5886:
4899:
2081:
2714:
2549:
While the example below is developed for the one-dimensional case, many of the arguments, and almost all of the notation, generalizes easily to any number of dimensions. Some of the formalism is also broad enough to handle related models, such as the
2513:
430:
2175:
4261:
4550:
5106:
2969:
3468:
4359:
3716:
1138:
4051:
3910:
1926:
4699:
675:
5498:
Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; Manolescu, Ioan; Tassion, Vincent (2017-09-05). "Discontinuity of the phase transition for the planar random-cluster and Potts models with $ q>4$ ".
3262:
1438:
2518:
A spin cluster is the union of neighbouring FK clusters with the same color: two neighbouring spin clusters have different colors, while two neighbouring FK clusters are colored independently.
1811:
3500:. It is very common in mathematical treatments to set ÎČ = 1, as it is easily regained by rescaling the interaction energy. This partition function is written as a function of the interaction
249:
2312:
4466:
4132:
4613:
3093:
1559:
4618:
The general solution for an arbitrary number of spins, and an arbitrary finite-range interaction, is given by the same general form. In this case, the precise expression for the matrix
3504:
to emphasize that it is only a function of the interaction, and not of any specific configuration of spins. The partition function, together with the
Hamiltonian, are used to define a
929:
3817:
1934:
2584:
2242:
1012:
3267:
This function can be seen to consist of two parts: the self-energy of a configuration of spins, plus the interaction energy of this set and all the other spins in the lattice. The
2799:
727:
465:
1191:
4938:
4982:
2347:
320:
1591:
850:
308:
558:
771:
2092:
1844:
1312:
1688:
1650:
1508:
1276:
1617:
532:
492:
6254:
876:
800:
4147:
1482:
2332:
1735:
1352:
1332:
1251:
1231:
1211:
961:
191:
5847:
3823:
2530:, and thus gains access to all of the mathematical techniques associated with this formalism. In particular, it can be solved exactly using the techniques of
5523:
Li, Zi-Qian; Yang, Li-Ping; Xie, Z. Y.; Tu, Hong-Hao; Liao, Hai-Jun; Xiang, T. (2020). "Critical properties of the two-dimensional $ q$ -state clock model".
2821:
3300:
6789:
4477:
134:
4280:
6613:
3098:
will be seen to describe the interaction between nearest neighbors. Of course, different functions give different interactions; so a function of
3514:
1020:
7216:
6209:
6746:
6726:
3956:
3732:. By endowing the configuration space with a probability measure built from a Hamiltonian in this way, the configuration space turns into a
47:. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is
118:
5314:
Mehta, Pankaj; Bukov, Marin; Wang, Ching-Hao; Day, Alexandre G. R.; Richardson, Clint; Fisher, Charles K.; Schwab, David J. (2019-05-30).
3832:
7130:
5902:
94:
4894:{\displaystyle P_{\gamma }(u)=\gamma \|\nabla u\|_{0}+\|u-f\|_{p}^{p}=\gamma \#\{i:u_{i}\neq u_{i+1}\}+\sum _{i=1}^{n}|u_{i}-f_{i}|^{p}}
1849:
5907:
5840:
3291:
939:
A generalization of the Potts model is often used in statistical inference and biophysics, particularly for modelling proteins through
7047:
5007:
In image processing, the Potts functional is related to the segmentation problem. However, in two dimensions the problem is NP-hard.
4658:
The Potts model has applications in signal reconstruction. Assume that we are given noisy observation of a piecewise constant signal
582:
6731:
2542:, which is the "ancestor" of the Potts model, in his 1924 PhD thesis). This section develops the mathematical formalism, based on
7057:
6741:
5742:
7099:
6996:
7286:
7276:
7122:
6814:
6799:
5833:
7186:
7150:
3185:
7454:
7191:
7103:
6301:
6202:
1743:
203:
7256:
5666:
3508:
on the Borel Ï-algebra in the following way: The measure of a cylinder set, i.e. an element of the base, is given by
2247:
1457:
Despite its simplicity as a model of a physical system, the Potts model is useful as a model system for the study of
5690:
Friedrich, F.; Kempe, A.; Liebscher, V.; Winkler, G. (2008). "Complexity
Penalized M-Estimation: Fast Computation".
2988:. Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a
7485:
7301:
7107:
7091:
7006:
6834:
6804:
6226:
5021:
4405:
3739:
Most thermodynamic properties can be expressed directly in terms of the partition function. Thus, for example, the
3161:
may depend on some or all of the spins; currently, only those that depend on a finite number are exactly solvable.
4066:
2076:{\displaystyle e^{J_{p}\delta (s_{i},s_{j})}=1+v\delta (s_{i},s_{j})\qquad {\text{ with }}\qquad v=e^{J_{p}}-1\ .}
7206:
7171:
7140:
7135:
6571:
6488:
4561:
3028:
2709:{\displaystyle Q^{\mathbf {Z} }=\{s=(\ldots ,s_{-1},s_{0},s_{1},\ldots ):s_{k}\in Q\;\forall k\in \mathbf {Z} \}}
1513:
58:, who described the model near the end of his 1951 Ph.D. thesis. The model was related to the "planar Potts" or "
881:
7490:
7145:
6774:
6769:
6576:
6473:
6101:
3749:
3004:
1357:
2187:
576:
was suggested by Potts in the course of his study of the model above and is defined by a simpler
Hamiltonian:
7459:
7236:
7072:
6971:
6956:
6495:
6368:
6284:
6195:
4137:
If neighboring spins are only allowed in certain specific configurations, then the state space is given by a
3726:
1441:
966:
7231:
7111:
5443:"Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $ $ {1 \le q \le 4}$ $ "
2244:. This is proportional to the partition function of the random cluster model with the open edge probability
1444:
in that couplings can be heterogeneous and non-local. There is no explicit lattice structure in this model.
7241:
5870:
2555:
126:
86:
7246:
6882:
2749:
683:
6844:
6428:
6373:
6289:
438:
7176:
5639:
Sokal, Alan D. (2005). "The multivariate Tutte polynomial (alias Potts model) for graphs and matroids".
7181:
7166:
6809:
6779:
6346:
6244:
5978:
5894:
5031:
2508:{\displaystyle e^{J_{p}\delta (s_{i},s_{j})}=(1-\delta (s_{i},s_{j}))+e^{J_{p}}\delta (s_{i},s_{j})\ .}
425:{\displaystyle H_{c}=J_{c}\sum _{\langle i,j\rangle }\cos \left(\theta _{s_{i}}-\theta _{s_{j}}\right)}
4907:
1928:
i.e. sets of nearest neighbor pairs of the same color. The transformation is done using the identity
7261:
7062:
6976:
6961:
6892:
6468:
6351:
6249:
4943:
4271:
7095:
6981:
6483:
6458:
6403:
6154:
6137:
1714:
1622:
For the clock model, there is evidence that the corresponding phase transitions are infinite order
1564:
940:
809:
257:
2170:{\displaystyle Z_{p}=\sum _{\omega }v^{\#{\text{edges}}(\omega )}q^{\#{\text{clusters}}(\omega )}}
1146:
537:
7495:
7396:
7386:
7201:
7077:
6859:
6784:
6598:
6463:
6319:
6274:
6001:
5991:
5917:
5036:
4138:
2728:
2527:
736:
7338:
7266:
6691:
6681:
6525:
6048:
4626:
2995:, however the natural topology of the q-adic numbers is finer than the above product topology.
2731:, may be used. Shifts get this name because there exists a natural operator on this space, the
1816:
110:
1281:
7361:
7343:
7323:
7318:
7037:
6869:
6849:
6696:
6639:
6478:
6388:
5933:
5856:
3740:
3505:
1710:
1667:
1629:
1487:
494:
is a coupling constant, determining the interaction strength. This model is now known as the
311:
20:
1596:
7480:
7436:
7391:
7381:
7067:
7042:
7011:
6991:
6829:
6751:
6736:
6603:
6076:
5597:
5542:
5337:
5280:
5225:
5173:
5118:
5079:
5046:
5016:
4256:{\displaystyle Z_{n}(c)=e^{-c\beta }|{\mbox{Fix}}\,\tau ^{n}|=e^{-c\beta }{\mbox{Tr}}A^{n}}
1706:
1661:
505:
470:
138:
4399:â {â1, 1} and only nearest neighbor spins interact. The interaction potential is given by
1690:, the model displays the phenomenon of 'interfacial adsorption' with intriguing critical
8:
7431:
7271:
7196:
7001:
6761:
6671:
6561:
6142:
6121:
4642:
3722:
855:
779:
44:
36:
5601:
5546:
5341:
5284:
5229:
5204:"Selection of sequence motifs and generative Hopfield-Potts models for protein families"
5177:
5122:
5083:
1464:
1256:
7401:
7366:
7281:
7251:
7082:
7021:
7016:
6839:
6676:
6341:
6279:
6218:
6169:
6081:
6058:
6039:
6019:
5983:
5754:
5723:
5715:
5672:
5644:
5621:
5566:
5532:
5500:
5480:
5454:
5423:
5394:"The self-dual point of the two-dimensional random-cluster model is critical for q â„ 1"
5366:
5327:
5315:
5268:
5249:
5215:
5203:
5142:
5026:
3733:
3489:
2964:{\displaystyle C_{m}=\{s\in Q^{\mathbf {Z} }:s_{m}=\xi _{0},\ldots ,s_{m+k}=\xi _{k}\}}
2317:
1720:
1653:
1337:
1317:
1236:
1216:
1196:
946:
176:
81:
The Potts model is related to, and generalized by, several other models, including the
2727:. For defining the Potts model, either this whole space, or a certain subset of it, a
7421:
6634:
6551:
6520:
6413:
6393:
6383:
6239:
6234:
6174:
6071:
6044:
5955:
5945:
5938:
5912:
5795:
5727:
5707:
5662:
5625:
5613:
5570:
5558:
5484:
5472:
5415:
5371:
5353:
5296:
5253:
5241:
5146:
5134:
5067:
4634:
3950:
constant and independent of any spin configuration). The partition function becomes
3916:
3721:
One can then extend by countable additivity to the full Ï-algebra. This measure is a
2531:
106:
98:
51:, and that it has a rich mathematical formulation that has been studied extensively.
