Random convex hulls and extreme value statistics. (English) Zbl 1188.82024
Summary: We study the statistical properties of convex hulls of \(N\) random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy’s formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of \(n\) independent random walks. In the continuum time limit this reduces to \(n\) independent planar Brownian trajectories for which we compute exactly, for all \(n\), the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [the authors, Convex hull of \(N\) planar Brownian motions: exact results and an application to ecology, Phys. Rev. Lett. 103, 140602 (2009)].
MSC:
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 60J65 | Brownian motion |
Software:
ismevReferences:
| [1] | Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964) · Zbl 0171.38503 |
| [2] | Acedo, L., Yuste, S.B.: Multiparticle random walks. Recent Res. Dev. Stat. Phys. 2, 83–106 (2002) |
| [3] | Affentranger, F.: The expected volume of a random polytope in a ball. J. Microsc. 151, 277–287 (1988) |
| [4] | Agrawal, H., Dhar, D.: Probability distribution of the sizes of largest erased-loops in loop-erased random walks. Phys. Rev. E 65, 031108 (2002) · doi:10.1103/PhysRevE.65.031108 |
| [5] | Akl, S., Toussaint, G.: Efficient convex hull algorithms for pattern recognition applications. In: International Conference on Pattern Recognition, pp. 483–487 (1978) |
| [6] | Aldous, D., Fristedt, B., Griffin, P.S., Pruitt, W.E.: The number of extreme points in the convex hull of a random sample. J. Appl. Probab. 28, 287–304 (1991) · Zbl 0742.60012 · doi:10.2307/3214867 |
| [7] | Ayari, S., Dubuc, S.: La formule de Cauchy sur la longueur d’une courbe. Can. Math. Bull. 40, 3–9 (1997) · Zbl 0906.51005 · doi:10.4153/CMB-1997-001-5 |
| [8] | Bárány, I.: Stochastic Geometry. C.I.M.E., Lecture Notes Math., vol. 1892. Springer, Berlin (2006) |
| [9] | Bárány, I., Reitzner, M.: Random polytopes. Preprint (2008). www.renyi.hu/\(\sim\)barany/cikkek/clt-pol2.pdf |
| [10] | Bárány, I., Vu, V.: Central limit theorems for Gaussian polytopes. Ann. Probab. 35(4), 1593–1621 (2007) · Zbl 1124.60014 · doi:10.1214/009117906000000791 |
| [11] | Barbier, E.: Note sur le problème de l’aiguille et le jeu du joint couvert. J. Math. Pures Appl. 5, 273–286 (1860) |
| [12] | Bartumeus, F., da Luz, M., Viswanathan, G., Catalan, J.: Animal search strategies: a quantitative random-walk analysis. Ecology (2005) |
| [13] | Baxter, G.: A combinatorial lemma for complex numbers. Ann. Math. Stat. 32(3), 901 (1961) · Zbl 0119.14201 · doi:10.1214/aoms/1177704985 |
| [14] | Ben Naim, E., Krapivsky, P.L., Majumdar, S.N.: Extremal properties of random trees. Phys. Rev. E 64, R035101 (2001) |
| [15] | Bena, I., Majumdar, S.N.: Universal extremal statistics in a freely expanding jepsen gas. Phys. Rev. E 75, 051103 (2007) · doi:10.1103/PhysRevE.75.051103 |
| [16] | Berg, H.: Random Walks in Biology. Princeton University Press, New York (1983) |
| [17] | Bhattacharya, B., Sen, S.: On a simple, practical, optimal, output-sensitive randomized planar convex hull algorithm. J. Algorithms 25, 173–193 (1997) · Zbl 0895.68060 |
| [18] | Biane, P., Letac, G.: The mean perimeter of some random plane convex sets generated by a Brownian motion. Preprint (2009). arXiv:0905.2256 · Zbl 1221.60115 |
