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Random Convex Hulls and Extreme Value Statistics

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Abstract

In this paper 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 [Phys. Rev. Lett. 103:140602, 2009].

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References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)

    MATH  Google Scholar 

  2. Acedo, L., Yuste, S.B.: Multiparticle random walks. Recent Res. Dev. Stat. Phys. 2, 83–106 (2002)

    Google Scholar 

  3. Affentranger, F.: The expected volume of a random polytope in a ball. J. Microsc. 151, 277–287 (1988)

    Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ayari, S., Dubuc, S.: La formule de Cauchy sur la longueur d’une courbe. Can. Math. Bull. 40, 3–9 (1997)

    MATH  MathSciNet  Google Scholar 

  8. Bárány, I.: Stochastic Geometry. C.I.M.E., Lecture Notes Math., vol. 1892. Springer, Berlin (2006)

    Google Scholar 

  9. Bárány, I., Reitzner, M.: Random polytopes. Preprint (2008). www.renyi.hu/~barany/cikkek/clt-pol2.pdf

  10. Bárány, I., Vu, V.: Central limit theorems for Gaussian polytopes. Ann. Probab. 35(4), 1593–1621 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Barbier, E.: Note sur le problème de l’aiguille et le jeu du joint couvert. J. Math. Pures Appl. 5, 273–286 (1860)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ben Naim, E., Krapivsky, P.L., Majumdar, S.N.: Extremal properties of random trees. Phys. Rev. E 64, R035101 (2001)

    Article  ADS  Google Scholar 

  15. Bena, I., Majumdar, S.N.: Universal extremal statistics in a freely expanding jepsen gas. Phys. Rev. E 75, 051103 (2007)

    ADS  Google Scholar 

  16. Berg, H.: Random Walks in Biology. Princeton University Press, New York (1983)

    Google Scholar 

  17. Bhattacharya, B., Sen, S.: On a simple, practical, optimal, output-sensitive randomized planar convex hull algorithm. J. Algorithms 25, 173–193 (1997)

    MathSciNet  Google Scholar 

  18. Biane, P., Letac, G.: The mean perimeter of some random plane convex sets generated by a Brownian motion. Preprint (2009). arXiv:0905.2256

  19. Biroli, G., Bouchaud, J.P., Potters, M.: Extreme value problems in random matrix theory and other disordered systems. JSTAT (P07019) (2007)

  20. Bouchaud, J.P., Mézard, M.: Universality classes for extreme value statistics. J. Phys. A., Math. Gen. 30, 7997–8015 (1997)

    Article  MATH  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  23. Bramson, M.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  24. Brown, B., Resnick, S.I.: Extreme values of independent stochastic processes. J. Appl. Probab. 14(4), 732–739 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  25. Brozius, H.: Convergence in mean of some characteristics of the convex hull. Adv. Appl. Probab. 21, 526–542 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  26. Brozius, H., de Haan, J.: On limiting laws for the convex hull of a sample. J. Appl. Probab. 24, 852 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  28. Buchta, C.: Zufallspolygone in konvexen Vielecken. J. Reine Angew. Math. 347, 212–220 (1983)

    MathSciNet  Google Scholar 

  29. Buchta, C.: Zufällige Polyeder. Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1985)

    Google Scholar 

  30. Burdzy, K.: Brownian motion in cones. Ann. Probab. 13, 1006–1010 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  ADS  Google Scholar 

  33. Cabo, J., Groeneboom, P.: Limit theorems for functionals of convex hulls. Probab. Theory Relat. Fields 100, 31–55 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  35. Carnal, H.: Die konvexe Hülle von n rotationssymmetrischverteilen Punkten. Z.  Wahrscheinlichkeitstheor. 15, 168–176 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  40. Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer, London (2001)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  45. Davis, R., Mulrow, E., Resnick, S.: The convex hull of a random sample in ℝd. Commun. Stat. Stoch. Models 3(1), 1–27 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    ADS  Google Scholar 

  47. Dean, D.S., Majumdar, S.N.: Large deviations of extreme eigenvalues of random matrices. Phys. Rev. Lett. 97, 160201 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  48. Dean, D.S., Majumdar, S.N.: Extreme value statistics of eigenvalues of Gaussian random matrices. Phys. Rev. E 77, 041108 (2008)

    MathSciNet  ADS  Google Scholar 

  49. Derrida, B.: Random-energy model—an exactly solvable model of disordered-systems. Phys. Rev. B 24, 2613–2626 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  50. Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and travelling waves. J. Stat. Phys. 51, 817–840 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  51. Devroye, L.: How to reduce the average complexity of convex hull finding algorithms. Comput. Math. Appl. 7, 299–308 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  53. Eddy, W.: A new convex hull algorithm for planar sets. ACM Trans. Math. Softw. 3(4), 398–403 (1977)

