330:Rendleman–Bartter
5120:Dirichlet process
5037:Branching process
5006:Telegraph process
4899:Geometric process
4879:Empirical process
4869:Diffusion process
4725:Branching process
4720:Bernoulli process
4536:Physics Letters B
4469:978-3-540-21316-1
4418:978-3-03719-022-7
4288:978-0-7167-1186-5
4281:, W. H. Freeman,
4093:978-0-8218-3677-4
4056:978-3-11-017237-9
3979:"Löwner equation"
3915:Annals of Physics
3888:10.1214/07-AOP364
3593:Comm. Math. Phys.
3345:
3313:fractal dimension
3272:Stanislav Smirnov
3253:fractal dimension
3249:Mandelbrot (1982)
3185:
3125:
3067:
3037:
2981:
2815:
2780:
2767:
2754:
2721:
2697:
2691:
2661:
2649:
2627:
2601:
2433:
2129:
2074:
2016:
1970:
1907:
1803:{\displaystyle D}
1670:
1555:
1507:
1453:
1397:
1346:{\displaystyle D}
1314:
1214:
1160:
1083:
1012:{\displaystyle D}
992:{\displaystyle D}
775:{\displaystyle D}
639:{\displaystyle D}
588:{\displaystyle D}
470:, the complement
444:), then for each
437:{\displaystyle D}
376:{\displaystyle D}
327:{\displaystyle D}
254:{\displaystyle D}
16:(Redirected from
5978:
5971:Complex analysis
5927:Machine learning
5814:Usual hypotheses
5697:Girsanov theorem
5682:Dynkin's formula
5447:Continuous paths
5355:Actuarial models
5295:Garman–Kohlhagen
5265:Black–Karasinski
5260:Black–Derman–Toy
5247:Financial models
5113:Fields and other
5042:Gaussian process
4991:Sigma-martingale
4795:Additive process
4697:
4690:
4683:
4674:
4668:
4653:
4647:
4639:
4613:
4596:
4568:
4551:
4542:(1–2): 135–138,
4523:
4514:
4504:
4485:Werner, Wendelin
4480:
4453:
4434:Werner, Wendelin
4429:
4409:10.4171/022-1/20
4394:
4373:
4356:
4346:
4319:
4307:
4306:
4291:
4280:
4266:
4242:(1–2): 103–121,
4233:
4220:
4195:
4172:Werner, Wendelin
4159:
4158:
4134:
4121:
4120:
4104:
4072:
4067:, archived from
4037:
4012:
3991:
3973:
3955:
3930:
3928:cond-mat/0503313
3906:
3881:
3872:(4): 1421–1452,
3850:
3849:
3846:10.1063/1.533190
3839:
3822:(3): 1338–1363.
3810:
3804:
3803:
3793:
3772:
3763:
3762:
3744:
3723:
3717:
3716:
3698:
3677:
3671:
3670:
3652:
3637:Math. Res. Lett.
3631:
3625:
3624:
3589:
3580:
3574:
3573:
3555:
3535:
3529:
3528:
3502:
3482:
3476:
3475:
3450:
3441:(6): 2127–2148,
3430:
3424:
3423:
3406:
3386:
3359:
3357:
3356:
3351:
3346:
3338:
3219:
3217:
3216:
3211:
3206:
3205:
3187:
3178:
3163:= 4, the RHS is
3155:
3153:
3152:
3147:
3136:
3135:
3126:
3124:
3117:
3116:
3106:
3098:
3081:
3080:
3068:
3060:
3049:
3047:
3046:
3041:
3039:
3030:
3007:
3005:
3004:
2999:
2982:
2979:
2971:
2931:
2929:
2928:
2923:
2894:
2893:
2884:
2883:
2864:
2862:
2861:
2856:
2841:
2839:
2838:
2833:
2831:
2827:
2826:
2825:
2820:
2816:
2814:
2813:
2804:
2803:
2794:
2781:
2773:
2768:
2760:
2755:
2747:
2740:
2739:
2730:
2729:
2722:
2720:
2719:
2710:
2709:
2700:
2698:
2696:
2692:
2690:
2682:
2671:
2662:
2657:
2654:
2650:
2642:
2633:
2628:
2620:
2612:
2611:
2602:
2599:
2591:
2574:was computed by
2573:
2571:
2570:
2565:
2563:
2555:
2554:
2542:
2541:
2526:
2525:
2480:showed that the
2447:
2445:
2444:
2439:
2434:
2432:
2424:
2389:
2152:
2150:
2149:
2144:
2130:
2125:
2085:
2083:
2082:
2077:
2075:
2070:
2055:
2053:
2052:
2047:
2045:
2044:
2027:
2025:
2024:
2019:
2017:
2012:
1981:
1979:
1978:
1973:
1971:
1960:
1959:
1950:
1936:
1935:
1918:
1916:
1915:
1910:
1908:
1897:
1896:
1887:
1873:
1872:
1853:
1851:
1850:
1845:
1833:
1831:
1830:
1825:
1809:
1807:
1806:
1801:
1781:
1779:
1778:
1773:
1771:
1770:
1754:
1752:
1751:
1746:
1744:
1743:
1724:
1722:
1721:
1716:
1696:
1695:
1671:
1668:
1633:
1632:
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1570:
1568:
1567:
1562:
1557:
1554:
1529:
1528:
1515:
1509:
1506:
1498:
1488:
1487:
1474:
1464:
1462:
1461:
1456:
1454:
1452:
1432:
1421:
1416:
1403:
1398:
1396:
1388:
1378:
1377:
1364:
1352:
1350:
1349:
1344:
1329:
1327:
1326:
1321:
1316:
1313:
1303:
1302:
1277:
1267:
1266:
1241:
1229:
1228:
1216:
1213:
1205:
1195:
1194:
1181:
1171:
1169:
1168:
1163:
1161:
1159:
1139:
1119:
1107:
1102:
1084:
1082:
1074:
1064:
1063:
1050:
1038:
1036:
1035:
1030:
1018:
1016:
1015:
1010:
998:
996:
995:
990:
978:
976:
975:
970:
949:driving function
946:
944:
943:
938:
921:
920:
904:
902:
901:
896:
879:
878:
862:
860:
859:
854:
836:
834:
833:
828:
810:
808:
807:
802:
781:
779:
778:
773:
761:
759:
758:
753:
751:
750:
734:
732:
731:
726:
724:
723:
707:
705:
704:
699:
697:
696:
672:
670:
669:
664:
662:
661:
645:
643:
642:
637:
625:
623:
622:
617:
615:
614:
594:
592:
591:
586:
570:
568:
567:
562:
529:
527:
526:
521:
486:
485:
469:
467:
466:
461:
443:
441:
440:
435:
423:
421:
420:
415:
382:
380:
379:
374:
362:
360:
359:
354:
333:
331:
330:
325:
313:
311:
310:
305:
293:
291:
290:
285:
283:
263:simply connected
260:
258:
257:
252:
164:Oded Schramm
106:, also known as
90:
88:
87:
82:
65:
64:
21:
5986:
5985:
5981:
5980:
5979:
5977:
5976:
5975:
5956:
5955:
5954:
5949:
5931:
5892:Queueing theory
5835:
5777:Skorokhod space
5640:
5631:Kunita–Watanabe
5602:
5568:Sanov's theorem
5538:Ergodic theorem
5511:
5507:Time-reversible
5425:
5388:Queueing models
5382:
5378:Sparre–Anderson
5368:Cramér–Lundberg
5349:
5335:SABR volatility
5241:
5198:
5150:Boolean network
5108:
5094:Renewal process
5025:
4974:Non-homogeneous
4964:Poisson process
4854:Contact process
4817:Brownian motion
4787:Continuous time
4781:
4775:Maximal entropy
4706:
4701:
4657:
4640:
4624:
4621:
4616:
4571:
4549:math-ph/0206028
4526:
4483:
4470:
4451:math.PR/0303354
4442:Springer-Verlag
4432:
4419:
4377:
4344:math.PR/9904022
4323:
4310:
4304:
4294:
4289:
4269:
4231:
4223:
4162:
4137:
4124:
4107:
4094:
4075:
4057:
4040:
4010:math-ph/0312056
3994:
3976:
3961:"Löwner method"
3958:
3909:
3863:
3859:
3857:Further reading
3854:
3853:
3812:
3811:
3807:
3784:(1B): 939–995.
3774:
3773:
3766:
3735:(4): 939–1024.
3725:
3724:
3720:
3679:
3678:
3674:
3633:
3632:
3628:
3587:
3582:
3581:
3577:
3537:
3536:
3532:
3484:
3483:
3479:
3432:
3431:
3427:
3391:Electron. Comm.
3388:
3387:
3383:
3378:
3366:
3320:
3319:
3303:
3291:
3269:
3246:
3238:
3226:Cardy's formula
3224:= 6, we obtain
3197:
3165:
3164:
3127:
3108:
3107:
3099:
3069:
3054:
3053:
3017:
3016:
2941:
2940:
2885:
2876:
2871:
2870:
2847:
2846:
2805:
2795:
2789:
2788:
2745:
2741:
2731:
2723:
2711:
2701:
2683:
2672:
2655:
2634:
2603:
2582:
2581:
2576:Schramm (2001a)
2546:
2533:
2517:
2512:
2511:
2506:
2499:
2497:
2425:
2390:
2377:
2376:
2346:
2334: = 6.