| [19] | Biroli, G., Bouchaud, J.P., Potters, M.: Extreme value problems in random matrix theory and other disordered systems. JSTAT (P07019) (2007) · Zbl 1244.82029 |
| [20] | Bouchaud, J.P., Mézard, M.: Universality classes for extreme value statistics. J. Phys. A., Math. Gen. 30, 7997–8015 (1997) · Zbl 0953.60040 · doi:10.1088/0305-4470/30/23/004 |
| [21] | Boyle, S., Lourenço, W., da Silva, L., Smith, A.: Home-range estimates vary with sample size and methods. Folia Primatol. 80, 33–42 (2009) · doi:10.1159/000201092 |
| [22] | Bräker, H., Hsing, T.: On the area and perimeter of a random convex hull in a bounded convex set. Probab. Theory Relat. Fields, 517–550 (1998) · Zbl 0912.60029 |
| [23] | Bramson, M.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581 (1978) · doi:10.1002/cpa.3160310502 |
| [24] | Brown, B., Resnick, S.I.: Extreme values of independent stochastic processes. J. Appl. Probab. 14(4), 732–739 (1977) · Zbl 0384.60055 · doi:10.2307/3213346 |
| [25] | Brozius, H.: Convergence in mean of some characteristics of the convex hull. Adv. Appl. Probab. 21, 526–542 (1989) · Zbl 0681.60025 · doi:10.2307/1427634 |
| [26] | Brozius, H., de Haan, J.: On limiting laws for the convex hull of a sample. J. Appl. Probab. 24, 852 (1987) · Zbl 0653.60011 · doi:10.2307/3214210 |
| [27] | Brunet, E., Derrida, B.: Statistics at the tip of a branching random walk and the delay of traveling waves. Europhys. Lett. 87, 60010 (2009) · doi:10.1209/0295-5075/87/60010 |
| [28] | Buchta, C.: Zufallspolygone in konvexen Vielecken. J. Reine Angew. Math. 347, 212–220 (1983) · Zbl 0514.52005 |
| [29] | Buchta, C.: Zufällige Polyeder. Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1985) · Zbl 0582.52005 |
| [30] | Burdzy, K.: Brownian motion in cones. Ann. Probab. 13, 1006–1010 (1985) · Zbl 0574.60053 · doi:10.1214/aop/1176992922 |
| [31] | Burdzy, K., San Martin, J.: Curvature of the convex hull of planar Brownian motion near its minimum point. Stoch. Process. Appl. 33(1), 89–103 (1989) · Zbl 0696.60041 · doi:10.1016/0304-4149(89)90068-9 |
| [32] | Burkhardt, T.W., Gyorgi, G., Moloney, N.R., Racz, Z.: Extreme statistics for time series: Distribution of the maximum relative to the initial value. Phys. Rev. E 76, 041119 (2007) · doi:10.1103/PhysRevE.76.041119 |
| [33] | Cabo, J., Groeneboom, P.: Limit theorems for functionals of convex hulls. Probab. Theory Relat. Fields 100, 31–55 (1994) · Zbl 0808.60019 · doi:10.1007/BF01204952 |
| [34] | Calka, P., Schreiber, T.: Large deviation probabilities for the number of vertices of random polytopes in the ball. Adv. Appl. Probab. 38, 47–58 (2006) · Zbl 1099.60009 · doi:10.1239/aap/1143936139 |
| [35] | Carnal, H.: Die konvexe Hülle von n rotationssymmetrischverteilen Punkten. Z. Wahrscheinlichkeitstheor. 15, 168–176 (1970) · Zbl 0193.46602 · doi:10.1007/BF00531885 |
| [36] | Carpentier, D., LeDoussal, P.: Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models. Phys. Rev. E 63, 026110 (2001) · doi:10.1103/PhysRevE.63.026110 |
| [37] | Cauchy, A.: La rectification des courbes. Mémoire de l’Académie des Sciences (1832) |
| [38] | Chassaing, P., Marckert, J.F., Yor, M.: A stochastically quasi-optimal search algorithm for the maximum of the simple random walk. Ann. Appl. Probab. 13(4), 1264–1295 (2003) · Zbl 1084.68145 · doi:10.1214/aoap/1069786499 |
