1000:. One of the main limitations of these networks is energy consumption where usually each node has a battery and an embedded form of energy harvesting. To reduce energy consumption in sensor networks, various sleep schemes have been suggested that entail having a subcollection of nodes go into a low energy-consuming sleep mode. These sleep schemes obviously affect the coverage and connectivity of sensor networks. Simple power-saving models have been proposed such as the simple uncoordinated 'blinking' model where (at each time interval) each node independently powers down (or up) with some fixed probability. Using the tools of percolation theory, a blinking Boolean Poisson model has been analyzed to study the
289:
969:
49:. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of
408:
focus of percolation theory is establishing the conditions when giant components exist in models, which has parallels with the study of random networks. If no big component exists, the model is said to be subcritical. The conditions of giant component criticality naturally depend on parameters of the model such as the density of the underlying point process.
407:
In the
BooleanâPoisson model, disks there can be isolated groups or clumps of disks that do not contact any other clumps of disks. These clumps are known as components. If the area (or volume in higher dimensions) of a component is infinite, one says it is an infinite or "giant" component. A major
88:
proposed a mathematical model in wireless networks that gave rise to the field of continuum percolation theory, thus generalizing discrete percolation. The underlying points of this model, sometimes known as the
Gilbert disk model, were scattered uniformly in the infinite plane
689:
416:
The excluded area of a placed object is defined as the minimal area around the object into which an additional object cannot be placed without overlapping with the first object. For example, in a system of randomly oriented homogeneous rectangles of length
885:
It has been shown via Monte-Carlo simulations that percolation threshold in both homogeneous and heterogeneous systems of rectangles or ellipses is dominated by the average excluded areas and can be approximated fairly well by the linear relation
45:). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of
955:
806:
68:
and performance in wireless networks. In addition to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study of
880:
461:
992:
Wireless networks are sometimes best represented with stochastic models owing to their complexity and unpredictability, hence continuum percolation have been used to develop
1518:
64:, which, with the rise of several wireless network technologies in recent years, has been generalized and studied in order to determine the theoretical bounds of
2053:
99:. Gilbert, who had noticed similarities between discrete and continuum percolation, then used concepts and techniques from the probability subject of
1877:
1013:
993:
2480:
1473:
115:
The exact names, terminology, and definitions of these models may vary slightly depending on the source, which is also reflected in the use of
2010:
1990:
996:. For example, the tools of continuous percolation theory and coverage processes have been used to study the coverage and connectivity of
892:
2394:
346:
to all other shapes and the underlying (Poisson) point process. This model is known as the germâgrain model where the underlying points
1281:
Balister, Paul; Sarkar, Amites; BollobĂĄs, BĂ©la (2008). "Percolation, connectivity, coverage and colouring of random geometric graphs".
2311:
1995:
2321:
2005:
1426:
Dousse, O.; Mannersalo, P.; Thiran, P. (2004). "Latency of wireless sensor networks with uncoordinated power saving mechanisms".
2363:
2260:
292:
Simulation of 4 PoissonâBoolean (constant-radius or
Gilbert disk) models as the density increases with largest clusters in red.
2550:
2540:
2386:
2078:
2063:
2450:
2414:
744:
2718:
2455:
835:
2367:
1565:
1466:
1023:
271:
2520:
343:
222:
1441:
Gui, C.; Mohapatra, P. (2004). "Power conservation and quality of surveillance in target tracking sensor networks".
2565:
2371:
2355:
2270:
2098:
2068:
1490:
1314:
Li, Jiantong; Ăstling, Mikael (2013). "Percolation thresholds of two-dimensional continuum systems of rectangles".
684:{\displaystyle A_{r}=2lw\left(1+{\frac {4}{\pi ^{2}}}\right)+{\frac {2}{\pi }}\left(l^{2}+w^{2}\right)=2l^{2}\left}
2470:
2435:
2404:
2399:
1835:
1752:
2409:
2038:
2033:
1840:
1737:
2723:
2500:
2336:
2235:
2220:
1759:
1632:
1548:
1459:
2495:
2375:
2505:
1115:
Dousse, O.; Baccelli, F.; Thiran, P. (2005). "Impact of interferences on connectivity in ad hoc networks".