2304:Brownian motion
2297:
2242:
2218:
2207:
2180:
2104:
2103:
2093:
2058:
2057:
2036:
2031:
2030:
1985:
1984:
1951:
1927:
1922:
1921:
1888:
1864:
1859:
1858:
1836:
1835:
1816:
1815:
1813:
1792:
1791:
1788:
1762:
1757:
1756:
1735:
1730:
1729:
1687:
1624:
1619:
1618:
1614:are related by
1596:
1595:
1576:
1575:
1520:
1519:
1499:
1479:
1475:
1467:
1466:
1433:
1404:
1389:
1369:
1365:
1358:
1357:
1335:
1334:
1294:
1278:
1258:
1242:
1220:
1206:
1186:
1182:
1174:
1173:
1140:
1120:
1075:
1055:
1051:
1044:
1043:
1021:
1020:
1001:
1000:
981:
980:
952:
951:
912:
907:
906:
870:
865:
864:
839:
838:
813:
812:
787:
786:
764:
763:
742:
737:
736:
715:
710:
709:
688:
683:
682:
653:
648:
647:
628:
627:
606:
601:
600:
577:
576:
532:
531:
477:
472:
471:
446:
445:
426:
425:
385:
384:
365:
364:
336:
335:
316:
315:
296:
295:
274:
273:
243:
242:
239:
233:
216:Markov property
184:Wendelin Werner
157:Markov property
118:
102:with parameter
56:
33:
32:
23:
22:
15:
12:
11:
5:
5984:
5982:
5974:
5973:
5968:
5958:
5957:
5951:
5950:
5948:
5947:
5942:
5940:List of topics
5936:
5933:
5932:
5930:
5929:
5924:
5919:
5914:
5909:
5904:
5899:
5897:Renewal theory
5894:
5889:
5884:
5879:
5874:
5869:
5864:
5862:Ergodic theory
5859:
5854:
5852:Control theory
5849:
5843:
5841:
5837:
5836:
5834:
5833:
5832:
5831:
5826:
5816:
5811:
5806:
5801:
5796:
5795:
5794:
5784:
5782:Snell envelope
5779:
5774:
5769:
5764:
5759:
5754:
5749:
5744:
5739:
5734:
5729:
5724:
5719:
5714:
5709:
5704:
5699:
5694:
5689:
5684:
5679:
5674:
5669:
5664:
5659:
5654:
5648:
5646:
5642:
5641:
5639:
5638:
5633:
5628:
5623:
5618:
5612:
5610:
5604:
5603:
5601:
5600:
5581:Borel–Cantelli
5570:
5565:
5560:
5555:
5550:
5545:
5540:
5535:
5530:
5525:
5519:
5517:
5516:Limit theorems
5513:
5512:
5510:
5509:
5504:
5499:
5494:
5489:
5484:
5479:
5474:
5469:
5464:
5459:
5454:
5449:
5444:
5439:
5433:
5431:
5427:
5426:
5424:
5423:
5418:
5413:
5408:
5403:
5398:
5392:
5390:
5384:
5383:
5381:
5380:
5375:
5370:
5365:
5359:
5357:
5351:
5350:
5348:
5347:
5342:
5337:
5332:
5327:
5322:
5317:
5312:
5307:
5302:
5297:
5292:
5287:
5282:
5277:
5272:
5267:
5262:
5257:
5251:
5249:
5243:
5242:
5240:
5239:
5234:
5229:
5224:
5219:
5214:
5208:
5206:
5200:
5199:
5197:
5196:
5191:
5186:
5185:
5184:
5179:
5169:
5164:
5159:
5154:
5153:
5152:
5147:
5137:
5135:Hopfield model
5132:
5127:
5122:
5116:
5114:
5110:
5109:
5107:
5106:
5101:
5096:
5091:
5086:
5081:
5080:
5079:
5074:
5069:
5064:
5054:
5052:Markov process
5049:
5044:
5039:
5033:
5031:
5027:
5026:
5024:
5023:
5021:Wiener sausage
5018:
5016:Wiener process
5013:
5008:
5003:
4998:
4996:Stable process
4993:
4988:
4986:Semimartingale
4983:
4978:
4977:
4976:
4971:
4961:
4956:
4951:
4946:
4941:
4936:
4931:
4929:Jump diffusion
4926:
4921:
4916:
4911:
4906:
4904:Hawkes process
4901:
4896:
4891:
4886:
4884:Feller process
4881:
4876:
4871:
4866:
4861:
4856:
4851:
4849:Cauchy process
4846:
4845:
4844:
4839:
4834:
4829:
4824:
4814:
4813:
4812:
4802:
4800:Bessel process
4797:
4791:
4789:
4783:
4782:
4780:
4779:
4778:
4777:
4772:
4767:
4762:
4752:
4747:
4742:
4737:
4732:
4727:
4722:
4716:
4714:
4708:
4707:
4702:
4700:
4699:
4692:
4685:
4677:
4671:
4670:
4655:
4620:
4619:External links
4617:
4615:
4614:
4594:hep-th/0210015
4587:(3): 493–521,
4577:Bernard, Denis
4569:
4532:Bernard, Denis
4524:
4481:
4468:
4460:10.1007/b96719
4430:
4417:
4375:
4321:
4308:
4292:
4287:
4267:
4221:
4186:(4): 401–411,
4160:
4135:
4122:
4105:
4092:
4073:
4055:
4038:
3997:J. Stat. Phys.
3992:
3974:
3956:
3907:
3860:
3858:
3855:
3852:
3851:
3837:10.1.1.39.7560
3816:J. Math. Phys.
3805:
3764:
3718:
3689:(2): 619–672.
3672:
3643:(6): 729–744.
3626:
3599:(1): 109–156.
3575:
3546:(3): 239–244.
3530:
3493:(3): 239–244.