| [39] | Coffman, E.G., Flajolet, P., Flato, L., Hofro, M.: The maximum of random walk and its application to rectangle packing. Prob. Eng. Inf. Sci. 12, 373–386 (1998) · Zbl 0958.60050 · doi:10.1017/S0269964800005258 |
| [40] | Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer, London (2001) · Zbl 0980.62043 |
| [41] | Comtet, A., Majumdar, S.N.: Precise asymptotics of a random walker’s maximum. J. Stat. Mech. (P06013) (2005) |
| [42] | Cranston, M., Hsu, P., March, P.: Smoothness of the convex hull of planar Brownian motion. Ann. Probab. 17(1), 144 (1989) · Zbl 0678.60073 · doi:10.1214/aop/1176991500 |
| [43] | Crofton, M.: On the theory of local probability, applied to straight lines at random in a plane. Trans. R. Soc. 158, 181–199 (1868) · JFM 01.0075.05 · doi:10.1098/rstl.1868.0008 |
| [44] | Daniels, H.E., Skyrme, T.H.R.: The maximum of a random walk whose mean path has a maximum. Adv. Appl. Probab. 17(1), 85–99 (1985) · Zbl 0552.60067 · doi:10.2307/1427054 |
| [45] | Davis, R., Mulrow, E., Resnick, S.: The convex hull of a random sample in \(\mathbb{R}\) d . Commun. Stat. Stoch. Models 3(1), 1–27 (1987) · Zbl 0621.60014 · doi:10.1080/15326348708807044 |
| [46] | Dean, D.S., Majumdar, S.N.: Extreme-value statistics of hierarchically correlated variables, deviation from Gumbel statistics and anomalous persistence. Phys. Rev. E 64, 046121 (2001) |
| [47] | Dean, D.S., Majumdar, S.N.: Large deviations of extreme eigenvalues of random matrices. Phys. Rev. Lett. 97, 160201 (2006) · Zbl 1228.82035 · doi:10.1103/PhysRevLett.97.160201 |
| [48] | Dean, D.S., Majumdar, S.N.: Extreme value statistics of eigenvalues of Gaussian random matrices. Phys. Rev. E 77, 041108 (2008) · doi:10.1103/PhysRevE.77.041108 |
| [49] | Derrida, B.: Random-energy model–an exactly solvable model of disordered-systems. Phys. Rev. B 24, 2613–2626 (1981) · Zbl 1323.60134 · doi:10.1103/PhysRevB.24.2613 |
| [50] | Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and travelling waves. J. Stat. Phys. 51, 817–840 (1988) · Zbl 1036.82522 · doi:10.1007/BF01014886 |
| [51] | Devroye, L.: How to reduce the average complexity of convex hull finding algorithms. Comput. Math. Appl. 7, 299–308 (1981) · Zbl 0456.68080 · doi:10.1016/0898-1221(81)90059-6 |
| [52] | Dobrushin, R.L., Kotecká, R., Shlosman, S.: Wulff Construction: a Global Shape from Local Interaction. Translations of Mathematical Monographs. American Mathematical Society, Providence (1992) · Zbl 0917.60103 |
| [53] | Eddy, W.: A new convex hull algorithm for planar sets. ACM Trans. Math. Softw. 3(4), 398–403 (1977) · Zbl 0374.68036 · doi:10.1145/355759.355766 |
| [54] | Eddy, W.: The distribution of the convex hull of a Gaussian sample. J. Appl. Probab. 17, 686–695 (1980) · Zbl 0435.60046 · doi:10.2307/3212962 |
| [55] | Eddy, W., Gale, J.: The convex hull of a spherically symmetric sample. Adv. Appl. Probab. 13, 751–763 (1981) · Zbl 0477.60031 · doi:10.2307/1426971 |
| [56] | Edelstein-Keshet, L.: Mathematical Models in Biology. Random House, New York (1988) · Zbl 0674.92001 |
| [57] | Efron, B.: The convex hull of a random set of points. Biometrika 52(3–4), 331 (1965) · Zbl 0138.41301 |
| [58] | El Bachir, M.: L’enveloppe convexe du mouvement brownien. Ph.D. thesis, Université Paul Sabatier, Toulouse, France (1983) |
