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Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

  • Tulasi Ram Reddy
  • Sreekar Vadlamani
  • D. Yogeshwaran
Article

Abstract

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of \(\alpha \)-mixing (for local statistics) and exponential \(\alpha \)-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.

Keywords

Clustering spin models Central limit theorem Cayley graphs Fast decaying covariance Exponentially quasi-local statistics Cubical complexes 

Mathematics Subject Classification

82B20 60G60 60F05 60D05 

Notes

Acknowledgements

DY is thankful for the discussions with Matthew Wright which led to his interest in this question and especially the applications to random cubical complexes. The authors are also thankful to numerous comments by anonymous referees that has lead to an improved presentation.

References

  1. 1.
    Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12, 1454–1508 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atkin, R.: An algebra for patterns on a complex. Int. J. Man-Mach. Stud. 6(3), 285–307 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Atkin, R.: An algebra for patterns on a complex. II. Int. J. Man-Mach. Stud. 8(5), 483–498 (1976)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baccelli, F., Haji-Mirsadeghi, M.O., Khezeli, A.: Dynamics on unimodular random graphs. arXiv:1608.05940 (2016)
  5. 5.
    Baryshnikov, Y., Yukich, J., et al.: Gaussian limits for random measures in geometric probability. Ann. Appl. Prob. 15(1A), 213–253 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beffara, V., Gayet, D.: Percolation of random nodal lines. arXiv:1605.08605 (2016)
  7. 7.
    Benjamini, I.: Coarse geometry and randomness, École d’Été de Probabilités de Saint-Flour, vol. 2100. Springer (2013)Google Scholar
  8. 8.
    Björklund, M., Gorodnik, A.: Central limit theorems for group actions which are exponentially mixing of all orders. arXiv:1706.09167 (2017)
  9. 9.
    Błaszczyszyn, B.: Factorial moment expansion for stochastic systems. Stoch. Proc. Appl. 56(2), 321–335 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Błaszczyszyn, B., Merzbach, E., Schmidt, V.: A note on expansion for functionals of spatial marked point processes. Stat. Probab. Lett. 36(3), 299–306 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Blaszczyszyn, B., Yogeshwaran, D., Yukich, J.E.: Limit theory for geometric statistics of point processes having fast decay of correlations. arXiv:1606.03988 (2018)
  12. 12.
    Bobrowski, O., Kahle, M.: Topology of random geometric complexes: a survey. arXiv:1409.4734 (2017)
  13. 13.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep.s 424(4), 175–308 (2006)MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. 14.
    Bolthausen, E., Cipriani, A., Kurt, N.: Exponential decay of covariances for the supercritical membrane model. Comm. Math. Phys. 353(3), 1217–1240 (2017)MathSciNetCrossRefzbMATHADSGoogle Scholar
  15. 15.
    Borcea, J., Brändén, P., Liggett, T.M.: Negative dependence and the geometry of polynomials. J. Am. Math. Soc. 22(2), 521–567 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bradley, R.: Equivalent mixing conditions for random fields. Ann. Probab. 21(4), 1921–1926 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bradley, R.: On quantiles and the central limit question for strongly mixing sequences. J. Theor. Probab. 10(2), 1921–1926 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bradley, R.: Basic properties of strong mixing conditions : a survey and some open questions. Probab. Surv. 2, 107–144 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bradley, R., Tone, C.: A central limit theorem for non-stationary strongly mixing random fields. J. Theor. Probab. 2, 107–144 (2015)Google Scholar
  20. 20.
    Bulinski, A., Spodarev, E.: Central limit theorems for weakly dependent random fields. In: Spodarev, E. (ed.) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, pp. 337–383. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Bulinski, A., Spodarev, E., Timmermann, F.: Central limit theorems for the excursion set volumes of weakly dependent random fields. Bernoulli 18(1), 100–118 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bulinski, A., Suquet, C.: Normal approximation for quasi-associated random fields. Stat. Probab. Lett. 54(2), 215–226 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cohen, G., Conze, J.P.: Almost mixing of all orders and clt for some \({\mathbb{Z}}^{d}\) actions on subgroups of \(\mathbb{F}_{p}^{{\mathbb{z}^{d}}}\). arXiv:1609.06484 (2016)