2510:
2146:
2749:
2744:
2108:
1692:
1637:
1553:
376:
of all the shapes forms a
Boolean germ-grain model. Typical choices for the grains include disks, random
2440:
2754:
2445:
2430:
2073:
2043:
1610:
1508:
2525:
2326:
2240:
2225:
2156:
1732:
1615:
1513:
127:
A number of well-studied models exist in continuum percolation, which are often based on homogeneous
2359:
2245:
1747:
1722:
1667:
339:
218:
1378:"Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses"
1129:
2660:
2650:
2465:
2341:
2123:
2048:
1862:
1727:
1583:
1538:
54:
2602:
2530:
1955:
1945:
1789:
1124:
1028:
977:
116:
61:
301:
The disk model can be generalized to more arbitrary shapes where, instead of a disk, a random
2625:
2607:
2587:
2582:
2301:
2133:
2113:
1960:
1903:
1742:
1652:
1001:
128:
104:
1428:
Proceedings of the 5th ACM International
Symposium on Mobile Ad Hoc Networking and Computing
2700:
2655:
2645:
2331:
2306:
2275:
2255:
2093:
2015:
2000:
1867:
1389:
1323:
1202:
384:
65:
1443:
Proceedings of the 10th Annual
International Conference on Mobile Computing and Networking
1377:
8:
2695:
2535:
2460:
2265:
2025:
1935:
1825:
811:
The excluded area theory states that the critical number density (percolation threshold)
275:
100:
1393:
1327:
1206:
2665:
2630:
2545:
2515:
2346:
2285:
2280:
2103:
1940:
1605:
1543:
1482:
1355:
1142:
976:
The applications of percolation theory are various and range from material sciences to
29:
21:
2685:
1898:
1815:
1784:
1677:
1657:
1647:
1503:
1498:
1405:
1347:
1339:
2490:
2141:
1359:
2705:
2592:
2475:
2351:
2088:
1845:
1820:
1769:
1620:
1573:
1397:
1331:
1241:
1210:
1173:
1146:
1134:
1097:
981:
285:, then the resulting model is sometimes known as the Gilbert disk (Boolean) model.
33:
1697:
1072:
Random
Networks for Communication: From Statistical Physics to Information Systems
2670:
2570:
2555:
2316:
2250:
1928:
1872:
1855:
1600:
96:
69:
37:
2485:
1717:
1401:
2675:
2640:
2560:
2166:
1913:
1830:
1799:
1794:
1774:
1764:
1707:
1682:
1662:
1627:
1595:
1578:
1335:
997:
1702:
1214:
2738:
2577:
2118:
1950:
1908:
1850:
1672:
1588:
1528:
1409:
1343:
1246:
1229:
1178:
1161:
1138:
1018:
387:
known as coverage processes. The above models can be extended from the plane
302:
236:, then the resulting mathematical structure is known as a random disk model.
85:
73:
50:
46:
2635:
2597:
2151:
2083:
1972:
1967:
1779:
1712:
1687:
1523:
1351:
1160:
Dousse, O.; Franceschetti, M.; Macris, N.; Meester, R.; Thiran, P. (2006).
2680:
2215:
2199:
2194:
2189:
2179:
1982:
1923:
1918:
1882:
1642:
1533:
274:), which is a commonly studied model in continuum percolation as well as
17:
2690:
2230:
2174:
2058:
2011:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1451:
968:
76:, while becoming a subject of mathematical interest in its own right.
2184:
373:
288:
1193:
Balberg, I. (1987). "Recent developments in continuum percolation".
1101:
950:{\displaystyle N_{\mathrm {c} }-N_{\mathrm {c} 0}\propto A_{r}^{-1}}
820:
of a system is inversely proportional to the average excluded area
377:
60:
Continuum percolation arose from an early mathematical model for
1159:
1266:
Stoyan, D.; Kendall, W. S.; Mecke, J.; Ruschendorf, L. (1995).
270:
is known as a
BooleanâPoisson model (also known as simply the
1265:
1090:
Journal of the
Society for Industrial and Applied Mathematics
244:
Given a random disk model, if the set union of all the disks
1991:
Autoregressive conditional heteroskedasticity (ARCH) model
972:
A Boolean model as a coverage model in a wireless network.
1065:
1063:
1061:
278:. If all the radii are set to some common constant, say,
1519:
Independent and identically distributed random variables
1004:
and connectivity effects of such a simple power scheme.
980:
systems. Often the work involves showing that a type of
1162:"Percolation in the signal to interference ratio graph"
1996:
Autoregressive integrated moving average (ARIMA) model
1425:
1280:
1274:
1261:
1259:
1257:
1058:
1045:
1043:
960:
with a proportionality constant in the range 3.1â3.5.