| [59] | Evans, M.R., Majumdar, S.N.: Condensation and extreme value statistics. JSTAT (P05004) (2008) |
| [60] | Evans, S.N.: On the Hausdorff dimension of Brownian cone points. Math. Proc. Camb. Philos. Soc. 98, 343–353 (1985) · Zbl 0583.60078 · doi:10.1017/S0305004100063519 |
| [61] | Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968) · Zbl 0155.23101 |
| [62] | Finch, S., Hueter, I.: Random convex hulls: a variance revisited. Adv. Appl. Probab. 36(4), 981–986 (1994) · Zbl 1068.60020 · doi:10.1239/aap/1103662954 |
| [63] | Fyodorov, Y.V., Bouchaud, J.P.: Freezing and extreme value statistics in a random energy model with logarithmically correlated potential. J. Phys. A., Math. Theor. 41, 372001 (2008) · Zbl 1214.82016 · doi:10.1088/1751-8113/41/37/372001 |
| [64] | Fyodorov, Y.V., LeDoussal, P., Rosso, A.: Statistical mechanics of logarithmic rem: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields. JSTAT (P10005) (2009) |
| [65] | Garcia-Garcia, R., Rosso, A., Schehr, G.: The longest excursion of fractional Brownian motion: numerical evidence of non-Markovian effects. arXiv:0911:1897 (2009) |
| [66] | Geffroy, J.: Contribution à la théorie des valeurs extrêmes. II. Publ. Inst. Stat. Univ. Paris 7(8), 3–65 (1959) · Zbl 0092.34901 |
| [67] | Geffroy, J.: Localisation asymptotique du polyèdre d’appui d’un échantillon Laplacien à k dimensions. Publ. Inst. Stat. Univ. Paris 10, 213–228 (1961) · Zbl 0102.35101 |
| [68] | Giuggioli, L., Abramson, G., Kenkre, V.M., Parmenter, R.R., Yates, T.L.: Theory of home range estimation from displacement measurements of animal populations. J. Theor. Biol. 240, 126–135 (2006) · doi:10.1016/j.jtbi.2005.09.002 |
| [69] | Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44(3), 423–453 (1943) · Zbl 0063.01643 · doi:10.2307/1968974 |
| [70] | Godrèche, C., Luck, J.M.: A record-driven growth process. JSTAT (P11006) (2008) · Zbl 0987.82008 |
| [71] | Godrèche, C., Majumdar, S.N., Schehr, G.: The longest excursion of stochastic processes in nonequilibrium systems. Phys. Rev. Lett. 102, 240602 (2009) |
| [72] | Goldman, A.: Le spectre de certaines mosaïques poissoniennes du plan et l’enveloppe convexe du pont brownien. Prob. Theor. Relat. Fields 105, 57–83 (1996) · Zbl 0858.35094 · doi:10.1007/BF01192071 |
| [73] | Goldman, A.: Sur une conjecture de D.G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. C. R. Acad. Sci. Paris 326, 233–237 (1998) · Zbl 0919.60028 |
| [74] | Graham, R.: An efficient algorithm for determining the convex hull of a finite planar set. Inf. Process. Lett. 1, 132–133 (1972) · Zbl 0236.68013 · doi:10.1016/0020-0190(72)90045-2 |
| [75] | Groeneboom, P.: Limit theorems for convex hulls. Probab. Theory Relat. Fields 79, 327–368 (1988) · Zbl 0635.60012 · doi:10.1007/BF00342231 |
| [76] | Gumbel, E.: Les valeurs extrêmes des distributions statistiques. Ann. Inst. Henri Poincaré 5(2), 115–158 (1935) · Zbl 0011.36102 |
| [77] | Gumbel, E.: Statistics of Extremes. Columbia University Press, New York (1958) · Zbl 0086.34401 |
| [78] | Gyorgi, G., Holdsworth, P.C.W., Portelli, B., Racz, Z.: Statistics of extremal intensities for Gaussian interfaces. Phys. Rev. E 68, 056116 (2003) |
| [79] | Gyorgi, G., Moloney, N.R., Ozogany, K., Racz, Z.: Maximal height statistics for 1/f {\(\alpha\)} signals. Phys. Rev. E 75, 021123 (2007) |