    Article  MATH  Google Scholar 

  54. Eddy, W.: The distribution of the convex hull of a Gaussian sample. J. Appl. Probab. 17, 686–695 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  55. Eddy, W., Gale, J.: The convex hull of a spherically symmetric sample. Adv. Appl. Probab. 13, 751–763 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  56. Edelstein-Keshet, L.: Mathematical Models in Biology. Random House, New York (1988)

    MATH  Google Scholar 

  57. Efron, B.: The convex hull of a random set of points. Biometrika 52(3–4), 331 (1965)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  61. Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968)

    MATH  Google Scholar 

  62. Finch, S., Hueter, I.: Random convex hulls: a variance revisited. Adv. Appl. Probab. 36(4), 981–986 (1994)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  69. Gnedenko, B.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44(3), 423–453 (1943)

    Article  MathSciNet  Google Scholar 

  70. Godrèche, C., Luck, J.M.: A record-driven growth process. JSTAT (P11006) (2008)

  71. Godrèche, C., Majumdar, S.N., Schehr, G.: The longest excursion of stochastic processes in nonequilibrium systems. Phys. Rev. Lett. 102, 240602 (2009)

    Article  ADS  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MATH  ADS  Google Scholar 

  74. Graham, R.: An efficient algorithm for determining the convex hull of a finite planar set. Inf. Process. Lett. 1, 132–133 (1972)

    Article  MATH  Google Scholar 

  75. Groeneboom, P.: Limit theorems for convex hulls. Probab. Theory Relat. Fields 79, 327–368 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  76. Gumbel, E.: Les valeurs extrêmes des distributions statistiques. Ann. Inst. Henri Poincaré 5(2), 115–158 (1935)

    MATH  MathSciNet  Google Scholar 

  77. Gumbel, E.: Statistics of Extremes. Columbia University Press, New York (1958)

    MATH  Google Scholar 

  78. Gyorgi, G., Holdsworth, P.C.W., Portelli, B., Racz, Z.: Statistics of extremal intensities for Gaussian interfaces. Phys. Rev. E 68, 056116 (2003)

    ADS  Google Scholar 

  79. Gyorgi, G., Moloney, N.R., Ozogany, K., Racz, Z.: Maximal height statistics for 1/f α signals. Phys. Rev. E 75, 021123 (2007)

    MathSciNet  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  84. Jewell, N., Romano, J.: Coverage problems and random convex hulls. J. Appl. Probab. 19, 546 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  85. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  87. Khoshnevisan, D.: Moment inequalities for functionals of the Brownian convex hull. Ann. Probab. 20(2), 627 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  88. Kirkpatrick, D.G., Seidel, R.: The ultimate planar convex hull algorithm? SIAM J. Comput. 15(1), 287–299 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  94. Larralde, H., Trunfio, P., Havlin, S., Stanley, H.E., Weiss, G.H.: Territory covered by N diffusing particles. Nature 355, 423–426 (1992)

    Article  ADS  Google Scholar 

  95. Le Gall, J.F.: Mouvement brownien, cônes et processus stables. Probab. Theory Relat. Fields 76, 587–627 (1987)

    Article  MATH  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    MathSciNet  ADS  Google Scholar 

  98. Letac, G.: Expected perimeter length. Am. Math. Mon. 85, 686 (1978)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  100. Lévy, P.: Sur certains processus stochastiques homogènes. Comput. Math. 7, 283–339 (1939)

    MATH  Google Scholar 

  101. Lévy, P.: Processus stochastiques et mouvement brownien. Gauthiers-Villars, Paris (1948)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  103. Majumdar, S.N.: Brownian functionals in physics and computer science. Curr. Sci. 89, 2075 (2005)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  105. Majumdar, S.N., Comtet, A.: Exact maximal height distribution of fluctuating interfaces. Phys. Rev. Lett. 92, 225501 (2004)

    Article  ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  108. Majumdar, S.N., Krapivsky, P.L.: Extremal paths on a random Cayley tree. Phys. Rev. E. 62, 7735 (2000)

    Article  ADS  Google Scholar 

  109. Majumdar, S.N., Krapivsky, P.L.: Extreme value statistics and traveling fronts: an application to computer science. Phys. Rev. E 65, 036127 (2002)

    ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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)

    MathSciNet  ADS  Google Scholar 

  112. Majumdar, S.N., Nechaev, S.K.: Exact asymptotic results for the Bernoulli matching model of sequence alignment. Phys. Rev. E 72, 020901 (2005)

    MathSciNet  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  114. Majumdar, S.N., Vergassola, M.: Large deviations of the maximum eigenvalue for Wishart and Gaussian random matrices. Phys. Rev. Lett. 102, 060601 (2009)