7226:
6877:
5676:
5427:
5269:"Simulation of biological cell sorting using a two-dimensional extended Potts model"
7441:
7328:
7211:
7087:
6824:
6581:
6556:
6505:
6356:
6309:
6066:
5787:
5699:
5654:
5605:
5550:
5464:
5405:
5361:
5345:
5288:
5233:
5181:
5126:
5087:
4377:
2805:
1703:
1657:
1458:
48:
6433:
6116:
5349:
5161:
3463:{\displaystyle Z_{n}(V)=\sum _{s_{0},\ldots ,s_{n}\in Q}\exp(-\beta H_{n}(C_{0}))}
7406:
7306:
7291:
7052:
6986:
6664:
6608:
6591:
6336:
6111:
6106:
6034:
6029:
5996:
5923:
5658:
2809:
730:
167:
163:
102:
7221:
6453:
5554:
5292:
5237:
4545:{\displaystyle M_{\sigma \sigma '}=\exp \left(\beta J_{p}\sigma \sigma '\right)}
141:, has been used to simulate static and kinetic phenomena in foam and biological
7411:
7376:
7296:
6902:
6649:
6566:
6535:
6530:
6510:
6500:
6443:
6418:
6398:
6363:
6331:
6314:
6159:
5875:
4056:
If all states are allowed, that is, the underlying set of states is given by a
2732:
2559:
2543:
2337:
Alternatively, instead of FK clusters, the model can be formulated in terms of
1623:
878:
standard Potts model is equivalent to the three-state vector Potts model, with
90:
32:
6438:
5775:
5468:
5410:
5393:
5130:
5091:
4354:{\displaystyle {\mbox{Fix}}\,\tau ^{n}=\{s\in Q^{\mathbf {Z} }:\tau ^{n}s=s\}}
170:
lattice, but is often generalized to other dimensions and lattice structures.
7474:
7313:
6854:
6686:
6644:
6586:
6408:
6324:
6264:
6147:
5799:
5711:
5703:
5617:
5476:
5419:
5357:
5138:
4993:
3927:
The simplest model is the model where there is no interaction at all, and so
2989:
142:
130:
75:
71:
55:
5442:
502:. Potts provided the location in two dimensions of the phase transition for
7371:
7333:
6887:
6819:
6708:
6703:
6515:
6448:
6423:
6259:
6164:
5743:"Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials"
5562:
5375:
5316:"A high-bias, low-variance introduction to Machine Learning for physicists"
5300:
5245:
5185:
3139:, that is, an infinite string of spins. In the above example, the function
2813:
114:
5441:
Duminil-Copin, Hugo; Sidoravicius, Vladas; Tassion, Vincent (2017-01-01).
3711:{\displaystyle \mu (C_{k})={\frac {1}{Z_{n}(V)}}\exp(-\beta H_{n}(C_{k}))}
1737:, and led to the rigorous proof of the critical temperature of the model.
1133:{\displaystyle H=\sum _{i<j}J_{ij}(s_{i},s_{j})+\sum _{i}h_{i}(s_{i}),}
39:. By studying the Potts model, one may gain insight into the behaviour of
7416:
6951:
6935:
6930:
6925:
6915:
6718:
6654:
6618:
6378:
6269:
5968:
4630:
4555:
with the index Ï, ÏâČ â {â1, 1}. The partition function is then given by
4389:
4267:
3497:
2539:
2535:
943:. This generalized Potts model consists of 'spins' that each may take on
803:
40:
28:
5825:
5719:
4625:
The goal of solving a model such as the Potts model is to give an exact
1813:, the relation amounts to transforming the sum over spin configurations
7426:
6966:
6910:
6794:
6747:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
6187:
5963:
5818:
5609:
4057:
3127:
the
Hamiltonian, but is used to build it. The argument to the function
2724:
63:
5819:"Code for efficiently computing Tutte, Chromatic and Flow Polynomials"
5791:
4992:. The parameter Îł > 0 controls the tradeoff between regularity and
4471:
This potential can be captured in a 2 Ă 2 matrix with matrix elements
4046:{\displaystyle Z_{n}(c)=e^{-c\beta }\sum _{s_{0},\ldots ,s_{n}\in Q}1}
6920:
6024:
5649:
5585:
5537:
5041:
3915:
which will show up as the logarithm of the leading eigenvalue of the
1694:
properties when fixing opposite boundaries in two different states .
59:
3905:{\displaystyle P(V)=\lim _{n\to \infty }{\frac {1}{n}}\log Z_{n}(V)}
3725:; it gives the likelihood of a given configuration occurring in the
3274:
limit of this function is the
Hamiltonian of the system; for finite
5928:
5505:
5497:
5459:
5332:
5220:
4678:, one seeks a minimizer of the corresponding inverse problem, the
2551:
2334:
can be an arbitrary complex number, rather than a natural integer.
561:
166:; the lattice is usually taken to be a two-dimensional rectangular
82:
5759:
1921:{\displaystyle \omega ={\Big \{}(i,j){\Big |}s_{i}=s_{j}{\Big \}}}
113:. Generalizations of the Potts model have also been used to model
6093:
3143:
just picked out two spins out of the infinite string: the values
1691:
5440:
5111:
Mathematical
Proceedings of the Cambridge Philosophical Society
3119:
will describe a next-nearest neighbor interaction. A function
2526:
The one dimensional Potts model may be expressed in terms of a
1713:. Understanding this relationship has helped develop efficient
670:{\displaystyle H_{p}=-J_{p}\sum _{(i,j)}\delta (s_{i},s_{j})\,}
194:
5780:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
5689:
4996:. There are fast algorithms for the exact minimization of the
2978:+1 spins match up exactly to a given, specific set of values Ο
5162:"Statistics of Two-Dimensional Lattices with Four Components"
2719:
be the set of all bi-infinite strings of values from the set
2184:
are the connected components of the union of closed segments
1461:. For example, for the standard ferromagnetic Potts model in
122:
16:
Model in statistical mechanics generalizing the Ising model
6727:
Autoregressive conditional heteroskedasticity (ARCH) model
2314:. An advantage of the random cluster formulation is that
1697:
6255:
Independent and identically distributed random variables
1717:
methods for numerical exploration of the model at small
1561:. The phase transition is continuous (second order) for
3123:
gives interaction energy between a set of spins; it is
1652:. Further use is found through the model's relation to
66:. The four-state Potts model is sometimes known as the
6732:
Autoregressive integrated moving average (ARIMA) model
4940:
forces piecewise constant solutions and the data term
4588:
4380:
specifying which neighboring spin values are allowed.
4285:
4237:
4195:
173:
Originally, Domb suggested that the spin takes one of
5776:"Fast approximate energy minimization via graph cuts"
4946:
4910:
4702:
4629:
for the partition function and an expression for the
4564:
4480:
4408:
4392:, where the spin can only take on one of two values,
4283:
4150:
4069:
3959:
3835:
3752:
3517:
3303:
3257:{\displaystyle H_{n}(s)=\sum _{k=0}^{n}V(\tau ^{k}s)}
3188:
3031:
3003:
The interaction between the spins is then given by a
2824:
2752:
2587:
2350:
2320:
2250:
2190:
2095:
1937:
1852:
1819:
1746:
1723:
1670:
1632:
1626:, and a continuous phase transition is observed when
1599:
1567:
1516:
1490:
1467:
1360:
1340:
1320:
1284:
1259:
1239:
1219:
1199:
1149:
1023:
969:
949:
884:
858:
812:
782:
739:
686:
585:
540:
508:
473:
441:
435:
with the sum running over the nearest neighbor pairs
323:
260:
206:
179:
5774:
Boykov, Y.; Veksler, O.; Zabih, R. (November 2001).
5392:
Beffara, Vincent; Duminil-Copin, Hugo (2012-08-01).
5391:
1702:
The Potts model has a close relation to the
Fortuin-
3480:
being the cylinder sets defined above. Here, ÎČ = 1/
2086:This leads to rewriting the partition function as
1014:(with no particular ordering). The Hamiltonian is,
5773:
5267:Graner, François; Glazier, James A. (1992-09-28).
4976:
4932:
4893:
4607:
4544:
4460:
4388:The simplest case of the interacting model is the
4353:
4255:
4126:
4045:
3904:
3811:
3710:
3462:
3256:
3087:
2963:
2793:
2708:
2565:
2507:
2326:
2306:
2236:
2169:
2075:
1920:
1838:
1805:
1729:
1682:
1644:
1611:
1585:
1553:
1502:
1476:
1432:
1346:
1326:
1306:
1270:
1245:
1225:
1205:
1185:
1132:
1006:
955:
923:
870:
844:
794:
765:
721:
669:
552:
526:
486:
459:
424:
302:
243:
193:possible values , distributed uniformly about the
185:
5747:Advances in Neural Information Processing Systems
5692:Journal of Computational and Graphical Statistics
5313:
5107:"Some generalized order-disorder transformations"
3285:
1913:
1883:
1861:
1806:{\displaystyle Z_{p}=\sum _{\{s_{i}\}}e^{-H_{p}}}
93:. The infinite-range Potts model is known as the
7472:
6614:Stochastic chains with memory of variable length
5817:Haggard, Gary; Pearce, David J.; Royle, Gordon.
5740:
4141:. The partition function may then be written as
3852:
2521:
1484:, a phase transition exists for all real values
244:{\displaystyle \theta _{s}={\frac {2\pi s}{q}},}
5586:"Interfacial adsorption in planar potts models"
2974:that is, the set of all possible strings where
2307:{\displaystyle p={\frac {v}{1+v}}=1-e^{-J_{p}}}
129:. A further generalization of these methods by
5816:
5201:
4653:
1664:found in combinatorics. For integer values of
78:, who considered an equivalent model in 1943.
62:", which was suggested to him by his advisor,
6203:
5841:
5741:KrĂ€henbĂŒhl, Philipp; Koltun, Vladlen (2011).
5266:
4461:{\displaystyle V(\sigma )=-J_{p}s_{0}s_{1}\,}
4060:, then the sum may be trivially evaluated as
5584:Selke, Walter; Huse, David A. (1983-06-01).
5202:Shimagaki, Kai; Weigt, Martin (2019-09-19).