| [80] | Haushofer, J., Bake, C.I., Livingstone, M.S., Kanwisher, N.: Privileged coding of convex shapes in human object-selective cortex. J. Neurophysiol. 100, 753–762 (2008) · doi:10.1152/jn.90310.2008 |
| [81] | Hilhorst, H.J., Calka, P., Schehr, G.: Sylvester’s question and the random acceleration process. J. Stat. Mech., Theory Exp. (P10010) (2008) |
| [82] | Hsing, T.: On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Probab. 4, 478–493 (1994) · Zbl 0806.60004 · doi:10.1214/aoap/1177005069 |
| [83] | Jarvis, R.A.: On the identification of the convex hull of a finite set of points in the plane. Inf. Process. Lett. 2, 18–21 (1973) · Zbl 0256.68041 · doi:10.1016/0020-0190(73)90020-3 |
| [84] | Jewell, N., Romano, J.: Coverage problems and random convex hulls. J. Appl. Probab. 19, 546 (1982) · Zbl 0504.60014 · doi:10.2307/3213513 |
| [85] | Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) · Zbl 0969.15008 · doi:10.1007/s002200050027 |
| [86] | Kearney, M.J., Majumdar, S.N.: On the area under a continuous time Brownian motion till its first-passage time. J. Phys. A, Math. Gen. 38, 4097–4104 (2005) · Zbl 1086.82016 · doi:10.1088/0305-4470/38/19/004 |
| [87] | Khoshnevisan, D.: Moment inequalities for functionals of the Brownian convex hull. Ann. Probab. 20(2), 627 (1992) · Zbl 0755.60064 · doi:10.1214/aop/1176989794 |
| [88] | Kirkpatrick, D.G., Seidel, R.: The ultimate planar convex hull algorithm? SIAM J. Comput. 15(1), 287–299 (1986) · Zbl 0589.68035 · doi:10.1137/0215021 |
| [89] | Krapivsky, P.L., Majumdar, S.N.: Traveling waves, front selection, and exact nontrivial exponents in a random fragmentation problem. Phys. Rev. Lett. 85, 5492 (2000) · doi:10.1103/PhysRevLett.85.5492 |
| [90] | Krug, J.: Records in a changing world. JSTAT (P07001) (2007) |
| [91] | Krug, J., Jain, K.: Breaking records in a evolutionary race. Physica A 358, 1–9 (2005) · doi:10.1016/j.physa.2005.06.002 |
| [92] | Lakshminarayan, A., Tomsovic, S., Bohigas, O., Majumdar, S.N.: Extreme statistics of complex random and quantum chaotic states. Phys. Rev. Lett. 100, 044103 (2008) · doi:10.1103/PhysRevLett.100.044103 |
| [93] | Larralde, H., Trunfio, P., Havlin, S., Stanley, H.E., Weiss, G.H.: Number of distinct sites visited by n random walkers. Phys. Rev. A 45(10), 7128–7139 (1992) · doi:10.1103/PhysRevA.45.7128 |
| [94] | Larralde, H., Trunfio, P., Havlin, S., Stanley, H.E., Weiss, G.H.: Territory covered by N diffusing particles. Nature 355, 423–426 (1992) · doi:10.1038/355423a0 |
| [95] | Le Gall, J.F.: Mouvement brownien, cônes et processus stables. Probab. Theory Relat. Fields 76, 587–627 (1987) · Zbl 0611.60076 · doi:10.1007/BF00960076 |
| [96] | LeDoussal, P., Monthus, C.: Exact solutions for the statistics of extrema of some random 1d landscapes, applications to the equilibrium and the dynamics of the toy model. Physica A 317, 140–198 (2003) · Zbl 1001.82045 · doi:10.1016/S0378-4371(02)01317-1 |
| [97] | LeDoussal, P., Wiese, K.J.: Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records. Phys. Rev. E 79, 051105 (2009) |
| [98] | Letac, G.: Expected perimeter length. Am. Math. Mon. 85, 686 (1978) |
| [99] | Letac, G.: An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk. J. Theor. Prob. 6(2), 385 (1993) · Zbl 0777.60067 · doi:10.1007/BF01047580 |
| [100] | Lévy, P.: Sur certains processus stochastiques homogènes. Comput. Math. 7, 283–339 (1939) · JFM 65.1346.02 |