1382:
Physica A: Statistical
Mechanics and Its Applications
895:
838:
747:
464:
225:
of all the other radii and all the underlying points
1421:
1419:
1289:
1153:
1114:
1069:
801:{\displaystyle A_{r}=2\pi ab+{\frac {C^{2}}{2\pi }}}
1254:
1083:
1081:
1040:
949:
875:{\displaystyle N_{\mathrm {c} }\propto A_{r}^{-1}}
874:
800:
683:
1416:
694:In a system of identical ellipses with semi-axes
28:is a branch of mathematics that extends discrete
2736:
1878:Stochastic chains with memory of variable length
1371:
1369:
1309:
1307:
1298:Introduction to the theory of coverage processes
1221:
1186:
1088:Gilbert, E. N. (1961). "Random plane networks".
1078:
1108:
1014:Stochastic geometry models of wireless networks
994:stochastic geometry models of wireless networks
107:existed for the infinite or "giant" component.
168:. For each point of the Poisson process (i.e.
110:
1467:
1366:
1304:
402:
1440:
1434:
1054:. Vol. 119. Cambridge University Press.
1074:. Vol. 24. Cambridge University Press.
2006:Autoregressiveâmoving-average (ARMA) model
1474:
1460:
738:, the average excluded areas is given by:
1375:
1313:
1245:
1177:
1128:
455:, the average excluded area is given by:
1481:
1268:Stochastic Geometry and Its Applications
967:
287:
156:that form a homogeneous Poisson process
1283:Handbook of Large-Scale Random Networks
1192:
1087:
1070:Franceschetti, M.; Meester, R. (2007).
1049:
411:
255:is taken, then the resulting structure
2737:
2312:Doob's martingale convergence theorems
1376:Li, Jiantong; Ăstling, Mikael (2016).
2064:Constant elasticity of variance (CEV)
2054:ChanâKarolyiâLongstaffâSanders (CKLS)
1455:
190:with its center located at the point
1295:
1227:
987:
383:Boolean models are also examples of
1117:IEEE/ACM Transactions on Networking
296:
13:
2551:Skorokhod's representation theorem
2332:Law of large numbers (weak/strong)
1024:Boolean model (probability theory)
917:
902:
845:
14:
2766:
2521:Martingale representation theorem
2566:Stochastic differential equation
2456:Doob's optional stopping theorem
2451:DoobâMeyer decomposition theorem
1270:. Vol. 2. Wiley Chichester.
239:
139:Consider a collection of points
122:
79:
2436:Convergence of random variables
2322:FisherâTippettâGnedenko theorem
1300:. Vol. 1. New York: Wiley.
963:
380:and segments of random length.
2034:Binomial options pricing model
1166:Journal of Applied Probability
361:and the random compact shapes
162:with constant (point) density
1:
2501:Kolmogorov continuity theorem
2337:Law of the iterated logarithm
1034:
305:(hence bounded and closed in
134:
2506:Kolmogorov extension theorem
2185:Generalized queueing network
1693:Interacting particle systems
26:continuum percolation theory
7:
1638:Continuous-time random walk
1402:10.1016/j.physa.2016.06.020
1007:
393:to general Euclidean space
111:Definitions and terminology
95:according to a homogeneous
10:
2771:
2646:Extreme value theory (EVT)
2446:Doob decomposition theorem
1738:OrnsteinâUhlenbeck process
1509:Chinese restaurant process
1336:10.1103/PhysRevE.88.012101
1230:"On continuum percolation"
403:Components and criticality
2714:
2618:
2526:Optional stopping theorem
2423:
2385:
2327:Large deviation principle
2294:
2208:
2165:
2132:
2079:HeathâJarrowâMorton (HJM)
2024:
2016:Moving-average (MA) model
2001:Autoregressive (AR) model
1981:
1891:
1826:Hidden Markov model (HMM)
1808:
1760:SchrammâLoewner evolution
1564:
1489:
1234:The Annals of Probability
1215:10.1080/13642818708215336
2441:Doléans-Dade exponential
2271:Progressively measurable
2069:CoxâIngersollâRoss (CIR)
1445:. ACM. pp. 129â143.
1430:. ACM. pp. 109â120.