3477:
3425:
3397:(6): 115–120,
3380:
3379:
3377:
3374:
3365:
3362:
3361:
3360:
3349:
3344:
3341:
3336:
3333:
3330:
3327:
3301:
3296:outlining the
3289:
3267:
3244:
3237:
3234:
3209:
3204:
3200:
3196:
3193:
3190:
3184:
3181:
3175:
3172:
3157:
3156:
3145:
3142:
3139:
3134:
3130:
3123:
3120:
3115:
3111:
3105:
3102:
3096:
3093:
3090:
3087:
3084:
3079:
3076:
3072:
3066:
3063:
3036:
3033:
3027:
3024:
3009:
3008:
2997:
2994:
2991:
2988:
2985:
2977:
2974:
2970:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2892:
2888:
2882:
2878:
2867:Gamma function
2854:
2843:
2842:
2830:
2824:
2819:
2812:
2808:
2802:
2798:
2792:
2787:
2784:
2779:
2776:
2771:
2766:
2763:
2758:
2753:
2750:
2744:
2738:
2734:
2728:
2718:
2714:
2708:
2704:
2695:
2689:
2686:
2681:
2678:
2675:
2669:
2666:
2660:
2653:
2648:
2645:
2640:
2637:
2631:
2626:
2623:
2618:
2615:
2610:
2606:
2597:
2594:
2590:
2562:
2558:
2553:
2549:
2545:
2540:
2536:
2532:
2529:
2524:
2520:
2502:
2498:
2493:
2490:
2478:Beffara (2008)
2451:Each value of
2449:
2448:
2437:
2431:
2428:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2387:
2384:
2367:central charge
2359:
2358:
2352:
2342:
2335:
2327:
2317:
2307:
2293:
2286:
2276:
2265:
2254:
2241:
2235:
2222:
2221:
2214:
2210:
2203:
2178:
2154:
2153:
2142:
2139:
2136:
2133:
2128:
2123:
2120:
2117:
2114:
2111:
2092:
2089:
2088:
2087:
2073:
2068:
2065:
2043:
2039:
2028:
2015:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1982:
1969:
1966:
1963:
1958:
1954:
1948:
1945:
1942:
1939:
1934:
1930:
1919:
1906:
1903:
1900:
1895:
1891:
1885:
1882:
1879:
1876:
1871:
1867:
1843:
1823:
1811:
1799:
1787:
1784:
1769:
1765:
1742:
1738:
1726:
1725:
1714:
1711:
1708:
1705:
1702:
1699:
1694:
1690:
1686:
1683:
1680:
1677:
1674:
1669: or
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1631:
1627:
1603:
1594:and the curve
1583:
1572:
1571:
1560:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1527:
1523:
1518:
1512:
1505:
1502:
1497:
1494:
1491:
1486:
1482:
1478:
1451:
1448:
1445:
1442:
1439:
1436:
1431:
1428:
1425:
1420:
1415:
1411:
1407:
1401:
1395:
1392:
1387:
1384:
1381:
1376:
1372:
1368:
1342:
1331:
1330:
1319:
1312:
1309:
1306:
1301:
1297:
1293:
1290:
1287:
1284:
1281:
1276:
1273:
1270:
1265:
1261:
1257:
1254:
1251:
1248:
1245:
1238:
1235:
1232:
1227:
1223:
1219:
1212:
1209:
1204:
1201:
1198:
1193:
1189:
1185:
1158:
1155:
1152:
1149:
1146:
1143:
1138:
1135:
1132:
1129:
1126:
1123:
1117:
1114:
1111:
1106:
1101:
1097:
1093:
1090:
1087:
1081:
1078:
1073:
1070:
1067:
1062:
1058:
1054:
1028:
1008:
988:
968:
965:
962:
959:
936:
933:
930:
927:
924:
919:
915:
894:
891:
888:
885:
882:
877:
873:
852:
849:
846:
826:
823:
820:
800:
797:
794:
771:
749:
745:
722:
718:
695:
691:
660:
656:
635:
613:
609:
584:
560:
557:
554:
551:
548:
545:
542:
539:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
484:
480:
459:
456:
453:
433:
413:
410:
407:
404:
401:
398:
395:
392:
372:
352:
349:
346:
343:
323:
303:
282:
270:complex domain
250:
235:Main article:
232:
229:
114:
80:
77:
74:
71:
68:
63:
59:
55:
52:
49:
46:
43:
40:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5983:
5972:
5969:
5967:
5964:
5963:
5961:
5946:
5943:
5941:
5938:
5937:
5934:
5928:
5925:
5923:
5920:
5918:
5915:
5913:
5910:
5908:
5905:
5903:
5900:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5880:
5878:
5875:
5873:
5870:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5845:
5844:
5842:
5838:
5830:
5827:
5825:
5822:
5821:
5820:
5817:
5815:
5812:
5810:
5807:
5805:
5802:
5800:
5799:Stopping time
5797:
5793:
5790:
5789:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5740:
5738:
5735:
5733:
5730:
5728:
5725:
5723:
5720:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5675:
5673:
5670:
5668:
5665:
5663:
5660:
5658:
5655:
5653:
5650:
5649:
5647:
5643:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5613:
5611:
5609:
5605:
5598:
5594:
5590:
5589:Hewitt–Savage
5586:
5582:
5578:
5574:
5573:Zero–one laws
5571:
5569:
5566:
5564:
5561:
5559:
5556:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5520:
5518:
5514:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5434:
5432:
5428:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5393:
5391:
5389:
5385:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5360:
5358:
5356:
5352:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5270:Black–Scholes
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5252:
5250:
5248:
5244:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5210:
5209:
5207:
5205:
5201:
5195:
5192:
5190:
5187:
5183:
5180:
5178:
5175:
5174:
5173:
5172:Point process
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5151:
5148:
5146:
5143:
5142:
5141:
5138:
5136:
5133:
5131:
5130:Gibbs measure
5128:
5126:
5123:
5121:
5118:
5117:
5115:
5111:
5105:
5102:
5100:
5097:
5095:
5092:
5090:
5087:
5085:
5082:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5059:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5034:
5032:
5028:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4975:
4972:
4970:
4967:
4966:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4919:Itô diffusion
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4894:Gamma process
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4855:
4852:
4850:
4847:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4819:
4818:
4815:
4811:
4808:
4807:
4806:
4803:
4801:
4798:
4796:
4793:
4792:
4790:
4788:
4784:
4776:
4773:
4771:
4768:
4766:
4765:Self-avoiding
4763:
4761:
4758:
4757:
4756:
4753:
4751:
4750:Moran process
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4717:
4715:
4713:
4712:Discrete time
4709:
4705:
4698:
4693:
4691:
4686:
4684:
4679:
4678:
4675:
4667:
4663:
4662:
4656:
4651:
4645:
4638:
4634:
4630:
4629:
4628:Tutorial: SLE
4623:
4622:
4618:
4612:
4608:
4604:
4600:
4595:
4590:
4586:
4582:
4578:
4574:
4573:Bauer, Michel
4570:
4567:
4563:
4559:
4555:
4550:
4545:
4541:
4537:
4533:
4529:
4528:Bauer, Michel
4525:
4522:
4518:
4513:
4508:
4503:
4498:
4494:
4490:
4486:
4482:
4479:
4475:
4471:
4465:
4461:
4457:
4452:
4447:
4443:
4439:
4435:
4431:
4428:
4424:
4420:
4414:
4410:
4406:
4402:
4398:
4393:
4388:
4384:
4380:
4379:Schramm, Oded
4376:
4372:
4368:
4364:
4360:
4355:
4350:
4345:
4340:
4336:
4332:
4331:
4326:
4325:Schramm, Oded
4322:
4317:
4313:
4309:
4303:
4302:
4297:
4296:Norris, J. R.
4293:
4290:
4284:
4279:
4278:
4272:
4268:
4265:
4261:
4257:
4253:
4249:
4245:
4241:
4237:
4230:
4226:
4222:
4219:
4215:
4211:
4207:
4203:
4199:
4194:
4189:
4185:
4181:
4177:
4173:
4169:
4168:Schramm, Oded
4165:
4161:
4157:
4152:
4148:
4144:
4140:
4136:
4133:
4132:
4127:
4123:
4119:
4114:
4110:
4106:
4103:
4099:
4095:
4089:
4085:
4081:
4080:
4074:
4070:
4066:
4062:
4058:
4052:
4048:
4044:
4039:
4036:
4032:
4028:
4024:
4020:
4016:
4011:
4006:
4002:
3998:
3993:
3990:
3986:
3985:
3980:
3975:
3972:
3968:
3967:
3962:
3957:
3954:
3950:
3946:
3942:
3938:
3934:
3929:
3924:
3921:(1): 81–118,
3920:
3916:
3912:
3908:
3905:
3901:
3897:
3893:
3889:
3885:
3880:
3875:
3871:
3867:
3862:
3861:
3856:
3847:
3843:
3838:
3833:
3829:
3825:
3821:
3818:
3817:
3809:
3806:
3801:
3797:
3792:
3787:
3783:
3780:
3779:
3771:
3769:
3765:
3760:
3756:
3752:
3748:
3743:
3738:
3734:
3731:
3730:
3722:
3719:
3714:
3710:
3706:
3702:
3697:
3692:
3688:
3685:
3684:
3683:Ann. of Math.
3676:
3673:
3668:
3664:
3660:
3656:
3651:
3646:
3642:
3639:
3638:
3630:
3627:
3622:
3618:
3614:
3610:
3606:
3602:
3598:
3595:
3594:
3586:
3579:
3576:
3571:
3567:
3563:
3559:
3554:
3549:
3545:
3541:
3534:
3531:
3526:
3522:
3518:
3514:
3510:
3506:
3501:
3496:
3492:
3488:
3481:
3478:
3474:
3470:
3466:
3462:
3458:
3454:
3449:
3444:
3440:
3436:
3429:
3426:
3422:
3418:
3414:
3410:
3405:
3400:
3396:
3392:
3385:
3382:
3375:
3373:
3371:
3363:
3347:
3342:
3339:
3334:
3331:
3328:
3325:
3318:
3317:
3316:
3314:
3310:
3305:
3299:
3295:
3287:
3283:
3281:
3277:
3273:
3265:
3261:
3256:
3254:
3250:
3242:
3235:
3233:
3231:
3227:
3223:
3202:
3198:
3191:
3188:
3182:
3179:
3173:
3170:
3162:
3143:
3140:
3137:
3132:
3121:
3118:
3113:
3109:
3103:
3100:
3094:
3088:
3082:
3077:
3074:
3064:
3061:
3052:
3051:
3050:
3034:
3031:
3025:
3022:
3014:
2992:
2989:
2986:
2983:
2975:
2964:
2958:
2955:
2952:
2946:
2939:
2938:
2937:
2935:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2890:
2886:
2880:
2877:
2868:
2828:
2822:
2817:
2810:
2806:
2800:
2796:
2790:
2785:
2782:
2777:
2774:
2769:
2764:
2761:
2756:
2751:
2748:
2742:
2736:
2732:
2726:
2716:
2712:
2706:
2702:
2687:
2684:
2679:
2676:
2673:
2658:
2646:
2643:
2629:
2624:
2621:
2616:
2608:
2604:
2595:
2580:
2579:
2578:
2577:
2556:
2551:
2547:
2543:
2538:
2534:
2530:
2527:
2522:
2518:
2509:
2505:
2496:
2491:
2489:
2487:
2483:
2479:
2475:
2473:
2470:
2466:
2462:
2458:
2454:
2435:
2429:
2426:
2418:
2415:
2412:
2403:
2400:
2397:
2394:
2385:
2382:
2375:
2374:
2373:
2371:
2368:
2364:
2356:
2353:
2350:
2345:
2340:
2336:
2333:
2328:
2325:
2321:
2318:
2315:
2311:
2308:
2305:
2301:
2296:
2291:
2287:
2284:
2280:
2277:
2274:
2270:
2266:
2263:
2259:
2255:
2252:
2248:
2244:
2243:
2240:
2236:
2234:
2230:
2228:
2219:
2217:
2211:
2208:
2206:
2200:
2199:
2198:
2196:
2191:
2189:
2185:
2181:
2173:
2171:
2167:
2163:
2159:
2137:
2131:
2126:
2121:
2115:
2109:
2102:
2101:
2100:
2098:
2090:
2071:
2066:
2063:
2041:
2037:
2029:
2013:
2008:
2005:
2002:
1996:
1990:
1983:
1967:
1964:
1961:
1956:
1952:
1946:
1940:
1932:
1928:
1920:
1904:
1901:
1898:
1893:
1889:
1883:
1877:
1869:
1865:
1857:
1856:
1855:
1841:
1821:
1797:
1785:
1783:
1767:
1763:
1740:
1736:
1706:
1700:
1692:
1688:
1684:
1678:
1672:
1661:
1655:
1652:
1643:
1637:
1629:
1625:
1617:
1616:
1615:
1601:
1581:
1558:
1548:
1542:
1539:
1533:
1525:
1521:
1516:
1510:
1503:
1492:
1484:
1480:
1449:
1446:
1440:
1434:
1426:
1413:
1409:
1405:
1399:
1393:
1382:
1374:
1370:
1356:
1355:
1354:
1340:
1317:
1307:
1299:
1295:
1291:
1285:
1279:
1271:
1263:
1259:
1255:
1249:
1243:
1233:
1225:
1221:
1217:
1210:
1199:
1191:
1187:
1156:
1153:
1147:
1141:
1136:
1133:
1127:
1121:
1112:
1099:
1095:
1091:
1088:
1085:
1079:
1068:
1060:
1056:
1042:
1041:
1040:
1026:
1006:
986:
963:
957:
950:
934:
931:
925:
917:
913:
892:
889:
883:
875:
871:
850:
847:
844:
824:
821:
818:
798:
795:
792:
783:
769:
747:
743:
720:
716:
693:
689:
680:
676:
675:Loewner (1923
658:
654:
633:
611:
607:
598:
582:
574:
552:
549:
546:
537:
511:
508:
505:
496:
493:
490:
487:
482:
478:
457:
454:
451:
431:
402:
399:
390:
370:
347:
341:
321:
301:
272:not equal to
271:
268:
264:
248:
238:
230:
228:
226:
222:
217:
212:
208:
204:
200:
196:
192:
191:scaling limit
187:
185:
181:
177:
173:
169:
165:
160:
158:
155:and a domain
154:
150:
146:
142:
138:
134:
130:
126:
122:
121:scaling limit
117:
113:
109:
105:
101:
97:
69:
61:
57:
50:
47:
41:
38:
29:
19:
5857:Econometrics
5819:Wiener space
5707:Itô integral
5608:Inequalities
5497:Self-similar
5467:Gauss–Markov
5457:Exchangeable
5437:Càdlàg paths
5373:Risk process
5325:LIBOR market
5194:Random graph
5189:Random field
5001:Superprocess
4980:
4939:Lévy process
4934:Jump process
4909:Hunt process
4745:Markov chain
4660:
4627:
4584:
4580:
4539:
4535:
4502:math/0307353
4492:
4488:
4437:
4392:math/0602151
4382:
4334:
4328:
4315:
4300:
4276:
4239:
4235:
4193:math/0010165
4183:
4179:
4146:
4142:
4130:
4078:
4069:the original
4046:
4000:
3996:
3982:
3964:
3918:
3914:
3879:math/0211322
3869:
3865:
3819:
3814:
3808:
3791:math/0112234
3781:
3778:Ann. Probab.
3776:
3732:
3727:
3721:
3696:math/0504586
3686:
3681:
3675:
3650:math/0109120
3640:
3635:
3629:
3596:
3591:
3578:
3543:
3539:
3533:
3490:
3486:
3480:
3448:math/0310210
3438:
3434:
3428:
3404:math/0107096
3394:
3390:
3384:
3367:
3308:
3306:
3284:
3276:Harry Kesten
3257:
3239:
3236:Applications
3221:
3160:
3158:
3010:
2844:
2507:
2503:
2500:
2494:
2485:
2476:
2464:
2460:
2459:, one value
2456:
2452:
2450:
2369:
2362:
2360:
2354:
2343:
2338:
2331:
2319:
2309:
2294:
2289:
2278:
2272:
2268:
2261:
2257:
2250:
2246:
2238:
2231:
2226:
2223:
2215:
2212:
2204:
2201:
2194:
2192:
2187:
2183:
2176:
2174:
2169:
2165:
2161:
2157:
2155:
2096:
2094:
1789:
1727:
1573:
1332:
948:
784:
424:a subset of
240:
221:Itô calculus
188:
161:
148:
144:
140:
136:
132:
128:
115:
111:
107:
103:
99:
93:
5902:Ruin theory
5840:Disciplines
5712:Itô's lemma
5487:Predictable
5162:Percolation
5145:Potts model
5140:Ising model
5104:White noise
5062:Differences
4924:Itô process
4864:Cox process
4760:Loop-erased
4755:Random walk
4495:: 145–190,
4337:: 221–288,
4225:Loewner, C.