| [101] | Lévy, P.: Processus stochastiques et mouvement brownien. Gauthiers-Villars, Paris (1948) |
| [102] | Lévy, P.: Le caractère universel de la courbe du mouvement brownien et la loi du logarithme itéré. Circ. Mat. Palermo Ser. 2.4, 337–366 (1955) · Zbl 0067.11201 |
| [103] | Majumdar, S.N.: Brownian functionals in physics and computer science. Curr. Sci. 89, 2075 (2005) · Zbl 1136.01013 |
| [104] | Majumdar, S.N., Bohigas, O., Lakshminarayan, A.: Exact minimum eigenvalue distribution of an entangled random pure state. J. Stat. Phys. 131, 33–49 (2008) · Zbl 1137.81007 · doi:10.1007/s10955-008-9491-5 |
| [105] | Majumdar, S.N., Comtet, A.: Exact maximal height distribution of fluctuating interfaces. Phys. Rev. Lett. 92, 225501 (2004) · doi:10.1103/PhysRevLett.92.225501 |
| [106] | Majumdar, S.N., Comtet, A.: Airy distribution function: from the area under a Brownian excursion to the maximal height of fluctuating interfaces. J. Stat. Phys. 119, 777–826 (2005) · Zbl 1170.82370 · doi:10.1007/s10955-005-3022-4 |
| [107] | Majumdar, S.N., Comtet, A., Ziff, R.M.: Unified solution of the expected maximum of a random walk and the discrete flux to a spherical trap. J. Stat. Phys. 122, 833–856 (2006) · Zbl 1092.82035 · doi:10.1007/s10955-005-9002-x |
| [108] | Majumdar, S.N., Krapivsky, P.L.: Extremal paths on a random Cayley tree. Phys. Rev. E. 62, 7735 (2000) · doi:10.1103/PhysRevE.62.7735 |
| [109] | Majumdar, S.N., Krapivsky, P.L.: Extreme value statistics and traveling fronts: an application to computer science. Phys. Rev. E 65, 036127 (2002) |
| [110] | Majumdar, S.N., Krapivsky, P.L.: Extreme value statistics and traveling fronts: various applications. Physica A, Stat. Mech. Appl. 318(1–2), 161–170 (2003) · Zbl 1009.62040 · doi:10.1016/S0378-4371(02)01422-X |
| [111] | Majumdar, S.N., Mallick, K., Sabhapandit, S.: Statistical properties of the final state in one-dimensional ballistic aggregation. Phys. Rev. E 79, 021109 (2009) · doi:10.1103/PhysRevE.79.021109 |
| [112] | Majumdar, S.N., Nechaev, S.K.: Exact asymptotic results for the Bernoulli matching model of sequence alignment. Phys. Rev. E 72, 020901 (2005) · doi:10.1103/PhysRevE.72.020901 |
| [113] | Majumdar, S.N., Randon-Furling, J., Kearney, M.J., Yor, M.: On the time to reach maximum for a variety of constrained Brownian motions. J. Phys. A, Math. Theor. 41, 365005 (2008) · Zbl 1195.82020 · doi:10.1088/1751-8113/41/36/365005 |
| [114] | Majumdar, S.N., Vergassola, M.: Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices. Phys. Rev. Lett. 102, 060601 (2009) · doi:10.1103/PhysRevLett.102.060601 |
| [115] | Majumdar, S.N., Ziff, R.M.: Universal record statistics of random walks and levy flights. Phys. Rev. Lett. 101, 050601 (2008) · Zbl 1228.82037 |
| [116] | Mayer, M., Molchanov, I.: Limit theorems for the diameter of a random sample in the unit ball. Extremes 10, 129–150 (2007) · Zbl 1164.62003 · doi:10.1007/s10687-007-0038-y |
| [117] | McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1976) · Zbl 0316.35053 · doi:10.1002/cpa.3160280302 |
| [118] | Meier, R., Ackermann, F., Herrmann, G., Posch, S., Sagerer, G.: Segmentation of molecular surfaces based on their convex hull. In: International Conference on Image Processing (ICIP’95), vol. 3, p. 3552 (1995) |
