On the spectral function of the Poisson-Voronoi cells
Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003) no. 6, p. 1057-1082
@article{AIHPB_2003__39_6_1057_0,
     author = {Goldman, Andr\'e and Calka, Pierre},
     title = {On the spectral function of the Poisson-Voronoi cells},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Elsevier},
     volume = {39},
     number = {6},
     year = {2003},
     pages = {1057-1082},
     doi = {10.1016/S0246-0203(03)00025-6},
     zbl = {1031.60009},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2003__39_6_1057_0}
}
Goldman, André; Calka, Pierre. On the spectral function of the Poisson-Voronoi cells. Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003) no. 6, pp. 1057-1082. doi : 10.1016/S0246-0203(03)00025-6. http://www.numdam.org/item/AIHPB_2003__39_6_1057_0/

[1] T.W. Anderson, The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6 (1955) 170-176. | MR 69229 | Zbl 0066.37402

[2] F. Baccelli, B. Błaszczyszyn, On a coverage process ranging from the Boolean model to the Poisson-Voronoi tessellation with applications to wireless communications, Adv. Appl. Probab. 33 (2) (2001) 293-323. | Zbl 0986.60010

[3] P. Calka, The explicit expression of the distribution of the number of sides of the typical Poisson-Voronoi cell, Adv. Appl. Probab. (2003), in press. | Zbl 1038.60008

[4] P. Calka, Precise formulas for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process, Adv. Appl. Probab. (2003), submitted for publication. | Zbl 1045.60005

[5] R. Cowan, S.N. Chiu, L. Holst, A limit theorem for the replication time of a DNA molecule, J. Appl. Probab. 32 (2) (1995) 296-303. | MR 1334888 | Zbl 0822.92008

[6] M.D. Donsker, S.R.S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (4) (1975) 525-565. | MR 397901 | Zbl 0333.60077

[7] B.V. Fedosov, Asymptotic formulae for eigenvalues of the Laplace operator for a polyhedron, Dokl. Akad. Nauk SSSR 157 (1964) 536-538. | MR 164129 | Zbl 0133.36101

[8] E.N. Gilbert, Random subdivisions of space into crystals, Ann. Math. Statist. 33 (1962) 958-972. | MR 144253 | Zbl 0242.60009

[9] A. Goldman, Le spectre de certaines mosaïques poissoniennes du plan et l'enveloppe convexe du pont brownien, Probab. Theory Related Fields 105 (1) (1996) 57-83. | MR 1389732 | Zbl 0858.35094

[10] A. Goldman, Sur une conjecture de D.G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne, Ann. Probab. 26 (4) (1998) 1727-1750. | MR 1675067 | Zbl 0936.60009

[11] A. Goldman, P. Calka, On the spectral function of the Johnson-Mehl and Voronoi tessellations, Preprint 00-02 of LaPCS, 2000.

[12] A. Goldman, P. Calka, Sur la fonction spectrale des cellules de Poisson-Voronoi, C. R. Acad. Sci. Paris Sér. I Math. 332 (9) (2001) 835-840. | Zbl 1008.60070

[13] W.A. Johnson, R.F. Mehl, Reaction kinetics in processes of nucleation and growth, Trans. Amer. Inst. Min. Engr. 135 (1939) 416-458.

[14] M. Kac, Can one hear the shape of a drum?, part II, Amer. Math. Monthly 73 (4) (1966) 1-23. | MR 201237 | Zbl 0139.05603

[15] I.N. Kovalenko, A simplified proof of a conjecture of D.G. Kendall concerning shapes of random polygons, J. Appl. Math. Stochastic Anal. 12 (4) (1999) 301-310. | MR 1736071 | Zbl 0959.60007

[16] S. Kumar, R.N. Singh, Thermal conductivity of polycristalline materials, J. Amer. Cer. Soc. 78 (3) (1995) 728-736.

[17] J.L. Meijering, Interface area, edge length, and number of vertices in crystal aggregates with random nucleation, Philips Res. Rep. 8 (1953). | Zbl 0053.33401

[18] J. Møller, Random Johnson-Mehl tessellations, Adv. Appl. Probab. 24 (4) (1992) 814-844. | Zbl 0768.60014

[19] J. Møller, Lectures on Random Voronoĭ Tessellations, Springer-Verlag, New York, 1994. | MR 1295245 | Zbl 0812.60016

[20] L. Muche, The Poisson Voronoi tessellation. III. Miles' formula, Math. Nachr. 191 (1998) 247-267. | MR 1621322 | Zbl 0906.60009

[21] A. Okabe, B. Boots, K. Sugihara, S.N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Chichester, 2000, With a foreword by D.G. Kendall. | MR 1770006 | Zbl 0877.52010

[22] E. Pielou, Mathematical Ecology, Wiley-Interscience, New York, 1977. | MR 434494 | Zbl 0259.92001

[23] M.H. Protter, Can one hear the shape of a drum? revisited, SIAM Rev. 29 (2) (1987) 185-197. | MR 889243 | Zbl 0645.35074

[24] R.T. Rockafellar, Convex Analysis, Princeton Math. Ser., 28, Princeton University Press, Princeton, NJ, 1970. | Zbl 0193.18401

[25] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, Wiley, Chichester, 1987, With a foreword by D.G. Kendall. | MR 895588 | Zbl 0838.60002

[26] D.W. Stroock, Probability Theory, an Analytic View, Cambridge University Press, Cambridge, 1993. | MR 1267569 | Zbl 0925.60004

[27] A.S. Sznitman, Brownian Motion, Obstacles and Random Media, Springer Monographs in Math., Springer-Verlag, Berlin, 1998. | MR 1717054 | Zbl 0973.60003

[28] R. Van De Weygaert, Fragmenting the Universe III. The construction and statistics of 3-D Voronoi tessellations, Astron. Astrophys. 283 (1994) 361-406. | MR 1290532

[29] M. Van Den Berg, S. Srisatkunarajah, Heat equation for a region in R2 with a polygonal boundary, J. London Math. Soc. (2) 37 (1) (1988) 119-127. | MR 921750 | Zbl 0609.35003

[30] H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Göttinger Nachr. (1911) 110-117. | JFM 42.0432.03