1195:Philosophical Magazine B
1139:10.1109/tnet.2005.845546
320:is placed on each point
2661:Mathematical statistics
2651:Large deviations theory
2481:Infinitesimal generator
2342:Maximal ergodic theorem
2261:Piecewise-deterministic
1863:Random dynamical system
1728:Markov additive process
129:Poisson point processes
55:random geometric graphs
2496:KarhunenâLoĂšve theorem
2431:CameronâMartin formula
2395:BurkholderâDavisâGundy
1790:Variance gamma process
1247:10.1214/aop/1176992809
1179:10.1239/jap/1152413741
1029:Percolation thresholds
984:occurs in the system.
978:wireless communication
973:
951:
876:
802:
685:
293:
117:point process notation
2626:Actuarial mathematics
2588:Uniform integrability
2583:Stratonovich integral
2511:LĂ©vyâProkhorov metric
2415:MarcinkiewiczâZygmund
2302:Central limit theorem
1904:Gaussian random field
1733:McKeanâVlasov process
1653:Dyson Brownian motion
1514:GaltonâWatson process
1052:Continuum Percolation
971:
952:
877:
803:
686:
291:
2701:Time series analysis
2656:Mathematical finance
2541:Reflection principle
1868:Regenerative process
1668:FlemingâViot process
1483:Stochastic processes
1050:Meester, R. (1996).
893:
836:
745:
462:
412:Excluded area theory
385:stochastic processes
329:. Again, each shape
208:has a random radius
66:information capacity
2696:Stochastic analysis
2536:Quadratic variation
2531:Prokhorov's theorem
2466:FeynmanâKac formula
1936:Markov random field
1584:Birthâdeath process
1394:2016PhyA..462..940L
1328:2013PhRvE..88a2101L
1285:. pp. 117â142.
1207:1987PMagB..56..991B
946:
871:
276:stochastic geometry
101:branching processes
84:In the early 1960s
2750:Probability theory
2745:Percolation theory
2666:Probability theory
2546:Skorokhod integral
2516:Malliavin calculus
2099:Korn-Kreer-Lenssen
1983:Time series models
1946:PitmanâYor process
974:
947:
929:
872:
854:
798:
681:
294:
30:percolation theory
22:probability theory
2755:Phase transitions
2732:
2731:
2686:Signal processing
2405:Doob's upcrossing
2400:Doob's martingale
2364:EngelbertâSchmidt
2307:Donsker's theorem
2241:Feller-continuous
2109:RendlemanâBartter
1899:Dirichlet process
1816:Branching process
1785:Telegraph process
1678:Geometric process
1658:Empirical process
1648:Diffusion process
1504:Branching process
1499:Bernoulli process
1316:Physical Review E
1296:Hall, P. (1988).
1228:Hall, P. (1985).
988:Wireless networks
796:
669:
641:
623:
595:
531:
513:
429:and aspect ratio
62:wireless networks
2762:
2706:Machine learning
2593:Usual hypotheses
2476:Girsanov theorem
2461:Dynkin's formula
2226:Continuous paths
2134:Actuarial models
2074:GarmanâKohlhagen
2044:BlackâKarasinski
2039:BlackâDermanâToy
2026:Financial models
1892:Fields and other
1821:Gaussian process
1770:Sigma-martingale
1574:Additive process
1476:
1469:
1462:
1453:
1452:
1447:
1446:
1438:
1432:
1431:
1423:
1414:
1413:
1373:
1364:
1363:
1311:
1302:
1301:
1293:
1287:
1286:
1278:
1272:
1271:
1263:
1252:
1251:
1249:
1240:(4): 1250â1266.