| [119] | Murphy, D., Noon, B.: Integrating scientific methods with habitat conservation planning: reserve design for northern spotted owls. Ecol. Appl. 2, 3–17 (1992) · doi:10.2307/1941885 |
| [120] | Nadal, C., Majumdar, S.N.: Non-intersecting Brownian interfaces and Wishart random matrices. Phys. Rev. E 79, 061117 (2009) · doi:10.1103/PhysRevE.79.061117 |
| [121] | Odlyzko, A.M.: Search for the maximum of a random walk. Random Struct. Algorithms 6, 275–295 (1995) · Zbl 0818.60065 · doi:10.1002/rsa.3240060215 |
| [122] | Preparata, F.P., Hong, S.J.: Convex hulls of finite sets of points in two and three dimensions. Commun. ACM 20(2), 87–93 (1977) · Zbl 0342.68030 · doi:10.1145/359423.359430 |
| [123] | Rambeau, J., Schehr, G.: Maximum relative height of one-dimensional interfaces: from Rayleigh to Airy distribution. JSTAT (P09004) (2009) |
| [124] | Randon-Furling, J., Majumdar, S.N.: Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time. J. Stat. Mech. (P10008) (2007) |
| [125] | Randon-Furling, J., Majumdar, S.N., Comtet, A.: Convex hull of N planar Brownian motions: exact results and an application to ecology. Phys. Rev. Lett. 103, 140602 (2009) · doi:10.1103/PhysRevLett.103.140602 |
| [126] | Raychaudhuri, S., Cranston, M., Przybyla, C., Shapir, Y.: Maximal height scaling of kinetically growing surfaces. Phys. Rev. Lett. 87, 136101 (2001) · doi:10.1103/PhysRevLett.87.136101 |
| [127] | Raynaud, H.: Sur le comportement asymptotique de l’enveloppe convexe d’un nuage de points tirés au hasard dans \(\mathbb{R}\) n . C. R. Acad. Sci. 261, 627–629 (1965) · Zbl 0138.12002 |
| [128] | Raynaud, H.: Sur l’enveloppe convexe des nuages de points aléatoires dans \(\mathbb{R}\) n . J. Appl. Probab. 7(1), 35–48 (1970) · Zbl 0192.53602 · doi:10.2307/3212146 |
| [129] | Reitzner, M.: Random polytopes and the Efron-Stein jackknife inequality. Ann. Probab. 31, 2136–2166 (2003) · Zbl 1058.60010 · doi:10.1214/aop/1068646381 |
| [130] | Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133, 483–507 (2005) · Zbl 1081.60008 · doi:10.1007/s00440-005-0441-8 |
| [131] | Reitzner, M.: The combinatorial structure of random polytopes. Adv. Math. 191, 178–208 (2005) · Zbl 1065.52004 · doi:10.1016/j.aim.2004.03.006 |
| [132] | Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. 2, 75–84 (1963) · Zbl 0118.13701 · doi:10.1007/BF00535300 |
| [133] | Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. 3, 138–147 (1964) · Zbl 0126.34103 · doi:10.1007/BF00535973 |
| [134] | Rossing, T.D. (ed.): Handbook of Acoustics. Springer, Berlin (2007) |
| [135] | Sabhapandit, S.: Statistical properties of a single-file diffusion front. J. Stat. Mech. (L05002) (2007) |
| [136] | Sabhapandit, S., Majumdar, S.N.: Density of near-extreme events. Phys. Rev. Lett. 98, 140201 (2007) · doi:10.1103/PhysRevLett.98.140201 |
| [137] | Sabhapandit, S., Majumdar, S.N., Redner, S.: Crowding at the front of marathon packs. J. Stat. Mech. (L03001) (2008) |
| [138] | Santaló, L.: Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications. Addison-Wesley, Reading (1976) · Zbl 0342.53049 |
| [139] | Schehr, G., LeDoussal, P.: Extreme value statistics from the real space renormalization group: Brownian motion, Bessel processes and continuous time random walks (2009). arXiv:0910:4913 |