    Article  ADS  Google Scholar 

  115. Majumdar, S.N., Ziff, R.M.: Universal record statistics of random walks and levy flights. Phys. Rev. Lett. 101, 050601 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  116. Mayer, M., Molchanov, I.: Limit theorems for the diameter of a random sample in the unit ball. Extremes 10, 129–150 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  117. McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1976)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  120. Nadal, C., Majumdar, S.N.: Non-intersecting Brownian interfaces and Wishart random matrices. Phys. Rev. E 79, 061117 (2009)

    MathSciNet  ADS  Google Scholar 

  121. Odlyzko, A.M.: Search for the maximum of a random walk. Random Struct. Algorithms 6, 275–295 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  126. Raychaudhuri, S., Cranston, M., Przybyla, C., Shapir, Y.: Maximal height scaling of kinetically growing surfaces. Phys. Rev. Lett. 87, 136101 (2001)

    Article  ADS  Google Scholar 

  127. Raynaud, H.: Sur le comportement asymptotique de l’enveloppe convexe d’un nuage de points tirés au hasard dans ℝn. C. R. Acad. Sci. 261, 627–629 (1965)

    MATH  MathSciNet  Google Scholar 

  128. Raynaud, H.: Sur l’enveloppe convexe des nuages de points aléatoires dans ℝn. J. Appl. Probab. 7(1), 35–48 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  129. Reitzner, M.: Random polytopes and the Efron-Stein jackknife inequality. Ann. Probab. 31, 2136–2166 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  130. Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133, 483–507 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  131. Reitzner, M.: The combinatorial structure of random polytopes. Adv. Math. 191, 178–208 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  132. Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. 2, 75–84 (1963)

    Article  MATH  Google Scholar 

  133. Rényi, A., Sulanke, R.: Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheor. 3, 138–147 (1964)

    Article  MATH  Google Scholar 

  134. Rossing, T.D. (ed.): Handbook of Acoustics. Springer, Berlin (2007)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  142. Schneider, R.: Random approximation of convex sets. J. Microsc. 151, 211 (1988)

    Google Scholar 

  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)

    Google Scholar 

  144. Shimura, M.: A limit theorem for conditional random walk. Nagoya Math. J. 95, 105–116 (1984)

    MATH  MathSciNet  Google Scholar 

  145. Shimura, M.: Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. 13, 433–443 (1985)

    MathSciNet  Google Scholar 

  146. Sirakov, N.M.: A new active convex hull model for image regions. J. Math. Imaging Vis. 26(3), 309–325 (2006)

    Article  MathSciNet  Google Scholar 

  147. Sire, C.: Probability distribution of the maximum of a smooth temporal signal. Phys. Rev. Lett. 98, 020601 (2007)

    Article  ADS  Google Scholar 

  148. Sire, C.: Contest based on a directed polymer in a random medium. Phys. Rev. E 78, 061106 (2008)

    MathSciNet  ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  151. Spitzer, F., Widom, H.: The circumference of a convex polygon. Proc. Am. Math. Soc. 12, 506–509 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  152. Takács, L.: Expected perimeter length. Am. Math. Mon. 87, 142 (1980)

    Article  Google Scholar 

  153. Toussaint, G.: A historical note on convex hull finding algorithms. Pattern Recognit. Lett. 3, 21–28 (1985)

    Article  ADS  Google Scholar 

  154. Tracy, C., Widom, H.: Level-spacing distributions and the airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  155. Valentine, F.: Convex Sets. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  157. Vu, V.: Central limit theorems for random polytopes in a smooth convex set. Adv. Math. 207, 221–243 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  158. Weil, W., Wieacker, J.A.: Handbook of Convex Geometry, vol. B, pp. 1391–1438. North Holland, Amsterdam (1993)

    Google Scholar 

  159. Wenger, R.: Randomized quick hull. Algorithmica 17, 322–329 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  160. Worton, B.J.: A convex hull-based estimator of home-range size. Biometrics 51(4), 1206–1215 (1995)

    Article  MATH  Google Scholar 

  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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  1. Julien Randon-Furling

    Present address: AG Rieger, Theoretische Physik, Universität des Saarlandes, PF 151150, 66041, Saarbrücken, Germany

Authors and Affiliations

  1. LPTMS, Univ. Paris-Sud 11, UMR CNRS 8626, 15 rue Georges Clemenceau, 91405, Orsay Cedex, France

    Satya N. Majumdar, Alain Comtet & Julien Randon-Furling

  2. Université Pierre et Marie Curie-Paris 6, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Alain Comtet

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  1. Satya N. Majumdar
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Correspondence to Satya N. Majumdar.

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Majumdar, S.N., Comtet, A. & Randon-Furling, J. Random Convex Hulls and Extreme Value Statistics. J Stat Phys 138, 955–1009 (2010). https://doi.org/10.1007/s10955-009-9905-z

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