1225:
1219:
1218:
1190:
1184:
1183:
1181:
1157:
1151:
1150:
1132:
1112:
1106:
1105:
1085:
1076:
1075:
1067:
1056:
1055:
1047:
982:phase transition
956:
954:
953:
948:
945:
937:
925:
924:
920:
907:
906:
905:
881:
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850:
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828:
819:
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799:
797:
795:
787:
786:
777:
757:
756:
737:
732:, and perimeter
731:
730:
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722:
719:
705:
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682:
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629:
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524:
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451:
450:
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428:
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398:
392:
367:
356:
337:
328:
319:
310:
297:Germ-grain model
284:
269:
254:
235:
216:
207:
198:
189:
181:), place a disk
180:
167:
161:
155:
149:
94:
44:
34:continuous space
2770:
2769:
2765:
2764:
2763:
2761:
2760:
2759:
2735:
2734:
2733:
2728:
2710:
2671:Queueing theory
2614:
2556:Skorokhod space
2419:
2410:KunitaâWatanabe
2381:
2347:Sanov's theorem
2317:Ergodic theorem
2290:
2286:Time-reversible
2204:
2167:Queueing models
2161:
2157:SparreâAnderson
2147:CramĂ©râLundberg
2128:
2114:SABR volatility
2020:
1977:
1929:Boolean network
1887:
1873:Renewal process
1804:
1753:Non-homogeneous
1743:Poisson process
1633:Contact process
1596:Brownian motion
1566:Continuous time
1560:
1554:Maximal entropy
1485:
1480:
1450:
1439:
1435:
1424:
1417:
1374:
1367:
1312:
1305:
1294:
1290:
1279:
1275:
1264:
1255:
1226:
1222:
1201:(6): 991â1003.
1191:
1187:
1158:
1154:
1113:
1109:
1102:10.1137/0109045
1086:
1079:
1068:
1059:
1048:
1041:
1037:
1010:
998:sensor networks
990:
966:
938:
933:
916:
915:
911:
901:
900:
896:
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891:
890:
863:
858:
844:
843:
839:
837:
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826:
821:
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707:
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695:
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659:
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582:
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523:
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469:
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430:
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405:
394:
388:
366:
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353:
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335:
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326:
321:
317:
312:
306:
299:
279:
267:
262:
256:
251:
245:
242:
232:
226:
217:(from a common
214:
209:
205:
200:
199:. If each disk
196:
191:
187:
182:
174:
169:
163:
157:
151:
146:
140:
137:
125:
113:
105:threshold value
103:to show that a
97:Poisson process
90:
82:
70:porous material
40:
38:Euclidean space
12:
11:
5:
2768:
2758:
2757:
2752:
2747:
2730:
2729:
2727:
2726:
2721:
2719:List of topics
2715:
2712:
2711:
2709:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2676:Renewal theory
2673:
2668:
2663:
2658:
2653:
2648:
2643:
2641:Ergodic theory
2638:
2633:
2631:Control theory
2628:
2622:
2620:
2616:
2615:
2613:
2612:
2611:
2610:
2605:
2595:
2590:
2585:
2580:
2575:
2574:
2573:
2563:
2561:Snell envelope
2558:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2427:
2425:
2421:
2420:
2418:
2417:
2412:
2407:
2402:
2397:
2391:
2389:
2383:
2382:
2380:
2379:
2360:BorelâCantelli
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2298:
2296:
2295:Limit theorems
2292:
2291:
2289:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2223:
2218:
2212:
2210:
2206:
2205:
2203:
2202:
2197:
2192:
2187:
2182:
2177:
2171:
2169:
2163:
2162:
2160:
2159:
2154:
2149:
2144:
2138:
2136:
2130:
2129:
2127:
2126:
2121:
2116:
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2030:
2028:
2022:
2021:
2019:
2018:
2013:
2008:
2003:
1998:
1993:
1987:
1985:
1979:
1978:
1976:
1975:
1970:
1965:
1964:
1963:
1958:
1948:
1943:
1938:
1933:
1932:
1931:
1926:
1916:
1914:Hopfield model
1911:
1906:
1901:
1895:
1893:
1889:
1888:
1886:
1885:
1880:
1875:
1870:
1865:
1860:
1859:
1858:
1853:
1848:
1843:
1833:
1831:Markov process
1828:
1823:
1818:
1812:
1810:
1806:
1805:
1803:
1802:
1800:Wiener sausage
1797:
1795:Wiener process
1792:
1787:
1782:
1777:
1775:Stable process
1772:
1767:
1765:Semimartingale
1762:
1757:
1756:
1755:
1750:
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1708:Jump diffusion
1705:
1700:
1695:
1690:
1685:
1683:Hawkes process
1680:
1675:
1670:
1665:
1663:Feller process
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1628:Cauchy process
1625:
1624:
1623:
1618:
1613:
1608:
1603:
1593:
1592:
1591:
1581:
1579:Bessel process
1576:
1570:
1568:
1562:
1561:
1559:
1558:
1557:
1556:
1551:
1546:
1541:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1495:
1493:
1487:
1486:
1479:
1478:
1471:
1464:
1456:
1449:
1448:
1433:
1415:
1365:
1303:
1288:
1273:
1253:
1220:
1185:
1172:(2): 552â562.