5159:
4960:
4947:
4921:
4911:
4824:
4786:
4763:
4750:
4738:
4728:
4348:
4305:
4127:{\displaystyle Z_{n}(c)=e^{-c\beta }q^{n+1}}
2958:
2873:
2703:
2603:
1833:
1820:
1778:
1765:
1001:
983:
454:
442:
364:
352:
97:. When the spins are taken to interact in a
5522:
4608:{\displaystyle Z_{n}(V)={\mbox{Tr}}\,M^{n}}
3088:{\displaystyle V(s)=-J\delta (s_{0},s_{1})}
1554:{\displaystyle \beta J=\log(1+{\sqrt {q}})}
6742:Autoregressiveâmoving-average (ARMA) model
6210:
6196:
5848:
5834:
5590:Zeitschrift fĂŒr Physik B: Condensed Matter
3822:Another important related quantity is the
2688:
934:
924:{\displaystyle J_{p}=-{\frac {3}{2}}J_{c}}
802:standard Potts model is equivalent to the
5855:
5758:
5648:
5583:
5536:
5518:
5516:
5504:
5458:
5409:
5365:
5331:
5219:
4594:
4457:
4291:
4270:or count of a set, and Fix is the set of
4201:
3812:{\displaystyle A_{n}(V)=-kT\log Z_{n}(V)}
3022:continuous function will do; for example
1433:{\displaystyle J_{ij}(k,k')=J_{ji}(k',k)}
806:and the 2-state vector Potts model, with
666:
6217:
2538:used combinatorial methods to solve the
2237:{\displaystyle \cup _{(i,j)\in \omega }}
1740:At the level of the partition function
1007:{\displaystyle s_{i}\in \{1,\dots ,q\}}
567:
7473:
7048:Doob's martingale convergence theorems
5513:
5447:Communications in Mathematical Physics
3922:
2578:} be a finite set of symbols, and let
1698:Relation with the random cluster model
1447:
6800:Constant elasticity of variance (CEV)
6790:ChanâKarolyiâLongstaffâSanders (CKLS)
6191:
5829:
5638:
5398:Probability Theory and Related Fields
5160:Ashkin, J.; Teller, E. (1943-09-01).
5104:
2998:
153:
5387:
5385:
5197:
5195:
4383:
2794:{\displaystyle \tau (s)_{k}=s_{k+1}}
1846:into a sum over edge configurations
1593:and discontinuous (first order) for
1452:
722:{\displaystyle \delta (s_{i},s_{j})}
101:manner, the model is related to the
460:{\displaystyle \langle i,j\rangle }
13:
7287:Skorokhod's representation theorem
7068:Law of large numbers (weak/strong)
5065:
4914:
4783:
4731:
4670:from the noisy observation vector
3862:
2689:
2148:
2124:
572:What is now known as the standard
547:
14:
7507:
7257:Martingale representation theorem
5810:
5382:
5192:
4984:couples the minimizing candidate
3278:, these are sometimes called the
7302:Stochastic differential equation
7192:Doob's optional stopping theorem
7187:DoobâMeyer decomposition theorem
5884:
5022:Critical three-state Potts model
4933:{\displaystyle \|\nabla u\|_{0}}
4320:
4274:of the iterated shift function:
2888:
2699:
2594:
125:, and statistical properties of
7172:Convergence of random variables
7058:FisherâTippettâGnedenko theorem
5767:
5734:
5683:
5632:
5577:
4977:{\displaystyle \|u-f\|_{p}^{p}}
4648:
3290:The corresponding finite-state
2566:Topology of the space of states
2037:
2031:
43:and certain other phenomena of
6770:Binomial options pricing model
5491:
5434:
5307:
5260:
5153:
5098:
5059:
4881:
4852:
4719:
4713:
4581:
4575:
4418:
4412:
4213:
4190:
4167:
4161:
4086:
4080:
3976:
3970:
3899:
3893:
3859:
3845:
3839:
3806:
3800:
3769:
3763:
3705:
3702:
3699:
3654:
3641:
3622:
3610:
3604:
3582:
3579:
3534:
3521:
3457:
3454:
3451:
3406:
3393:
3374:
3320:
3314:
3286:Partition function and measure
3251:
3235:
3205:
3199:
3082:
3056:
3041:
3035:
2867:
2835:
2763:
2756:
2666:
2612:
2496:
2470:
2444:
2441:
2415:
2403:
2395:
2369:
2231:
2219:
2208:
2196:
2162:
2156:
2138:
2132:
2028:
2002:
1982:
1956:
1878:
1866:
1548:
1532:
1427:
1410:
1391:
1374:
1314:is the energetic cost of spin
1301:
1295:
1193:is the energetic cost of spin
1180:
1163:
1124:
1111:
1085:
1059:
716:
690:
663:
637:
629:
617:
544:
1:
7237:Kolmogorov continuity theorem
7073:Law of the iterated logarithm
5641:Surveys in Combinatorics 2005
5350:10.1016/j.physrep.2019.03.001
5105:Potts, R. B. (January 1952).
5052:
2522:Measure-theoretic description
1586:{\displaystyle 1\leq q\leq 4}
1510:, with the critical point at
1442:Sherrington-Kirkpatrick model
845:{\displaystyle J_{p}=-2J_{c}}
303:{\displaystyle s=0,1,...,q-1}
148:
7242:Kolmogorov extension theorem
6921:Generalized queueing network
6429:Interacting particle systems
5871:Principle of maximum entropy
5659:10.1017/CBO9780511734885.009
3157:. In general, the function
1186:{\displaystyle J_{ij}(k,k')}
733:, which equals one whenever
553:{\displaystyle q\to \infty }
467:over all lattice sites, and
158:The Potts model consists of
31:, is a model of interacting
7:
6374:Continuous-time random walk
5555:10.1103/PhysRevE.101.060105
5293:10.1103/PhysRevLett.69.2013
5238:10.1103/PhysRevE.100.032128
5010:
4654:Signal and image processing
1440:. This model resembles the
766:{\displaystyle s_{i}=s_{j}}
105:, which is used to discuss
10:
7512:
7382:Extreme value theory (EVT)
7182:Doob decomposition theorem
6474:OrnsteinâUhlenbeck process
6245:Chinese restaurant process
5895:Statistical thermodynamics
5753:. Curran Associates, Inc.
5032:Square-lattice Ising model
2812:for this topology are the
27:, a generalization of the
7450:
7354:
7262:Optional stopping theorem
7159:
7121:
7063:Large deviation principle
7030:
6944:
6901:
6868:
6815:HeathâJarrowâMorton (HJM)
6760:
6752:Moving-average (MA) model
6737:Autoregressive (AR) model
6717:
6627:
6562:Hidden Markov model (HMM)
6544:
6496:SchrammâLoewner evolution
6300:
6225:
6130:
6092:
6057:
6012:
5954:
5893:
5882:
5863:
5469:10.1007/s00220-016-2759-8
5411:10.1007/s00440-011-0353-8
5131:10.1017/S0305004100027419
5092:10.1103/RevModPhys.54.235
5072:Reviews of Modern Physics
3280:finite state Hamiltonians
1839:{\displaystyle \{s_{i}\}}
310:and that the interaction
54:The model is named after
7177:Doléans-Dade exponential
7007:Progressively measurable
6805:CoxâIngersollâRoss (CIR)
6155:Condensed matter physics
6138:Statistical field theory
5704:10.1198/106186008X285591
5066:Wu, F. Y. (1982-01-01).
2546:, behind this solution.
1715:Markov chain Monte Carlo
1307:{\displaystyle h_{i}(k)}
941:direct coupling analysis
7486:Exactly solvable models
7397:Mathematical statistics
7387:Large deviations theory
7217:Infinitesimal generator
7078:Maximal ergodic theorem
6997:Piecewise-deterministic
6599:Random dynamical system
6464:Markov additive process
6013:Mathematical approaches
6002:Lennard-Jones potential
5918:thermodynamic potential
5273:Physical Review Letters
4693:), which is defined by
4622:is a bit more complex.
4139:subshift of finite type
2804:This set has a natural
2729:subshift of finite type
2723:. This set is called a
2528:subshift of finite type
1683:{\displaystyle q\geq 3}
1645:{\displaystyle q\leq 4}
1503:{\displaystyle q\geq 1}
935:Generalized Potts model
7232:KarhunenâLoĂšve theorem
7167:CameronâMartin formula
7131:BurkholderâDavisâGundy
6526:Variance gamma process
6049:conformal field theory
5186:10.1103/PhysRev.64.178
4978:
4934:
4895:
4850:
4627:closed-form expression
4609:
4546:
4462:
4355:
4257:
4128:
4047:
3906:
3813:
3712:
3464:
3258:
3231:
3089:
2965:
2795:
2710:
2509:
2341:, using the identity
2328:
2308:
2238:
2171:
2077:
1922:
1840:
1807:
1731:
1684:
1646:
1613:
1612:{\displaystyle q>4}
1587:
1555:
1504:
1478:
1434:
1348:
1328:
1308:
1272:
1247:
1227:
1207:
1187:
1134:
1008:
957:
925:
872:
846:
796:
767:
723:
671:
554:
528:
488:
461:
426:
304:
245:
187:
111:quantum chromodynamics
7491:Statistical mechanics
7362:Actuarial mathematics
7324:Uniform integrability
7319:Stratonovich integral
7247:LĂ©vyâProkhorov metric
7151:MarcinkiewiczâZygmund
7038:Central limit theorem
6640:Gaussian random field
6469:McKeanâVlasov process
6389:Dyson Brownian motion
6250:GaltonâWatson process
5964:Ferromagnetism models
5857:Statistical mechanics
4979:
4935:
4896:
4830:
4610:
4547:
4463:
4356:
4258:
4129:
4048:
3907:
3814:
3741:Helmholtz free energy
3713:
3465:
3259:
3211:
3090:
2966:
2796:
2711:
2510:
2329:
2309:
2239:
2172:
2078:
1923:
1841:
1808:
1732:
1711:statistical mechanics
1685:
1662:chromatic polynomials
1647:
1614:
1588:
1556:
1505:
1479:
1435:
1349:
1329:
1309:
1273:
1248:
1228:
1208:
1188:
1135:
1009:
958:
926:
873:
847:
797:
768:
724:
672:
555:
529:
527:{\displaystyle q=3,4}
489:
487:{\displaystyle J_{c}}
462:
427:
305:
246:
188:
162:that are placed on a
21:statistical mechanics
7437:Time series analysis
7392:Mathematical finance
7277:Reflection principle
6604:Regenerative process
6404:FlemingâViot process
6219:Stochastic processes
5643:. pp. 173â226.
5047:Cellular Potts model
5017:Random cluster model
4944:
4908:
4700:
4562:
4478:
4406:
4281:
4148:
4067:
3957:
3833:
3824:topological pressure
3750:
3515:
3301:
3186:
3164:Define the function
3029:
2822:
2750:
2585:
2348:
2318:
2248:
2188:
2093:
1935:
1850:
1817:
1744:
1721:
1707:random cluster model
1668:
1630:
1597:
1565:
1514:
1488:
1465:
1358:
1338:
1318:
1282:
1257:
1237:
1217:
1197:
1147:
1021:
967:
947:
882:
856:
810:
780:
773:and zero otherwise.