  24. 24.
    Derriennic, Y.: Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the central limit theorem. Discret. Contin. Dyn. Syst. 15(1), 143–158 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Doukhan, P.: Properties and examples. In: Mixing, Lecture Notes in Statistics, vol. 85. Springer, New York (1994)Google Scholar
  26. 26.
    Dousse, J., Féray, V.: Weighted dependency graphs and the Ising model. arXiv:1610.05082 (2016)
  27. 27.
    Duminil-Copin, H.: Graphical representations of lattice spin models. Lecture notes of Cours Peccot du Collège de France. Spartacus. http://www.ihes.fr/~duminil/publi/2016Peccot.pdf (2015)
  28. 28.
    Edelsbrunner, H., Harer, J.: Computational Topology, An Introduction. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  29. 29.
    Estrada, E., Rodriguez-Velazquez, J.A.: Complex networks as hypergraphs. arXiv:physics/0505137 (2005)
  30. 30.
    Féray, V.: Weighted dependency graphs. arXiv:1605.03836 (2016)
  31. 31.
    Forman, R.: A user’s guide to discrete Morse theory. Lothar. Combin. 48, 35 (2002)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Franceschetti, M., Meester, R.: Random Networks for Communication: From Statistical Physics to Information Systems, vol. 24. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2017)CrossRefzbMATHGoogle Scholar
  34. 34.
    Funaki, T.: Stochastic interface models. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, pp. 103–274. Springer, New York (2005)Google Scholar
  35. 35.
    Giacomin, G.: Aspects of statistical mechanics of random surfaces. IHP Lecture notes. https://www.lpma-paris.fr/modsto/_media/users/giacomin/ihp.pdf (2001)
  36. 36.
    Goldstein, L., Wiroonsri, N.: Stein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process. arXiv:1603.05322 (2016)
  37. 37.
    Göring, D., Klatt, M., Stegmann, C., Mecke, K.: Morphometric analysis in gamma-ray astronomy using Minkowski functionals-source detection via structure quantification. Astron. Astrophys. 555, A38 (2013)CrossRefGoogle Scholar
  38. 38.
    Gray, S.B.: Local properties of binary images in two dimensions. IEEE Transac. Comput. 20(5), 551–561 (1971)CrossRefzbMATHGoogle Scholar
  39. 39.
    Grimmett, G.: Probability on Graphs: Random Processes on Graphs and Lattices, vol. 1. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  40. 40.
    Gromov, M.: Groups of polynomial growth and expanding maps. J. Tits. Publ. Math. de l’I.H.E.S. 53, 53–78 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Gross, L.: Decay of correlations in classical lattice models at high temperature. Commun. Math. Phys. 68(1), 9–27 (1979)MathSciNetCrossRefzbMATHADSGoogle Scholar
  42. 42.
    Grote, J., Thäle, C.: Gaussian polytopes: a cumulant-based approach. arXiv:1602.06148 (2016)
  43. 43.
    Haenggi, M.: Interference in lattice networks. arXiv:1004.0027 (2010)
  44. 44.
    Hegerfeldt, G.C.: Noncommutative analogs of probabilistic notions and results. J. Funct. Anal. 64(3), 436–456 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Heinrich, L.: Asymptotic methods in statistics of random point processes. In: Spodarev, E. (ed.) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, pp. 115–150. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  46. 46.
    Hilfer, R.: Local porosity theory and stochastic reconstruction for porous media. In: Mecke, K.R., Stoyan, D. (eds.) Statistical Physics and Spatial Statistics, pp. 203–241. Springer, Berlin (2000)CrossRefGoogle Scholar
  47. 47.
    Hiraoka, Y., Tsunoda, K.: Limit theorems on random cubical homology. arXiv:1612.08485 (2016)
  48. 48.
    Holley, R.A., Stroock, D.W.: Applications of the stochastic Ising model to the Gibbs states. Commun. Math. Phys. 48(3), 249–265 (1976)MathSciNetCrossRefADSGoogle Scholar
  49. 49.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series, vol. 51. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  50. 50.
    Ioffe, D., Velenik, Y.: A note on the decay of correlations under \(\delta \)-pinning. Probab. Theory Relat. Fields 116(3), 379–389 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Janson, S.: Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. Ann. Prob. 16(1), 305–312 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  53. 53.