1152:
1123:(2): 425â436.
1107:
1096:(4): 533â543.
1077:
1057:
1038:
1036:
1033:
1032:
1031:
1026:
1021:
1016:
1009:
1006:
989:
986:
965:
962:
958:
957:
944:
941:
936:
932:
928:
923:
919:
914:
910:
904:
899:
883:
882:
869:
866:
861:
857:
853:
847:
842:
824:
816:
809:
808:
794:
791:
785:
781:
775:
772:
769:
766:
763:
760:
755:
751:
692:
691:
679:
674:
666:
662:
658:
653:
650:
646:
640:
637:
632:
628:
620:
616:
612:
607:
604:
600:
594:
591:
585:
579:
575:
571:
568:
564:
558:
554:
550:
545:
541:
536:
530:
527:
522:
518:
510:
506:
502:
497:
494:
490:
486:
483:
480:
477:
472:
468:
413:
410:
404:
401:
364:
351:
333:
324:
315:
298:
295:
265:
258:
249:
241:
238:
230:
212:
203:
194:
185:
172:
144:
136:
133:
124:
121:
112:
109:
81:
78:
74:semiconductors
9:
6:
4:
3:
2:
2767:
2756:
2753:
2751:
2748:
2746:
2743:
2742:
2740:
2725:
2722:
2720:
2717:
2716:
2713:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2623:
2621:
2617:
2609:
2606:
2604:
2601:
2600:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2578:Stopping time
2576:
2572:
2569:
2568:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2428:
2426:
2422:
2416:
2413:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2392:
2390:
2388:
2384:
2377:
2373:
2369:
2368:HewittâSavage
2365:
2361:
2357:
2353:
2352:Zeroâone laws
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2299:
2297:
2293:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2213:
2211:
2207:
2201:
2198:
2196:
2193:
2191:
2188:
2186:
2183:
2181:
2178:
2176:
2173:
2172:
2170:
2168:
2164:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2139:
2137:
2135:
2131:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2049:BlackâScholes
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2031:
2029:
2027:
2023:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1988:
1986:
1984:
1980:
1974:
1971:
1969:
1966:
1962:
1959:
1957:
1954:
1953:
1952:
1951:Point process
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1930:
1927:
1925:
1922:
1921:
1920:
1917:
1915:
1912:
1910:
1909:Gibbs measure
1907:
1905:
1902:
1900:
1897:
1896:
1894:
1890:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1838:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1813:
1811:
1807:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1754:
1751:
1749:
1746:
1745:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1698:ItĂŽ diffusion
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1673:Gamma process
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1598:
1597:
1594:
1590:
1587:
1586:
1585:
1582:
1580:
1577:
1575:
1572:
1571:
1569:
1567:
1563:
1555:
1552:
1550:
1547:
1545:
1544:Self-avoiding
1542:
1540:
1537:
1536:
1535:
1532:
1530:
1529:Moran process
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1496:
1494:
1492:
1491:Discrete time
1488:
1484:
1477:
1472:
1470:
1465:
1463:
1458:
1457:
1454:
1444:
1437:
1429:
1422:
1420:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1372:
1370:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1322:(1): 012101.