737:
684:
583:
568:Standard Potts model
538:
506:
471:
439:
321:
258:
204:
177:
139:cellular Potts model
7432:Stochastic analysis
7272:Quadratic variation
7267:Prokhorov's theorem
7202:FeynmanâKac formula
6672:Markov random field
6320:Birthâdeath process
6143:elementary particle
5908:partition functions
5602:1983ZPhyB..50..113S
5547:2020PhRvE.101f0105L
5342:2019PhR...810....1M
5285:1992PhRvL..69.2013G
5230:2019PhRvE.100c2128S
5178:1943PhRv...64..178A
5123:1952PCPS...48..106P
5084:1982RvMP...54..235W
5004:-Potts functional.
4973:
4776:
4643:thermodynamic limit
3923:Free field solution
3727:configuration space
3723:probability measure
3005:continuous function
1709:, another model in
1448:Physical properties
871:{\displaystyle q=3}
795:{\displaystyle q=2}
560:, this becomes the
68:AshkinâTeller model
45:solid-state physics
37:crystalline lattice
7402:Probability theory
7282:Skorokhod integral
7252:Malliavin calculus
6835:Korn-Kreer-Lenssen
6719:Time series models
6682:PitmanâYor process
6170:information theory
6077:correlation length
6072:Critical exponents
6059:Critical phenomena
6040:stochastic process
6020:Boltzmann equation
5913:equations of state
5610:10.1007/BF01304093
5027:Chiral Potts model
4974:
4959:
4930:
4891:
4762:
4682:-Potts functional
4635:equilibrium states
4605:
4592:
4542:
4458:
4351:
4289:
4266:where card is the
4253:
4241:
4199:
4124:
4043:
4039:
3902:
3866:
3809:
3734:canonical ensemble
3708:
3490:Boltzmann constant
3460:
3367:
3292:partition function
3254:
3085:
3018:on this topology.
2999:Interaction energy
2961:
2791:
2706:
2532:transfer operators
2505:
2324:
2304:
2234:
2167:
2118:
2073:
1918:
1836:
1803:
1782:
1727:
1680:
1642:
1609:
1583:
1551:
1500:
1477:{\displaystyle 2d}
1474:
1430:
1344:
1324:
1304:
1271:{\displaystyle k'}
1268:
1243:
1223:
1203:
1183:
1130:
1100:
1045:
1004:
953:
921:
868:
842:
792:
763:
719:
667:
633:
550:
524:
496:vector Potts model
484:
457:
422:
368:
300:
241:
183:
154:Vector Potts model
7468:
7467:
7422:Signal processing
7141:Doob's upcrossing
7136:Doob's martingale
7100:EngelbertâSchmidt
7043:Donsker's theorem
6977:Feller-continuous
6845:RendlemanâBartter
6635:Dirichlet process
6552:Branching process
6521:Telegraph process
6414:Geometric process
6394:Empirical process
6384:Diffusion process
6240:Branching process
6235:Bernoulli process
6185:
6184:
6175:Boltzmann machine
6045:mean-field theory
5946:Maxwell relations
5792:10.1109/34.969114
5786:(11): 1222â1239.
5525:Physical Review E
5279:(13): 2013â2016.
5208:Physical Review E
5068:"The Potts model"
4904:The jump penalty
4591:
4384:Interacting model
4288:
4240:
4198:
3998:
3919:of the solution.
3917:transfer operator
3875:
3851:
3614:
3326:
2501:
2327:{\displaystyle q}
2273:
2154:
2130:
2109:
2069:
2035:
1760:
1730:{\displaystyle q}
1656:problems and the
1546:
1459:phase transitions
1453:Phase transitions
1347:{\displaystyle k}
1327:{\displaystyle i}
1246:{\displaystyle j}
1226:{\displaystyle k}
1206:{\displaystyle i}
1091:
1030:
956:{\displaystyle q}
909:
612:
347:
236:
186:{\displaystyle q}
7503:
7442:Machine learning
7329:Usual hypotheses
7212:Girsanov theorem
7197:Dynkin's formula
6962:Continuous paths
6870:Actuarial models
6810:GarmanâKohlhagen
6780:BlackâKarasinski
6775:BlackâDermanâToy
6762:Financial models
6628:Fields and other
6557:Gaussian process
6506:Sigma-martingale
6310:Additive process
6212:
6205:
6198:
6189:
6188:
6067:Phase transition
5888:
5887:
5850:
5843:
5836:
5827:
5826:
5822:
5804:
5803:
5771:
5765:
5764:
5762:
5738:
5732:
5731:
5687:
5681:
5680:
5652:
5636:
5630:
5629:
5581:
5575:
5574:
5540:
5520:
5511:
5510:
5508:
5495:
5489:
5488:
5462:
5438:
5432:
5431:
5413:
5389:
5380:
5379:
5369:
5335:
5311:
5305:
5304:
5264:
5258:
5257:
5223:
5199:
5190:
5189:
5172:(5â6): 178â184.
5157:
5151:
5150:
5102:
5096:
5095:
5063:
4983:
4981:
4980:
4975:
4972:
4967:
4939:
4937:
4936:
4931:
4929:
4928:
4900:
4898:
4897:
4892:
4890:
4889:
4884:
4878:
4877:
4865:
4864:
4855:
4849:
4844:
4823:
4822:
4804:
4803:
4775:
4770:
4746:
4745:
4712:
4711:
4637:in the limit of
4614:
4612:
4611:
4606:
4604:
4603:
4593:
4589:
4574:
4573:
4551:
4549:
4548:
4543:
4541:
4537:
4536:
4525:
4524:
4498:
4497:
4496:
4467:
4465:
4464:
4459:
4456:
4455:
4446:
4445:
4436:
4435:
4378:adjacency matrix
4360:
4358:
4357:
4352:
4338:
4337:
4325:
4324:
4323:
4301:
4300:
4290:
4286:
4262:
4260:
4259:
4254:
4252:
4251:
4242:
4238:
4235:
4234:
4216:
4211:
4210:
4200:
4196:
4193:
4188:
4187:
4160:
4159:
4133:
4131:
4130:
4125:
4123:
4122:
4107:
4106:
4079:
4078:
4052:
4050:
4049:
4044:
4038:
4031:
4030:
4012:
4011:
3997:
3996:
3969:
3968:
3911:
3909:
3908:
3903:
3892:
3891:
3876:
3868:
3865:
3818:
3816:
3815:
3810:
3799:
3798:
3762:
3761:
3717:
3715:
3714:
3709:
3698:
3697:
3679:
3678:
3666:
3665:
3653:
3652:
3640:
3639:
3615:
3613:
3603:
3602:
3589:
3578:
3577:
3559:
3558:
3546:
3545:
3533:
3532:
3469:
3467:
3466:
3461:
3450:
3449:
3431:
3430:
3418:
3417:
3405:
3404:
3392:
3391:
3366:
3359:
3358:
3340:
3339:
3313:
3312:
3273:
3263:
3261:
3260:
3255:
3247:
3246:
3230:
3225:
3198:
3197:
3094:
3092:
3091:
3086:
3081:
3080:
3068:
3067:
2970:
2968:
2967:
2962:
2957:
2956:
2944:
2943:
2919:
2918:
2906:
2905:
2893:
2892:
2891:
2866:
2865:
2847:
2846:
2834:
2833:
2806:product topology
2800:
2798:
2797:
2792:
2790:
2789:
2771:
2770:
2715:
2713:
2712:
2707:
2702:
2681:
2680:
2659:
2658:
2646:
2645:
2633:
2632:
2599:
2598:
2597:
2556:Heisenberg model
2514:
2512:
2511:
2506:
2499:
2495:
2494:
2482:
2481:
2466:
2465:
2464:
2463:
2440:
2439:
2427:
2426:
2399:
2398:
2394:
2393:
2381:
2380:
2365:
2364:
2333:
2331:
2330:
2325:
2313:
2311:
2310:
2305:
2303:
2302:
2301:
2300:
2274:
2272:
2258:
2243:
2241:
2240:
2235:
2218:
2217:
2176:
2174:
2173:
2168:
2166:
2165:
2155:
2152:
2142:
2141:
2131:
2128:
2117:
2105:
2104:
2082:
2080:
2079:
2074:
2067:
2060:
2059:
2058:
2057:
2036:
2034: with
2033:
2027:
2026:
2014:
2013:
1986:
1985:
1981:
1980:
1968:
1967:
1952:
1951:
1927:
1925:
1924:
1919:
1917:
1916:
1910:
1909:
1897:
1896:
1887:
1886:
1865:
1864:
1845:
1843:
1842:
1837:
1832:
1831:
1812:
1810:
1809:
1804:
1802:
1801:
1800:
1799:
1781:
1777:
1776:
1756:
1755:
1736:
1734:
1733:
1728:
1689:
1687:
1686:
1681:
1651:
1649:
1648:
1643:
1618:
1616:
1615:
1610:
1592:
1590:
1589:
1584:
1560:
1558:
1557:
1552:
1547:
1542:
1509:
1507:
1506:
1501:
1483:
1481:
1480:
1475:
1439:
1437:
1436:
1431:
1420:
1409:
1408:
1390:
1373:
1372:
1353:
1351:
1350:
1345:
1333:
1331:
1330:
1325:
1313:
1311:
1310:
1305:
1294:
1293:
1277:
1275:
1274:
1269:
1267:
1252:
1250:
1249:
1244:
1232:
1230:
1229:
1224:
1212:
1210:
1209:
1204:
1192:
1190:
1189:
1184:
1179:
1162:
1161:
1139:
1137:
1136:
1131:
1123:
1122:
1110:
1109:
1099:
1084:
1083:
1071:
1070:
1058:
1057:
1044:
1013:
1011:
1010:
1005:
979:
978:
962:
960:
959:
954:
930:
928:
927:
922:
920:
919:
910:
902:
894:
893:
877:
875:
874:
869:
851:
849:
848:
843:
841:
840:
822:
821:
801:
799:
798:
793:
772:
770:
769:
764:
762:
761:
749:
748:
728:
726:
725:
720:
715:
714:
702:
701:
676:
674:
673:
668:
662:
661:
649:
648:
632:
611:
610:
595:
594:
559:
557:
556:
551:
533:
531:
530:
525:
493:
491:
490:
485:
483:
482:
466:
464:
463:
458:
431:
429:
428:
423:
421:
417:
416:
415:
414:
413:
396:
395:
394:
393:
367:
346:
345:
333:
332:
309:
307:
306:
301:
250:
248:
247:
242:
237:
232:
221:
216:
215:
192:
190:
189:
184:
87:Heisenberg model
49:exactly solvable
7511:
7510:
7506:
7505:
7504:
7502:
7501:
7500:
7471:
7470:
7469:
7464:
7446:
7407:Queueing theory
7350:
7292:Skorokhod space
7155:
7146:KunitaâWatanabe
7117:
7083:Sanov's theorem
7053:Ergodic theorem
7026:
7022:Time-reversible
6940:
6903:Queueing models
6897:
6893:SparreâAnderson
6883:CramĂ©râLundberg
6864:
6850:SABR volatility
6756:
6713:
6665:Boolean network
6623:
6609:Renewal process
6540:
6489:Non-homogeneous
6479:Poisson process
6369:Contact process
6332:Brownian motion
6302:Continuous time
6296:
6290:Maximal entropy
6221:
6216:
6186:
6181:
6126:
6088:
6053:
6035:BBGKY hierarchy
6030:Vlasov equation
6008:
5997:depletion force
5990:Particles with
5950:
5889:
5885:
5880:
5859:
5854:
5813:
5808:
5807:
5772:
5768:
5739:
5735:
5688:
5684:
5669:
5637:
5633:
5582:
5578:
5521:
5514:
5496:
5492:
5439:
5435:
5390:
5383:
5320:Physics Reports
5312:
5308:
5265:
5261:
5200:
5193:
5166:Physical Review
5158:
5154:
5103:
5099:
5064:
5060:
5055:
5013:
4968:
4963:
4945:
4942:
4941:
4924:
4920:
4909:
4906:
4905:
4885:
4880:
4879:
4873:
4869:
4860:
4856:
4851:
4845:
4834:
4812:
4808:
4799:
4795:
4771:
4766:
4741:
4737:
4707:
4703:
4701:
4698:
4697:
4688:
4656:
4651:
4599:
4595:
4587:
4569:
4565:
4563:
4560:
4559:
4529:
4520:
4516:
4512:
4508:
4489:
4485:
4481:
4479:
4476:
4475:
4451:
4447:
4441:
4437:
4431:
4427:
4407:
4404:
4403:
4397:
4386:
4333:
4329:
4319:
4318:
4314:
4296:
4292:
4284:
4282:
4279:
4278:
4247:
4243:
4236:
4224:
4220:
4212:
4206:
4202:
4194:
4189:
4177:
4173:
4155:
4151:
4149:
4146:
4145:
4112:
4108:
4096:
4092:
4074:
4070:
4068:
4065:
4064:
4026:
4022:
4007:
4003:
4002:
3986:
3982:
3964:
3960:
3958:
3955:
3954:
3940:
3925:
3887:
3883:
3867:
3855:
3834:
3831:
3830:
3794:
3790:
3757:
3753:
3751:
3748:
3747:
3693:
3689:
3674:
3670:
3661:
3657:
3648:
3644:
3635:
3631:
3598:
3594:
3593:
3588:
3573:
3569:
3554:
3550:
3541:
3537:
3528:
3524:
3516:
3513:
3512:
3479:
3445:
3441:
3426:
3422:
3413:
3409:
3400:
3396:
3387:
3383:
3354:
3350:
3335:
3331:
3330:
3308:
3304:
3302:
3299:
3298:
3288:
3268:
3242:
3238:
3226:
3215:
3193:
3189:
3187:
3184:
3183:
3169:
3156:
3149:
3118:
3111:
3104:
3076:
3072:
3063:
3059:
3030:
3027:
3026:
3001:
2987:
2981:
2952:
2948:
2933:
2929:
2914:
2910:
2901:
2897:
2887:
2886:
2882:
2861:
2857:
2842:
2838:
2829:
2825:
2823:
2820:
2819:
2779:
2775:
2766:
2762:
2751:
2748:
2747:
2698:
2676:
2672:
2654:
2650:
2641:
2637:
2625:
2621:
2593:
2592:
2588:
2586:
2583:
2582:
2568:
2524:
2490:
2486:
2477:
2473:
2459:
2455:
2454:
2450:
2435:
2431:
2422:
2418:
2389:
2385:
2376:
2372:
2360:
2356:
2355:
2351:
2349:
2346:
2345:
2319:
2316:
2315:
2296:
2292:
2288:
2284:
2262:
2257:
2249:
2246:
2245:
2195:
2191:
2189:
2186:
2185:
2151:
2147:
2143:
2127:
2123:
2119:
2113:
2100:
2096:
2094:
2091:
2090:
2053:
2049:
2048:
2044:
2032:
2022:
2018:
2009:
2005:
1976:
1972:
1963:
1959:
1947:
1943:
1942:
1938:
1936:
1933:
1932:
1912:
1911:
1905:
1901:
1892:
1888:
1882:
1881:
1860:
1859:
1851:
1848:
1847:
1827:
1823:
1818:
1815:
1814:
1795:
1791:
1787:
1783:
1772:
1768:
1764:
1751:
1747:
1745:
1742:
1741:
1722:
1719:
1718:
1700:
1669:
1666:
1665:
1631:
1628:
1627:
1624:BKT transitions
1598:
1595:
1594:
1566:
1563:
1562:
1541:
1515:
1512:
1511:
1489:
1486:
1485:
1466:
1463:
1462:
1455:
1450:
1413:
1401:
1397:
1383:
1365:
1361:
1359:
1356:
1355:
1339:
1336:
1335:
1334:being in state
1319:
1316:
1315:
1289:
1285:
1283:
1280:
1279:
1260:
1258:
1255:
1254:
1238:
1235:
1234:
1218:
1215:
1214:
1213:being in state
1198:
1195:
1194:
1172:
1154:
1150:
1148:
1145:
1144:
1118:
1114:
1105:
1101:
1095:
1079:
1075:
1066:
1062:
1050:
1046:
1034:
1022:
1019:
1018:
974:
970:
968:
965:
964:
948:
945:
944:
937:
915:
911:
901:
889:
885:
883:
880:
879:
857:
854:
853:
836:
832:
817:
813:
811:
808:
807:
781:
778:
777:
757:
753:
744:
740:
738:
735:
734:
731:Kronecker delta
710:
706:
697:
693:
685:
682:
681:
657:
653:
644:
640:
616:
606:
602:
590:
586:
584:
581:
580:
570:
539:
536:
535:
534:. In the limit
507:
504:
503:
478:
474:
472:
469:
468:
440:
437:
436:
409:
405:
404:
400:
389:
385:
384:
380:
379:
375:
351:
341:
337:
328:
324:
322:
319:
318:
259:
256:
255:
222:
220:
211:
207:
205:
202:
201:
178:
175:
174:
156:
151:
137:, known as the
135:Francois Graner
103:flux tube model
17:
12:
11:
5:
7509:
7499:
7498:
7496:Lattice models
7493:
7488:
7483:
7466:
7465:
7463:
7462:
7457:
7455:List of topics
7451:
7448:
7447:
7445:
7444:
7439:
7434:
7429:
7424:
7419:
7414:
7412:Renewal theory
7409:
7404:
7399:
7394:
7389:
7384:
7379:
7377:Ergodic theory
7374:
7369:
7367:Control theory
7364:
7358:
7356:
7352:
7351:
7349:
7348:
7347:
7346:
7341:
7331:
7326:
7321:
7316:
7311:
7310:
7309:
7299:
7297:Snell envelope
7294:
7289:
7284:
7279:
7274:
7269:
7264:
7259:
7254:
7249:
7244:
7239:
7234:
7229:
7224:
7219:
7214:
7209:
7204:
7199:
7194:
7189:
7184:
7179:
7174:
7169:
7163:
7161:
7157:
7156:
7154:
7153:
7148:
7143:
7138:
7133:
7127:
7125:
7119:
7118:
7116:
7115:
7096:BorelâCantelli
7085:
7080:
7075:
7070:
7065:
7060:
7055:
7050:
7045:
7040:
7034:
7032:
7031:Limit theorems
7028:
7027:
7025:
7024:
7019:
7014:
7009:
7004:
6999:
6994:
6989:
6984:
6979:
6974:
6969:
6964:
6959:
6954:
6948:
6946:
6942:
6941:
6939:
6938:
6933:
6928:
6923:
6918:
6913:
6907:
6905:
6899:
6898:
6896:
6895:
6890:
6885:
6880:
6874:
6872:
6866:
6865:
6863:
6862:
6857:
6852:
6847:
6842:
6837:
6832:
6827:
6822:
6817:
6812:
6807:
6802:
6797:
6792:
6787:
6782:
6777:
6772:
6766:
6764:
6758:
6757:
6755:
6754:
6749:
6744:
6739:
6734:
6729:
6723:
6721:
6715:
6714:
6712:
6711:
6706:
6701:
6700:
6699:
6694:
6684:
6679:
6674:
6669:
6668:
6667:
6662:
6652:
6650:Hopfield model
6647:
6642:
6637:
6631:
6629:
6625:
6624:
6622:
6621:
6616:
6611:
6606:
6601:
6596:
6595:
6594:
6589:
6584:
6579:
6569:
6567:Markov process
6564:
6559:
6554:
6548:
6546:
6542:
6541:
6539:
6538:
6536:Wiener sausage
6533:
6531:Wiener process
6528:
6523:
6518:
6513:
6511:Stable process
6508:
6503:
6501:Semimartingale
6498:
6493:
6492:
6491:
6486:
6476:
6471:
6466:
6461:
6456:
6451:
6446:
6444:Jump diffusion
6441:
6436:
6431:
6426:
6421:
6419:Hawkes process
6416:
6411:
6406:
6401:
6399:Feller process
6396:
6391:
6386:
6381:
6376:
6371:
6366:
6364:Cauchy process
6361:
6360:
6359:
6354:
6349:
6344:
6339:
6329:
6328:
6327:
6317:
6315:Bessel process
6312:
6306:
6304:
6298:
6297:
6295:
6294:
6293:
6292:
6287:
6282:
6277:
6267:
6262:
6257:
6252:
6247:
6242:
6237:
6231:
6229:
6223:
6222:
6215:
6214:
6207:
6200:
6192:
6183:
6182:
6180:
6179:
6178:
6177:
6172:
6167:
6160:Complex system
6157:
6152:
6151:
6150:
6145:
6134:
6132:
6128:
6127:
6125:
6124:
6119:
6114:
6109:
6104:
6098:
6096:
6090:
6089:
6087:
6086:
6085:
6084:
6079:
6069:
6063:
6061:
6055:
6054:
6052:
6051:
6042:
6037:
6032:
6027:
6022:
6016:
6014:
6010:
6009:
6007:
6006:
6005:
6004:
5999:
5988:
5987:
5986:
5981:
5976:
5971:
5960:
5958:
5952:
5951:
5949:
5948:
5943:
5942:
5941:
5936:
5931:
5926:
5915:
5910:
5905:
5899:
5897:
5891:
5890:
5883:
5881:
5879:
5878:
5876:ergodic theory
5873:
5867:
5865:
5861:
5860:
5853:
5852:
5845:
5838:
5830:
5824:
5823:
5812:
5811:External links
5809:
5806:
5805:
5766:
5733:
5698:(1): 201â224.