3911:Cardy, John
3294:Peano curve
3260:percolation
3230:percolation
3013:Itô's lemma
2314:Ising model
2202:Chordal SLE
180:Greg Lawler
145:chordal SLE
5960:Categories
5912:Statistics
5692:Filtration
5593:Kolmogorov
5577:Blumenthal
5502:Stationary
5442:Continuous
5430:Properties
5315:Hull–White
5057:Martingale
4944:Local time
4832:Fractional
4810:pure birth
4256:49.0714.01
4236:Math. Ann.
3376:References
3364:Simulation
2213:Radial SLE
149:radial SLE
5824:Classical
4837:Geometric
4827:Excursion
4611:119596360
4264:121752388
4174:(2001b),
4149:: 35–54,
4118:0712.3256
3989:EMS Press
3971:EMS Press
3832:CiteSeerX
3759:119677336
3742:1008.1378
3621:118713698
3553:0909.4499
3525:0764-4442
3500:0909.4499
3340:κ
3258:Critical
3192:
3183:π
3174:−
3129:∂
3071:∂
3062:κ
2976:γ
2853:Γ
2786:−
2765:κ
2688:κ
2680:κ
2677:−
2665:Γ
2659:π
2647:κ
2636:Γ
2596:γ
2557:∈
2430:κ
2416:−
2413:κ
2404:κ
2398:−
2127:κ
2110:ζ
1991:γ
1899:−
1842:ζ
1822:ζ
1701:γ
1673:ζ
1656:γ
1638:ζ
1602:γ
1582:ζ
1543:ζ
1540:−
1501:∂
1477:∂
1447:−
1435:ζ
1419:′
1391:∂
1367:∂
1292:−
1280:ζ
1244:ζ
1208:∂
1184:∂
1154:−
1142:ζ
1122:ζ
1105:′
1089:−
1077:∂
1053:∂
1027:γ
958:ζ
822:≥
796:∈
538:γ
497:γ
494:∖
455:≥
406:∞
391:γ
342:γ
302:γ
51:
42:
5945:Category
5829:Abstract
5363:Bühlmann
4969:Compound
4644:citation
4566:16790280
4371:17164604
4314:(1975),
4298:(2010),
4273:(1982),
4227:(1923),
3953:17747133
3713:14742163
3243:used SLE
2086:removed.
5452:Ergodic
5340:Vašíček
5182:Poisson
4842:Meander
4521:2178043
4478:2079672
4427:2334202
4397:Bibcode
4363:1776084
4218:5877745
4210:1849257
4102:2129588
4065:2087784
4035:7239233
4015:Bibcode
3933:Bibcode
3896:2435854
3824:Bibcode
3667:6837772
3601:Bibcode
3558:Bibcode
3505:Bibcode
3473:9055859
3465:3481779
3421:3481779
3409:Bibcode
3262:on the
2932:is the
2865:is the
1786:Example
595:by the
166: (
139:, with
5792:Tanaka
5477:Mixing
5472:Markov
5345:Wilkie
5310:Ho–Lee
5305:Heston
5077:Super-
4822:Bridge
4770:Biased
4609:
4564:
4519:
4476:
4466:
4425:
4415:
4369:
4361:
4285:
4262:
4254:
4216:
4208:
4100:
4090:
4063:
4053:
4033:
3951:
3904:226992
3902:
3894:
3834:
3757:
3711:
3665:
3619:
3523:
3471:
3463:
3419:
2845:where
2186:() in
2156:where
1728:where
999:. If
294:, and
201:, the
197:, the
98:, the
5645:Tools
5421:M/M/c
5416:M/M/1
5411:M/G/1
5401:Fluid
5067:Local
4607:S2CID
4589:arXiv
4562:S2CID
4544:arXiv
4497:arXiv
4446:arXiv
4387:arXiv
4367:S2CID
4339:arXiv
4305:(PDF)
4260:S2CID
4232:(PDF)
4214:S2CID
4188:arXiv
4113:arXiv
4031:S2CID
4005:arXiv
3949:S2CID
3923:arXiv
3900:S2CID
3874:arXiv
3786:arXiv
3755:S2CID
3737:arXiv
3709:S2CID
3691:arXiv
3663:S2CID
3645:arXiv
3617:S2CID
3588:(PDF)
3548:arXiv
3495:arXiv
3469:S2CID
3461:JSTOR
3443:arXiv
3417:JSTOR
3399:arXiv
3255:4/3.
2488:/8).