1321:
1317:
1310:
1308:
1299:
1292:
1284:
1277:
1269:
1262:
1260:
1258:
1248:
1243:
1239:
1235:
1231:
1224:
1216:
1212:
1208:
1204:
1200:
1196:
1189:
1180:
1175:
1171:
1167:
1163:
1156:
1148:
1144:
1140:
1136:
1131:
1130:10.1.1.5.3971
1126:
1122:
1118:
1111:
1103:
1099:
1095:
1091:
1084:
1082:
1073:
1066:
1064:
1062:
1053:
1046:
1044:
1039:
1030:
1027:
1025:
1022:
1020:
1019:Random graphs
1017:
1015:
1012:
1011:
1005:
1003:
999:
995:
985:
983:
979:
970:
961:
942:
939:
934:
930:
926:
921:
912:
908:
897:
889:
888:
887:
867:
864:
859:
855:
851:
840:
832:
831:
830:
827:
815:
792:
789:
783:
779:
773:
770:
767:
764:
761:
758:
753:
749:
741:
740:
739:
736:
726:
718:
710:
704:
698:
677:
672:
664:
660:
656:
651:
648:
644:
638:
635:
630:
626:
618:
614:
610:
605:
602:
598:
592:
589:
583:
577:
573:
569:
566:
562:
556:
552:
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543:
539:
534:
528:
525:
520:
516:
508:
504:
500:
495:
492:
488:
484:
481:
478:
475:
470:
466:
458:
457:
456:
449:
441:
433:
427:
421:
409:
400:
397:
391:
386:
381:
379:
375:
371:
360:
354:
345:
341:
338:has a common
336:
327:
318:
309:
304:
290:
286:
282:
277:
273:
272:Boolean model
268:
261:
252:
240:Boolean model
237:
233:
224:
220:
215:
206:
197:
188:
179:
175:
166:
160:
154:
150:in the plane
147:
132:
130:
123:Common models
120:
118:
108:
106:
102:
98:
93:
87:
86:Edgar Gilbert
80:Early history
77:
75:
71:
67:
63:
58:
56:
52:
51:random graphs
48:
47:point process
43:
39:
35:
31:
27:
23:
19:
2636:Econometrics
2598:Wiener space
2486:ItĂŽ integral
2387:Inequalities
2276:Self-similar
2246:GaussâMarkov
2236:Exchangeable
2216:CĂ dlĂ g paths
2152:Risk process
2104:LIBOR market
1973:Random graph
1968:Random field
1780:Superprocess
1718:LĂ©vy process
1713:Jump process
1688:Hunt process
1524:Markov chain
1442:
1436:
1427:
1385:
1381:
1319:
1315:
1297:
1291:
1282:
1276:
1267:
1237:
1233:
1223:
1198:
1194:
1188:
1169:
1165:
1155:
1120:
1116:
1110:
1093:
1089:
1071:
1051:
991:
975:
964:Applications
959:
884:
822:
813:
810:
734:
724:
716:
708:
702:
696:
693:
447:
439:
431:
425:
419:
415:
406:
395:
389:
382:
369:
358:
349:
340:distribution
331:
322:
313:
307:
300:
280:
263:
259:
247:
243:
228:
219:distribution
210:
201:
192:
183:
177:
170:
164:
158:
152:
142:
138:
126:
114:
91:
83:
59:
41:
25:
15:
2681:Ruin theory
2619:Disciplines
2491:ItĂŽ's lemma
2266:Predictable
1941:Percolation
1924:Potts model
1919:Ising model
1883:White noise
1841:Differences
1703:ItĂŽ process
1643:Cox process
1539:Loop-erased
1534:Random walk
1388:: 940â950.
344:independent
223:independent
18:mathematics
2739:Categories
2691:Statistics
2471:Filtration
2372:Kolmogorov
2356:Blumenthal
2281:Stationary
2221:Continuous
2209:Properties
2094:HullâWhite
1836:Martingale
1723:Local time
1611:Fractional
1589:pure birth
1035:References
706:and ratio
221:) that is
135:Disk model
2603:Classical
1616:Geometric
1606:Excursion
1410:0378-4371
1344:1539-3755
1125:CiteSeerX
940:−
927:∝
909:−
865:−
852:∝
793:π
765:π
639:π
615:π
529:π
505:π
374:set union
2724:Category
2608:Abstract
2142:BĂŒhlmann
1748:Compound
1360:21438506
1352:23944408
1008:See also
423:, width
368:are the
357:are the
311:) shape
2231:Ergodic
2119:VaĆĄĂÄek
1961:Poisson
1621:Meander
1390:Bibcode
1324:Bibcode
1203:Bibcode
1147:1514941
1002:latency
729:
713:
452:
436:
378:polygon
303:compact
36:(often
2571:Tanaka
2256:Mixing
2251:Markov
2124:Wilkie
2089:HoâLee
2084:Heston
1856:Super-
1601:Bridge
1549:Biased
1408:
1358:
1350:
1342:
1145:
1127:
372:. The
370:grains
283:> 0
2424:Tools
2200:M/M/c
2195:M/M/1
2190:M/G/1
2180:Fluid
1846:Local
1356:S2CID
1143:S2CID
359:germs
2376:LĂ©vy
2175:Bulk
2059:Chen
1851:Sub-
1809:Both
1406:ISSN
1348:PMID
1340:ISSN
1170:2006
700:and
342:and
72:and
53:and
20:and
1956:Cox
1398:doi
1386:462
1332:doi
1242:doi
1211:doi
1174:doi
1135:doi
1098:doi
32:to
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1080:^
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