5682:
5667:
5631:
5596:(2): 113â116.
5576:
5512:
5490:
5433:
5404:(3): 511â542.
5381:
5306:
5259:
5191:
5152:
5117:(1): 106â109.
5097:
5078:(1): 235â268.
5057:
5056:
5054:
5051:
5050:
5049:
5044:
5039:
5037:Minimal models
5034:
5029:
5024:
5019:
5012:
5009:
4971:
4966:
4962:
4958:
4955:
4952:
4949:
4927:
4923:
4919:
4916:
4913:
4902:
4901:
4888:
4883:
4876:
4872:
4868:
4863:
4859:
4854:
4848:
4843:
4840:
4837:
4833:
4829:
4826:
4821:
4818:
4815:
4811:
4807:
4802:
4798:
4794:
4791:
4788:
4785:
4782:
4779:
4774:
4769:
4765:
4761:
4758:
4755:
4752:
4749:
4744:
4740:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4710:
4706:
4686:
4655:
4652:
4650:
4647:
4616:
4615:
4602:
4598:
4586:
4583:
4580:
4577:
4572:
4568:
4553:
4552:
4540:
4535:
4532:
4528:
4523:
4519:
4515:
4511:
4507:
4504:
4501:
4495:
4492:
4488:
4484:
4469:
4468:
4454:
4450:
4444:
4440:
4434:
4430:
4426:
4423:
4420:
4417:
4414:
4411:
4395:
4385:
4382:
4362:
4361:
4350:
4347:
4344:
4341:
4336:
4332:
4328:
4322:
4317:
4313:
4310:
4307:
4304:
4299:
4295:
4264:
4263:
4250:
4246:
4233:
4230:
4227:
4223:
4219:
4215:
4209:
4205:
4192:
4186:
4183:
4180:
4176:
4172:
4169:
4166:
4163:
4158:
4154:
4135:
4134:
4121:
4118:
4115:
4111:
4105:
4102:
4099:
4095:
4091:
4088:
4085:
4082:
4077:
4073:
4054:
4053:
4042:
4037:
4034:
4029:
4025:
4021:
4018:
4015:
4010:
4006:
4001:
3995:
3992:
3989:
3985:
3981:
3978:
3975:
3972:
3967:
3963:
3938:
3924:
3921:
3913:
3912:
3901:
3898:
3895:
3890:
3886:
3882:
3879:
3874:
3871:
3864:
3861:
3858:
3854:
3850:
3847:
3844:
3841:
3838:
3820:
3819:
3808:
3805:
3802:
3797:
3793:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3760:
3756:
3719:
3718:
3707:
3704:
3701:
3696:
3692:
3688:
3685:
3682:
3677:
3673:
3669:
3664:
3660:
3656:
3651:
3647:
3643:
3638:
3634:
3630:
3627:
3624:
3621:
3618:
3612:
3609:
3606:
3601:
3597:
3592:
3587:
3584:
3581:
3576:
3572:
3568:
3565:
3562:
3557:
3553:
3549:
3544:
3540:
3536:
3531:
3527:
3523:
3520:
3477:
3471:
3470:
3459:
3456:
3453:
3448:
3444:
3440:
3437:
3434:
3429:
3425:
3421:
3416:
3412:
3408:
3403:
3399:
3395:
3390:
3386:
3382:
3379:
3376:
3373:
3370:
3365:
3362:
3357:
3353:
3349:
3346:
3343:
3338:
3334:
3329:
3325:
3322:
3319:
3316:
3311:
3307:
3287:
3284:
3265:
3264:
3253:
3250:
3245:
3241:
3237:
3234:
3229:
3224:
3221:
3218:
3214:
3210:
3207:
3204:
3201:
3196:
3192:
3167:
3154:
3147:
3131:is an element
3116:
3109:
3102:
3096:
3095:
3084:
3079:
3075:
3071:
3066:
3062:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3000:
2997:
2983:
2979:
2972:
2971:
2960:
2955:
2951:
2947:
2942:
2939:
2936:
2932:
2928:
2925:
2922:
2917:
2913:
2909:
2904:
2900:
2896:
2890:
2885:
2881:
2878:
2875:
2872:
2869:
2864:
2860:
2856:
2853:
2850:
2845:
2841:
2837:
2832:
2828:
2802:
2801:
2788:
2785:
2782:
2778:
2774:
2769:
2765:
2761:
2758:
2755:
2733:shift operator
2717:
2716:
2705:
2701:
2697:
2694:
2691:
2687:
2684:
2679:
2675:
2671:
2668:
2665:
2662:
2657:
2653:
2649:
2644:
2640:
2636:
2631:
2628:
2624:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2596:
2591:
2567:
2564:
2560:N-vector model
2544:measure theory
2523:
2520:
2516:
2515:
2504:
2498:
2493:
2489:
2485:
2480:
2476:
2472:
2469:
2462:
2458:
2453:
2449:
2446:
2443:
2438:
2434:
2430:
2425:
2421:
2417:
2414:
2411:
2408:
2405:
2402:
2397:
2392:
2388:
2384:
2379:
2375:
2371:
2368:
2363:
2359:
2354:
2323:
2299:
2295:
2291:
2287:
2283:
2280:
2277:
2271:
2268:
2265:
2261:
2256:
2253:
2233:
2230:
2227:
2224:
2221:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2194:
2178:
2177:
2164:
2161:
2158:
2150:
2146:
2140:
2137:
2134:
2126:
2122:
2116:
2112:
2108:
2103:
2099:
2084:
2083:
2072:
2066:
2063:
2056:
2052:
2047:
2043:
2040:
2030:
2025:
2021:
2017:
2012:
2008:
2004:
2001:
1998:
1995:
1992:
1989:
1984:
1979:
1975:
1971:
1966:
1962:
1958:
1955:
1950:
1946:
1941:
1915:
1908:
1904:
1900:
1895:
1891:
1885:
1880:
1877:
1874:
1871:
1868:
1863:
1858:
1855:
1835:
1830:
1826:
1822:
1798:
1794:
1790:
1786:
1780:
1775:
1771:
1767:
1763:
1759:
1754:
1750:
1726:
1699:
1696:
1679:
1676:
1673:
1641:
1638:
1635:
1608:
1605:
1602:
1582:
1579:
1576:
1573:
1570:
1550:
1545:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1499:
1496:
1493:
1473:
1470:
1454:
1451:
1449:
1446:
1429:
1426:
1423:
1419:
1416:
1412:
1407:
1404:
1400:
1396:
1393:
1389:
1386:
1382:
1379:
1376:
1371:
1368:
1364:
1343:
1323:
1303:
1300:
1297:
1292:
1288:
1266:
1263:
1242:
1222:
1202:
1182:
1178:
1175:
1171:
1168:
1165:
1160:
1157:
1153:
1141:
1140:
1129:
1126:
1121:
1117:
1113:
1108:
1104:
1098:
1094:
1090:
1087:
1082:
1078:
1074:
1069:
1065:
1061:
1056:
1053:
1049:
1043:
1040:
1037:
1033:
1029:
1026:
1003:
1000:
997:
994:
991:
988:
985:
982:
977:
973:
952:
936:
933:
918:
914:
908:
905:
900:
897:
892:
888:
867:
864:
861:
839:
835:
831:
828:
825:
820:
816:
791:
788:
785:
760:
756:
752:
747:
743:
718:
713:
709:
705:
700:
696:
692:
689:
678:
677:
665:
660:
656:
652:
647:
643:
639:
636:
631:
628:
625:
622:
619:
615:
609:
605:
601:
598:
593:
589:
569:
566:
549:
546:
543:
523:
520:
517:
514:
511:
481:
477:
456:
453:
450:
447:
444:
433:
432:
420:
412:
408:
403:
399:
392:
388:
383:
378:
374:
371:
366:
363:
360:
357:
354:
350:
344:
340:
336:
331:
327:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
252:
251:
240:
235:
231:
228:
225:
219:
214:
210:
182:
155:
152:
150:
147:
91:N-vector model
15:
9:
6:
4:
3:
2:
7508:
7497:
7494:
7492:
7489:
7487:
7484:
7482:
7479:
7478:
7476:
7461:
7458:
7456:
7453:
7452:
7449:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7393:
7390:
7388:
7385:
7383:
7380:
7378:
7375:
7373:
7370:
7368:
7365:
7363:
7360:
7359:
7357:
7353:
7345:
7342:
7340:
7337:
7336:
7335:
7332:
7330:
7327:
7325:
7322:
7320:
7317:
7315:
7314:Stopping time
7312:
7308:
7305:
7304:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7278:
7275:
7273:
7270:
7268:
7265:
7263:
7260:
7258:
7255:
7253:
7250:
7248:
7245:
7243:
7240:
7238:
7235:
7233:
7230:
7228:
7225:
7223:
7220:
7218:
7215:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7164:
7162:
7158:
7152:
7149:
7147:
7144:
7142:
7139:
7137:
7134:
7132:
7129:
7128:
7126:
7124:
7120:
7113:
7109:
7105:
7104:HewittâSavage
7101:
7097:
7093:
7089:
7088:Zeroâone laws
7086:
7084:
7081:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7059:
7056:
7054:
7051:
7049:
7046:
7044:
7041:
7039:
7036:
7035:
7033:
7029:
7023:
7020:
7018:
7015:
7013:
7010:
7008:
7005:
7003:
7000:
6998:
6995:
6993:
6990:
6988:
6985:
6983:
6980:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6949:
6947:
6943:
6937:
6934:
6932:
6929:
6927:
6924:
6922:
6919:
6917:
6914:
6912:
6909:
6908:
6906:
6904:
6900:
6894:
6891:
6889:
6886:
6884:
6881:
6879:
6876:
6875:
6873:
6871:
6867:
6861:
6858:
6856:
6853:
6851:
6848:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6811:
6808:
6806:
6803:
6801:
6798:
6796:
6793:
6791:
6788:
6786:
6785:BlackâScholes
6783:
6781:
6778:
6776:
6773:
6771:
6768:
6767:
6765:
6763:
6759:
6753:
6750:
6748:
6745:
6743:
6740:
6738:
6735:
6733:
6730:
6728:
6725:
6724:
6722:
6720:
6716:
6710:
6707:
6705:
6702:
6698:
6695:
6693:
6690:
6689:
6688:
6687:Point process
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6666:
6663:
6661:
6658:
6657:
6656:
6653:
6651:
6648:
6646:
6645:Gibbs measure
6643:
6641:
6638:
6636:
6633:
6632:
6630:
6626:
6620:
6617:
6615:
6612:
6610:
6607:
6605:
6602:
6600:
6597:
6593:
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6574:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6550:
6549:
6547:
6543:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6519:
6517:
6514:
6512:
6509:
6507:
6504:
6502:
6499:
6497:
6494:
6490:
6487:
6485:
6482:
6481:
6480:
6477:
6475:
6472:
6470:
6467:
6465:
6462:
6460:
6457:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6434:ItĂŽ diffusion
6432:
6430:
6427:
6425:
6422:
6420:
6417:
6415:
6412:
6410:
6409:Gamma process
6407:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6358:
6355:
6353:
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6334:
6333:
6330:
6326:
6323:
6322:
6321:
6318:
6316:
6313:
6311:
6308:
6307:
6305:
6303:
6299:
6291:
6288:
6286:
6283:
6281:
6280:Self-avoiding
6278:
6276:
6273:
6272:
6271:
6268:
6266:
6265:Moran process
6263:
6261:
6258:
6256:
6253:
6251:
6248:
6246:
6243:
6241:
6238:
6236:
6233:
6232:
6230:
6228:
6227:Discrete time
6224:
6220:
6213:
6208:
6206:
6201:
6199:
6194:
6193:
6190:
6176:
6173:
6171:
6168:
6166:
6163:
6162:
6161:
6158:
6156:
6153:
6149:
6148:superfluidity
6146:
6144:
6141:
6140:
6139:
6136:
6135:
6133:
6129:
6123:
6120:
6118:
6115:
6113:
6110:
6108:
6105:
6103:
6100:
6099:
6097:
6095:
6091:
6083:
6080:
6078:
6075:
6074:
6073:
6070:
6068:
6065:
6064:
6062:
6060:
6056:
6050:
6046:
6043:
6041:
6038:
6036:
6033:
6031:
6028:
6026:
6023:
6021:
6018:
6017:
6015:
6011:
6003:
6000:
5998:
5995:
5994:
5993:
5989:
5985:
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5966:
5965:
5962:
5961:
5959:
5957:
5953:
5947:
5944:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5921:
5919:
5916:
5914:
5911:
5909:
5906:
5904:
5901:
5900:
5898:
5896:
5892:
5877:
5874:
5872:
5869:
5868:
5866:
5862:
5858:
5851:
5846:
5844:
5839:
5837:
5832:
5831:
5828:
5820:
5815:
5814:
5801:
5797:
5793:
5789:
5785:
5781:
5777:
5770:
5761:
5756:
5752:
5748:
5744:
5737:
5729:
5725:
5721:
5717:
5713:
5709:
5705:
5701:
5697:
5693:
5686:
5678:
5674:
5670:
5668:9780521615235
5664:
5660:
5656:
5651:
5646:
5642:
5635:
5627:
5623:
5619:
5615:
5611:
5607:
5603:
5599:
5595:
5591:
5587:
5580:
5572:
5568:
5564:
5560:
5556:
5552:
5548:
5544:
5539:
5534:
5531:(6): 060105.
5530:
5526:
5519:
5517:
5507:
5502:
5494:
5486:
5482:
5478:
5474:
5470:
5466:
5461:
5456:
5453:(1): 47â107.
5452:
5448:
5444:
5437:
5429:
5425:
5421:
5417:
5412:
5407:
5403:
5399:
5395:
5388:
5386:
5377:
5373:
5368:
5363:
5359:
5355:
5351:
5347:
5343:
5339:
5334:
5329:
5325:
5321:
5317:
5310:
5302:
5298:
5294:
5290:
5286:
5282:
5278:
5274:
5270:
5263:
5255:
5251:
5247:
5243:
5239:
5235:
5231:
5227:
5222:
5217:
5214:(3): 032128.
5213:
5209:
5205:
5198:
5196:
5187:
5183:
5179:
5175:
5171:
5167:
5163:
5156:
5148:
5144:
5140:
5136:
5132:
5128:
5124:
5120:
5116:
5112:
5108:
5101:
5093:
5089:
5085:
5081:
5077:
5073:
5069:
5062:
5058:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5014:
5008:
5005:
5003:
4999:
4995:
4994:data fidelity
4991:
4987:
4969:
4964:
4956:
4953:
4950:
4925:
4917:
4886:
4874:
4870:
4866:
4861:
4857:
4846:
4841:
4838:
4835:
4831:
4827:
4819:
4816:
4813:
4809:
4805:
4800:
4796:
4792:
4789:
4780:
4777:
4772:
4767:
4759:
4756:
4753:
4747:
4742:
4734:
4725:
4722:
4716:
4708:
4704:
4696:
4695:
4694:
4692:
4685:
4681:
4677:
4673:
4669:
4666:. To recover
4665:
4661:
4646:
4644:
4640:
4636:
4632:
4628:
4623:
4621:
4600:
4596:
4584:
4578:
4570:
4566:
4558:
4557:
4556:
4538:
4533:
4530:
4526:
4521:
4517:
4513:
4509:
4505:
4502:
4499:
4493:
4490:
4486:
4482:
4474:
4473:
4472:
4452:
4448:
4442:
4438:
4432:
4428:
4424:
4421:
4415:
4409:
4402:
4401:
4400:
4398:
4391:
4381:
4379:
4375:
4371:
4367:
4345:
4342:
4339:
4334:
4330:
4326:
4315:
4311:
4308:
4302:
4297:
4293:
4277:
4276:
4275:
4273:
4269:
4248:
4244:
4231:
4228:
4225:
4221:
4217:
4207:
4203:
4184:
4181:
4178:
4174:
4170:
4164:
4156:
4152:
4144:
4143:
4142:
4140:
4119:
4116:
4113:
4109:
4103:
4100:
4097:
4093:
4089:
4083:
4075:
4071:
4063:
4062:
4061:
4059:
4040:
4035:
4032:
4027:
4023:
4019:
4016:
4013:
4008:
4004:
3999:
3993:
3990:
3987:
3983:
3979:
3973:
3965:
3961:
3953:
3952:
3951:
3949:
3945:
3941:
3934:
3930:
3920:
3918:
3896:
3888:
3884:
3880:
3877:
3872:
3869:
3856:
3848:
3842:
3836:
3829:
3828:
3827:
3826:, defined as
3825:
3803:
3795:
3791:
3787:
3784:
3781:
3778:
3775:
3772:
3766:
3758:
3754:
3746:
3745:
3744:
3742:
3737:
3735:
3731:
3728:
3724:
3694:
3690:
3686:
3683:
3680:
3675:
3671:
3667:
3662:
3658:
3649:
3645:
3636:
3632:
3628:
3625:
3619:
3616:
3607:
3599:
3595:
3590:
3585:
3574:
3570:
3566:
3563:
3560:
3555:
3551:
3547:
3542:
3538:
3529:
3525:
3518:
3511:
3510:
3509:
3507:
3503:
3499:
3495:
3491:
3487:
3483:
3476:
3446:
3442:
3438:
3435:
3432:
3427:
3423:
3419:
3414:
3410:
3401:
3397:
3388:
3384:
3380:
3377:
3371:
3368:
3363:
3360:
3355:
3351:
3347:
3344:
3341:
3336:
3332:
3327:
3323:
3317:
3309:
3305:
3297:
3296:
3295:
3293:
3283:
3281:
3277:
3271:
3248:
3243:
3239:
3232:
3227:
3222:
3219:
3216:
3212:
3208:
3202:
3194:
3190:
3182:
3181:
3180:
3178:
3174:
3170:
3162:
3160:
3153:
3146:
3142:
3138:
3134:
3130:
3126:
3122:
3115:
3108:
3101:
3077:
3073:
3069:
3064:
3060:
3053:
3050:
3047:
3044:
3038:
3032:
3025:
3024:
3023:
3021:
3017:
3013:
3009:
3006:
2996:
2994:
2992:
2986:
2977:
2953:
2949:
2945:
2940:
2937:
2934:
2930:
2926:
2923:
2920:
2915:
2911:
2907:
2902:
2898:
2894:
2883:
2879:
2876:
2870:
2862:
2858:
2854:
2851:
2848:
2843:
2839:
2830:
2826:
2818:
2817:
2816:
2815:
2814:cylinder sets
2811:
2807:
2786:
2783:
2780:
2776:
2772:
2767:
2759:
2753:
2746:
2745:
2744:
2742:
2738:
2734:
2730:
2726:
2722:
2695:
2692:
2685:
2682:
2677:
2673:
2669:
2663:
2660:
2655:
2651:
2647:
2642:
2638:
2634:
2629:
2626:
2622:
2618:
2615:
2609:
2606:
2600:
2589:
2581:
2580:
2579:
2577:
2573:
2563:
2561:
2557:
2553:
2547:
2545:
2541:
2537:
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2533:
2529:
2519:
2502:
2491:
2487:
2483:
2478:
2474:
2467:
2460:
2456:
2451:
2447:
2436:
2432:
2428:
2423:
2419:
2412:
2409:
2406:
2400:
2390:
2386:
2382:
2377:
2373:
2366:
2361:
2357:
2352:
2344:
2343:
2342:
2340:
2339:spin clusters
2335:
2321:
2297:
2293:
2289:
2285:
2281:
2278:
2275:
2269:
2266:
2263:
2259:
2254:
2251:
2228:
2225:
2222:
2214:
2211:
2205:
2202:
2199:
2192:
2183:
2159:
2144:
2135:
2120:
2114:
2110:
2106:
2101:
2097:
2089:
2088:
2087:
2070:
2064:
2061:
2054:
2050:
2045:
2041:
2038:
2023:
2019:
2015:
2010:
2006:
1999:
1996:
1993:
1990:
1987:
1977:
1973:
1969:
1964:
1960:
1953:
1948:
1944:
1939:
1931:
1930:
1929:
1906:
1902:
1898:
1893:
1889:
1875:
1872:
1869:
1856:
1853:
1828:
1824:
1796:
1792:
1788:
1784:
1773:
1769:
1761:
1757:
1752:
1748:
1738:
1724:
1716:
1712:
1708:
1705:
1695:
1693:
1677:
1674:
1671:
1663:
1659:
1655:
1639:
1636:
1633:
1625:
1620:
1606:
1603:
1600:
1580:
1577:
1574:
1571:
1568:
1543:
1538:
1535:
1529:
1526:
1523:
1520:
1517:
1497:
1494:
1491:
1471:
1468:
1460:
1445:
1443:
1424:
1421:
1417:
1414:
1405:
1402:
1398:
1394:
1387:
1384:
1380:
1377:
1369:
1366:
1362:
1341:
1321:
1298:
1290:
1286:
1264:
1261:
1240:
1220:
1200:
1176:
1173:
1169:
1166:
1158:
1155:
1151:
1127:
1119:
1115:
1106:
1102:
1096:
1092:
1088:
1080:
1076:
1072:
1067:
1063:
1054:
1051:
1047:
1041:
1038:
1035:
1031:
1027:
1024:
1017:
1016:
1015:
998:
995:
992:
989:
986:
980:
975:
971:
950:
942:
932:
916:
912:
906:
903:
898:
895:
890:
886:
865:
862:
859:
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833:
829:
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823:
818:
814:
805:
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786:
783:
774:
758:
754:
750:
745:
741:
732:
711:
707:
703:
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694:
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650:
645:
641:
634:
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620:
613:
607:
603:
599:
596:
591:
587:
579:
578:
577:
575:
565:
563:
541:
521:
518:
515:
512:
509:
501:
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475:
451:
448:
445:
418:
410:
406:
401:
397:
390:
386:
381:
376:
372:
369:
361:
358:
355:
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342:
338:
334:
329:
325:
317:
316:
315:
313:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
238:
233:
229:
226:
223:
217:
212:
208:
200:
199:
198:
196:
180:
171:
169:
165:
161:
146:
144:
143:morphogenesis
140:
136:
132:
131:James Glazier
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
79:
77:
76:Edward Teller
73:
72:Julius Ashkin
69:
65:
61:
57:
56:Renfrey Potts
52:
50:
46:
42:
38:
34:
30:
26:
22:
7372:Econometrics
7334:Wiener space
7222:ItĂŽ integral
7123:Inequalities
7012:Self-similar
6982:GaussâMarkov
6972:Exchangeable
6952:CĂ dlĂ g paths
6888:Risk process
6840:LIBOR market
