Comptes Rendus
Probabilités
Diffraction et mesure de Palm des processus ponctuels
Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 57-62.

En faisant appel à la notion de mesure de Palm, nous établissons l'existence de la mesure de diffraction pour tout processus ponctuel stationnaire et ergodique. Nous obtenons des caractérisations précises de ces mesures dans le cas de processus particuliers : sous-ensembles aléatoires de d , ensembles obtenus par la méthode « cut-and-project ».

Using the notion of Palm measure, we prove the existence of the diffraction measure of all stationary and ergodic point processes. We get precise expressions of those measures in the case of specific processes: stochastic subsets of d , sets obtained by the “cut-and-project” method.

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DOI : 10.1016/S1631-073X(02)00029-8

Jean-Baptiste Gouéré 1

1 LaPCS, Université Claude Bernard Lyon I, bâtiment recherche [B], 50, avenue Tony-Garnier, Domaine de Gerland, 69366 Lyon cedex 07, France
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Jean-Baptiste Gouéré. Diffraction et mesure de Palm des processus ponctuels. Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 57-62. doi : 10.1016/S1631-073X(02)00029-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00029-8/

[1] M. Baake, Diffraction of weighted lattice subsets, Preprint, 2002

[2] M. Baake; M. Höffe Diffraction of random tilings: some rigorous results, J. Statist. Phys., Volume 99 (2000) no. 1–2, pp. 216-261

[3] M. Baake; R.V. Moody Diffractive point sets with entropy, J. Phys. A, Volume 31 (1998) no. 45, pp. 9023-9039

[4] R. Burton; R. Pemantle Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab., Volume 21 (1993) no. 3, pp. 1329-1371

[5] I.P. Cornfeld; S.V. Fomin; Ya.G. Sinaı̆ Ergodic Theory, Springer-Verlag, New York, 1982 (Translated from Russian by A.B. Sosinskiı̆)

[6] M.E. Fisher; J. Stephenson Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers, Phys. Rev., Volume 132 (1963) no. 2, pp. 1411-1431

[7] A. Hof Diffraction by aperiodic structures at high temperatures, J. Phys. A, Volume 28 (1995) no. 1, pp. 57-62

[8] A. Hof On diffraction by aperiodic structures, Comm. Math. Phys., Volume 169 (1995) no. 1, pp. 25-43

[9] O. Kallenberg Random Measures, Akademie-Verlag, Berlin, 1986

[10] R. Kenyon Local statistics of lattice dimers, Ann. Inst. H. Poincaré Probab. Statist., Volume 33 (1997) no. 5, pp. 591-618

[11] Y. Meyer Quasicrystals, Diophantine approximation and algebraic numbers, Beyond Quasicrystals, Les Houches, 1994, Springer, Berlin, 1995, pp. 3-16

[12] J. Møller Lectures on Random Voronoı̆ Tessellations, Springer-Verlag, New York, 1994

[13] R.V. Moody Meyer sets and their duals, The Mathematics of Long-Range Aperiodic Order, Waterloo, ON, 1995, Kluwer Academic, Dordrecht, 1997, pp. 403-441

[14] R.V. Moody, Model sets: a survey, Preprint, 2001

[15] R.V. Moody Uniform distribution in model sets, Canadian Math. Bull., Volume 45 (2002) no. 1, pp. 123-130

[16] J. Neveu Processus ponctuels, École d'Été de Probabilités de Saint-Flour, VI-1976, Lecture Notes in Math., 598, Springer-Verlag, Berlin, 1977, pp. 249-445

[17] M. Schlottmann Generalized model sets and dynamical systems, Directions in Mathematical Quasicrystals, American Mathematical Society, Providence, RI, 2000, pp. 143-159

[18] D. Shechtman; I. Blech; D. Gratias; J.W. Cahn Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., Volume 53 (1984), pp. 1951-1953

[19] B. Solomyak Spectrum of dynamical systems arising from Delone sets, Quasicrystals and Discrete Geometry, Toronto, ON, 1995, American Mathematical Society, Providence, RI, 1998, pp. 265-275

[20] N. Wiener The ergodic theorem, Duke Math., Volume 5 (1939), pp. 1-18

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