| [140] | Schehr, G., Majumdar, S.N.: Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models. Phys. Rev. E 73, 056103 (2006) · doi:10.1103/PhysRevE.73.056103 |
| [141] | Schehr, G., Majumdar, S.N., Comtet, A., Randon-Furling, J.: Exact distribution of the maximal height of p vicious walkers. Phys. Rev. Lett. 101, 150601 (2008) · Zbl 1228.82038 · doi:10.1103/PhysRevLett.101.150601 |
| [142] | Schneider, R.: Random approximation of convex sets. J. Microsc. 151, 211 (1988) · Zbl 1256.52004 |
| [143] | Seidel, R.: Convex hull computations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 361–375. CRC Press, Boca Raton (1997) · Zbl 0907.68189 |
| [144] | Shimura, M.: A limit theorem for conditional random walk. Nagoya Math. J. 95, 105–116 (1984) · Zbl 0561.60037 |
| [145] | Shimura, M.: Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. 13, 433–443 (1985) · Zbl 0582.60048 |
| [146] | Sirakov, N.M.: A new active convex hull model for image regions. J. Math. Imaging Vis. 26(3), 309–325 (2006) · Zbl 1478.94085 · doi:10.1007/s10851-006-9004-6 |
| [147] | Sire, C.: Probability distribution of the maximum of a smooth temporal signal. Phys. Rev. Lett. 98, 020601 (2007) · doi:10.1103/PhysRevLett.98.020601 |
| [148] | Sire, C.: Contest based on a directed polymer in a random medium. Phys. Rev. E 78, 061106 (2008) |
| [149] | Sire, C., Majumdar, S.N., Dean, D.S.: Exact solution of a model of time-dependent evolutionary dynamics in a rugged fitness landscape. JSTAT (L07001) (2006) |
| [150] | Snyder, T., Steele, J.: Convex hulls of random walks. Proc. Am. Math. Soc. 117(4), 1165 (1993) · Zbl 0770.60011 · doi:10.1090/S0002-9939-1993-1169048-2 |
| [151] | Spitzer, F., Widom, H.: The circumference of a convex polygon. Proc. Am. Math. Soc. 12, 506–509 (1961) · Zbl 0099.38501 · doi:10.1090/S0002-9939-1961-0130616-7 |
| [152] | Takács, L.: Expected perimeter length. Am. Math. Mon. 87, 142 (1980) · doi:10.2307/2322010 |
| [153] | Toussaint, G.: A historical note on convex hull finding algorithms. Pattern Recognit. Lett. 3, 21–28 (1985) · doi:10.1016/0167-8655(85)90038-8 |
| [154] | Tracy, C., Widom, H.: Level-spacing distributions and the airy kernel. Commun. Math. Phys. 159, 151–174 (1994) · Zbl 0789.35152 · doi:10.1007/BF02100489 |
| [155] | Valentine, F.: Convex Sets. McGraw-Hill, New York (1964) · Zbl 0129.37203 |
| [156] | Vivo, P., Majumdar, S.N., Bohigas, O.: Large deviations of the maximum eigenvalue in Wishart random matrices. J. Phys. A, Math. Theor. 40, 4317–4337 (2007) · Zbl 1115.15019 · doi:10.1088/1751-8113/40/16/005 |
| [157] | Vu, V.: Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207, 221–243 (2006) · Zbl 1111.52010 · doi:10.1016/j.aim.2005.11.011 |
| [158] | Weil, W., Wieacker, J.A.: Handbook of Convex Geometry, vol. B, pp. 1391–1438. North Holland, Amsterdam (1993) · Zbl 0788.52002 |
| [159] | Wenger, R.: Randomized quick hull. Algorithmica 17, 322–329 (1997) · Zbl 0865.68123 · doi:10.1007/BF02523195 |
| [160] | Worton, B.J.: A convex hull-based estimator of home-range size. Biometrics 51(4), 1206–1215 (1995) · Zbl 0875.62501 · doi:10.2307/2533254 |
| [161] | Yaacoub, F., Hamam, Y., Abche, A., Fares, C.: Convex hull in medical simulations: a new hybrid approach. In: 32nd Annual Conference of IEEE Industrial Electronics Society, IECON’06, pp. 3308–3313 (2006) |
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