1465:or
1333:When
1172:or
599:. If
261:is a
5597:Lévy
5396:Bulk
5280:Chen
5072:Sub-
5030:Both
4666:MSRI
4650:link
4464:ISBN
4413:ISBN
4283:ISBN
4088:ISBN
4051:ISBN
3521:ISSN
3159:For
3011:and
2869:and
2337:For
2288:For
2267:For
1790:Let
1755:and
905:or
863:are
383:and
267:open
182:and
168:2000
5177:Cox
4599:doi
4585:239
4554:doi
4540:543
4507:doi
4456:doi
4405:doi
4349:doi
4335:118
4252:JFM
4244:doi
4198:doi
4151:doi
4023:doi
4001:115
3941:doi
3919:318
3884:doi
3842:doi
3796:doi
3747:doi
3701:doi
3687:171
3655:doi
3609:doi
3597:109
3566:doi
3544:333
3513:doi
3491:333
3453:doi
3270:by
3189:arg
762:to
708:of
646:to
575:to
530:of
241:If
112:SLE
94:In
39:log
5962::
5595:,
5591:,
5587:,
5583:,
5579:,
4664:,
4646:}}
4642:{{
4635:,
4631:,
4605:,
4597:,
4583:,
4575:;
4560:,
4552:,
4538:,
4530:;
4517:MR
4515:,
4505:,
4491:,
4474:MR
4472:,
4462:,
4454:,
4423:MR
4421:,
4411:,
4403:,
4395:,
4365:,
4359:MR
4357:,
4347:,
4333:,
4258:,
4250:,
4240:89
4238:,
4234:,
4212:,
4206:MR
4204:,
4196:,
4182:,
4178:,
4170:;
4166:;
4147:46
4145:,
4128:,
4098:MR
4096:,
4086:,
4061:MR
4059:,
4029:,
4021:,
4013:,
3999:,
3987:,
3981:,
3969:,
3963:,
3947:,
3939:,
3931:,
3917:,
3898:,
3892:MR
3890:,
3882:,
3870:36
3868:,
3840:.
3830:.
3820:41
3794:.
3782:32
3767:^
3753:.
3745:.
3733:26
3707:.
3699:.
3661:.
3653:.
3615:.
3607:.
3590:.
3564:.
3556:.
3542:.
3519:.
3511:.
3503:.
3489:.
3467:,
3459:,
3451:,
3439:33
3437:,
3415:,
3407:,
3395:33
3393:,
3304:.
3232:.
3144:0.
3026::=
2965::=
2474:)
2197::
811:,
782:.
265:,
205:,
159:.
48:Im
5599:)
5575:(
4696:e
4689:t
4682:v
4652:)
4601::
4591::
4556::
4546::
4509::
4499::
4493:2
4458::
4448::
4407::
4399::
4389::
4351::
4341::
4246::
4200::
4190::
4184:8
4153::
4115::
4025::
4017::
4007::
3943::
3935::
3925::
3886::
3876::
3848:.
3844::
3826::
3802:.
3798::
3788::
3761:.
3749::
3739::
3715:.
3703::
3693::
3669:.
3657::
3647::
3641:8
3623:.
3611::
3603::
3572:.
3568::
3560::
3550::
3527:.
3515::
3507::
3497::
3455::
3445::
3411::
3401::
3348:.
3343:8
3335:+
3332:1
3329:=
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3290:2
3268:6
3245:6
3222:κ
3208:)
3203:0
3199:z
3195:(
3180:1
3171:1
3161:κ
3141:=
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3119:+
3114:2
3110:w
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3101:4
3095:+
3092:)
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2996:]
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2973:[
2969:P
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2829:)
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2668:(
2652:)
2644:4
2639:(
2630:+
2625:2
2622:1
2617:=
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2605:z
2593:[
2589:P
2561:H
2552:0
2548:z
2544:=
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2504:κ
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2436:.
2427:2
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2419:6
2410:(
2407:)
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2392:(
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2363:κ
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2067:i
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1994:(
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1962:+
1957:2
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1947:=
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1941:z
1938:(
1933:t
1929:g
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1884:=
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1812:0
1798:D
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1707:t
1704:(
1698:(
1693:t
1689:g
1685:=
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1679:t
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1665:)
1662:t
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1647:)
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1559:.
1552:)
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1537:)
1534:z
1531:(
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1522:g
1517:2
1511:=
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1406:2
1400:=
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1380:(
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1371:f
1341:D
1318:.
1311:)
1308:z
1305:(
1300:t
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1256:+
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1250:t
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1237:)
1234:z
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1218:=
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1200:z
1197:(
1192:t
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1151:)
1148:t
1145:(
1137:z
1134:+
1131:)
1128:t
1125:(
1116:)
1113:z
1110:(
1100:t
1096:f
1092:z
1086:=
1080:t
1072:)
1069:z
1066:(
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1007:D
987:D
967:)
964:t
961:(
935:z
932:=
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926:z
923:(
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914:g
893:z
890:=
887:)
884:z
881:(
876:0
872:f
851:0
848:=
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793:z
770:D
748:t
744:D
721:t
717:f
694:t
690:g
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655:D
634:D
612:t
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583:D
559:)
556:]
553:t
550:,
547:0
544:[
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518:)
515:]
512:t
509:,
506:0
503:[
500:(
491:D
488:=
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479:D
458:0
452:t
432:D
412:)
409:)
403:,
400:0
397:(
394:(
371:D
351:)
348:0
345:(
322:D
281:C
249:D
141:κ
137:U
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129:κ
116:κ
110:(
104:κ
79:)
76:)
73:)
70:z
67:(
62:t
58:g
54:(
45:(
20:)
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