6709:Random graph
6704:Random field
6659:
6516:Superprocess
6454:LĂ©vy process
6449:Jump process
6424:Hunt process
6260:Markov chain
6131:Applications
6082:size scaling
5973:
5783:
5779:
5769:
5750:
5746:
5736:
5695:
5691:
5685:
5650:math/0503607
5640:
5634:
5593:
5589:
5579:
5538:1912.11416v3
5528:
5524:
5493:
5450:
5446:
5436:
5401:
5397:
5323:
5319:
5309:
5276:
5272:
5262:
5211:
5207:
5169:
5165:
5155:
5114:
5110:
5100:
5075:
5071:
5061:
5006:
5001:
4997:
4989:
4988:to the data
4985:
4903:
4690:
4683:
4679:
4675:
4671:
4667:
4663:
4659:
4657:
4649:Applications
4638:
4631:Gibbs states
4624:
4619:
4617:
4554:
4470:
4393:
4387:
4373:
4369:
4365:
4363:
4272:fixed points
4265:
4136:
4055:
3947:
3943:
3936:
3932:
3928:
3926:
3914:
3821:
3743:is given by
3738:
3729:
3720:
3501:
3493:
3485:
3481:
3474:
3472:
3294:is given by
3289:
3279:
3275:
3269:
3266:
3176:
3172:
3165:
3163:
3158:
3151:
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3140:
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3124:
3120:
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3099:
3097:
3019:
3015:
3011:
3007:
3002:
2993:-adic number
2990:
2984:
2975:
2973:
2803:
2743:, acting as
2740:
2736:
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2571:
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2179:
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1739:
1701:
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1253:is in state
1142:
938:
775:
679:
573:
571:
499:
495:
434:
314:is given by
253:
197:, at angles
172:
159:
157:
115:grain growth
80:
67:
53:
41:ferromagnets
24:
18:
7481:Spin models
7417:Ruin theory
7355:Disciplines
7227:ItĂŽ's lemma
7002:Predictable
6677:Percolation
6660:Potts model
6655:Ising model
6619:White noise
6577:Differences
6439:ItĂŽ process
6379:Cox process
6275:Loop-erased
6270:Random walk
6122:von Neumann
5992:force field
5984:percolation
4390:Ising model
4268:cardinality
3498:temperature
2574:= {1, ...,
2540:Ising model
2536:Ernst Ising
2182:FK clusters
1654:percolation
1233:while spin
804:Ising model
574:Potts model
500:clock model
312:Hamiltonian
117:in metals,
107:confinement
99:non-Abelian
60:clock model
29:Ising model
25:Potts model
7475:Categories
7427:Statistics
7207:Filtration
7108:Kolmogorov
7092:Blumenthal
7017:Stationary
6957:Continuous
6945:Properties
6830:HullâWhite
6572:Martingale
6459:Local time
6347:Fractional
6325:pure birth
5979:Heisenberg
5506:1611.09877
5460:1505.04159
5333:1803.08823
5221:1905.11848
5053:References
4058:full shift
2725:full shift
2180:where the
149:Definition
119:coarsening
64:Cyril Domb
7339:Classical
6352:Geometric
6342:Excursion
6102:Boltzmann
6025:H-theorem
5903:Ensembles
5800:1939-3539
5760:1210.5644
5728:117951377
5712:1061-8600
5626:121502987
5618:1431-584X
5571:209460838
5485:119153736
5477:1432-0916
5420:1432-2064
5358:0370-1573
5326:: 1â124.
5254:167217593
5147:122689941
5139:1469-8064
5042:Z N model
4961:‖
4954:−
4948:‖
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4915:∇
4912:‖
4867:−
4832:∑
4806:≠
4784:#
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4757:−
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4726:γ
4709:γ
4641:â â, the
4531:σ
4527:σ
4514:β
4506:
4491:σ
4487:σ
4425:−
4416:σ
4331:τ
4312:∈
4294:τ
4232:β
4226:−
4204:τ
4185:β
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4104:β
4098:−
4033:∈
4017:…
4000:∑
3994:β
3988:−
3881:
3863:∞
3860:→
3788:
3776:−
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3629:β
3626:−
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3564:…
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3436:…
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3361:∈
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3328:∑
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3054:δ
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2696:∈
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2468:δ
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2212:∈
2193:∪
2160:ω
2149:#
2136:ω
2125:#
2115:ω
2111:∑
2062:−
2000:δ
1954:δ
1854:ω
1789:−
1762:∑
1704:Kasteleyn
1675:≥
1637:≤
1578:≤
1572:≤
1530:
1518:β
1495:≥
1093:∑
1032:∑
993:…
981:∈
899:−
827:−
688:δ
635:δ
614:∑
600:−
548:∞
545:→
455:⟩
443:⟨
402:θ
398:−
382:θ
373:
365:⟩
353:⟨
349:∑
295:−
227:π
209:θ
168:Euclidean
95:Kac model
7460:Category
7344:Abstract
6878:BĂŒhlmann
6484:Compound
6112:Tsallis
5720:27594299
5677:17904893
5563:32688489
5428:55391558
5376:31404441
5301:10046374
5246:31639992
5011:See also
5000:and the
4534:′
4494:′
3484:, where
3171: :
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2982:, ..., Ο
2558:and the
2552:XY model
2153:clusters
1418:′
1388:′
1354:. Note:
1265:′
1177:′
963:states:
562:XY model
127:proteins
89:and the
83:XY model
70:, after
6967:Ergodic
6855:VaĆĄĂÄek
6697:Poisson
6357:Meander
6107:Shannon
6094:Entropy
5598:Bibcode
5543:Bibcode
5367:6688775
5338:Bibcode
5281:Bibcode
5226:Bibcode
5174:Bibcode
5119:Bibcode
5080:Bibcode
4376:is the
4372:matrix
3506:measure
3496:is the
3488:is the
1692:wetting
729:is the
498:or the
164:lattice
7307:Tanaka
6992:Mixing
6987:Markov
6860:Wilkie
6825:HoâLee
6820:Heston
6592:Super-
6337:Bridge
6285:Biased
5956:Models
5864:Theory
5798:
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1278:, and
1143:where
852:. The
680:where
254:where
195:circle
85:, the
23:, the
7160:Tools
6936:M/M/c
6931:M/M/1
6926:M/G/1
6916:Fluid
6582:Local
6165:chaos
6117:RĂ©nyi
5974:Potts
5969:Ising
5755:arXiv
5724:S2CID
5716:JSTOR
5673:S2CID
5645:arXiv
5622:S2CID
5567:S2CID
5533:arXiv
5501:arXiv
5481:S2CID
5455:arXiv
5424:S2CID
5328:arXiv
5250:S2CID
5216:arXiv
5143:S2CID
3473:with
2129:edges
1658:Tutte
160:spins
123:foams
35:on a
33:spins
7112:LĂ©vy
6911:Bulk
6795:Chen
6587:Sub-
6545:Both
6047:and
5796:ISSN
5708:ISSN
5663:ISBN
5614:ISSN
5559:PMID
5473:ISSN
5416:ISSN
5372:PMID
5354:ISSN
5297:PMID
5242:PMID
5135:ISSN
4364:The
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3150:and
3112:and
2810:base
2570:Let
1660:and
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776:The
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6692:Cox
5788:doi
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4674:in
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4633:or
4503:exp
4287:Fix
4197:Fix
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3402:0
3398:C
3394:(
3389:n
3385:H
3375:(
3364:Q
3356:n
3352:s
3348:,
3342:,
3337:0
3333:s
3324:=
3321:)
3318:V
3315:(
3310:n
3306:Z
3276:n
3270:n
3252:)
3249:s
3244:k
3236:(
3233:V
3228:n
3223:0
3220:=
3217:k
3209:=
3206:)
3203:s
3200:(
3195:n
3191:H
3177:R
3173:Q
3168:n
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3141:V
3137:Q
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3121:V
3117:2
3114:s
3110:1
3107:s
3103:0
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3083:)
3078:1
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3051:J
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3039:s
3036:(
3033:V
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2980:0
2976:k
2959:}
2954:k
2946:=
2941:k
2938:+
2935:m
2931:s
2927:,
2921:,
2916:0
2908:=
2903:m
2899:s
2895::
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2871:=
2868:]
2863:k
2855:,
2849:,
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2836:[
2831:m
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2773:=
2768:k
2764:)
2760:s
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2741:Q
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2721:Q
2704:}
2700:Z
2693:k
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2024:j
2020:s
2016:,
2011:i
2007:s
2003:(
1997:v
1994:+
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1988:=
1983:)
1978:j
1974:s
1970:,
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1914:}
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1862:{
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1834:}
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218:=
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