    Kahle, M.: Topology of random simplicial complexes: a survey. AMS Contemp. Math. 620, 201–222 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Klamt, S., Haus, U.U., Theis, F.: Hypergraphs and cellular network. PLoS Comput. Biol. 5(5), e1000385 (2009)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Klatt, M.A.: Morphometry of random spatial structures in physics. Ph.D. thesis. https://opus4.kobv.de/opus4-fau/frontdoor/index/index/docId/7654. Friedrich-Alexander-Universität Erlangen-Nürnberg (2016)
  56. 56.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  57. 57.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Visi. Graph. Image Process. 48(3), 357–393 (1989)CrossRefGoogle Scholar
  58. 58.
    Kopper, C., Magnen, J., Rivasseau, V.: Mass generation in the large N Gross–Neveu-model. Commun. Math. Phys. 169(1), 121–180 (1995)MathSciNetCrossRefzbMATHADSGoogle Scholar
  59. 59.
    Kraetzl, M., Laubenbacher, R., Gaston, M.E.: Combinatorial and algebraic approaches to network analysis. DSTO Internal Report (2001)Google Scholar
  60. 60.
    Krokowski, K., Thäle, C., et al.: Multivariate central limit theorems for rademacher functionals with applications. Elec. J. Prob. 22, 919–963 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Künsch, H.: Decay of correlations under Dobrushin’s uniqueness condition and its applications. Commun. Math. Phys. 82(2), 207–222 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    de La Harpe, P.: Topics in Geometric Group Theory. University of Chicago Press, Chicago (2000)zbMATHGoogle Scholar
  63. 63.
    Lachieze-Rey, R., Schulte, M., Yukich, J.E.: Normal approximation for stabilizing functionals. arXiv:1702.00726 (2017)
  64. 64.
    Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42. Cambridge University Press, New York (2016)CrossRefzbMATHGoogle Scholar
  66. 66.
    Lyons, R., Steif, J.E.: Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120(3), 515–575 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Malyshev, V.A.: The central limit theorem for Gibbsian random fields. Sov. Math. Dokl. 16, 1141–1145 (1975)Google Scholar
  68. 68.
    Martin, P.A., Yalcin, T.: The charge fluctuations in classical Coulomb systems. J. Stat. Phys. 22(4), 435–463 (1980)MathSciNetCrossRefADSGoogle Scholar
  69. 69.
    Michoel, T., Nachtergaele, B.: Alignment and integration of complex networks by hypergraph-based spectral cl. Phys. Rev. E 86(5), 056,111 (2012)CrossRefGoogle Scholar
  70. 70.
    Munkres, J.: Elements of Algebraic Topology. Addison-Wesley, Boston (1984)zbMATHGoogle Scholar
  71. 71.
    Nazarov, F., Sodin, M.: Correlation functions for random complex zeroes: strong clustering and local universality. Commun. Math. Phys. 310(1), 75–98 (2012)MathSciNetCrossRefzbMATHADSGoogle Scholar
  72. 72.
    Pansu, P.: Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergod. Theory Dyn. Syst. 3(3), 415–445 (1983)CrossRefzbMATHGoogle Scholar
  73. 73.
    Peccati, G., Taqqu, M.S.: Wiener Chaos: Moments, Cumulants and Diagrams, vol. 1. Springer, Milan (2011)CrossRefzbMATHGoogle Scholar
  74. 74.
    Peligrad, M.: Maximum of partial sums and in invariance principle for a class of weak dependent random variables. Proc. AMS 126(4), 1181–1189 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Penrose, M.: Random Geometric Graphs, Oxford Studies in Probability, vol. 5. Oxford University Press, Oxford (2003)CrossRefGoogle Scholar
  76. 76.
    Penrose, M.D.: A central limit theorem with applications to percolation, epidemics and Boolean models. Ann. Probab. 29(4), 1515–1546 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Penrose, M.D., Yukich, J.E.: Limit theory for point processes in manifolds. Ann. Appl. Prob. 23(6), 2161–2211 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Penrose, O., Lebowitz, J.L.: On the exponential decay of correlation functions. Commun. Math. Phys. 39(3), 165–184 (1974)MathSciNetCrossRefADSGoogle Scholar
  79. 79.
    Pete, G.: Probability and geometry on groups. Lecture notes for a graduate course. http://math.bme.hu/~gabor/PGG.pdf (2017)
  80. 80.
    Roe, J.: Lectures on Coarse Geometry, vol. 31. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  81. 81.
    Saha, P.K., Strand, R., Borgefors, G.: Digital topology and geometry in medical imaging: a survey. IEEE Transac. Med. Imaging 34(9), 1940–1964 (2015)CrossRefGoogle Scholar
  82. 82.
    Saulis, L., Statulevicius, V.: Limit Theorems for Large Deviations. Kluwer Academic, Dordrecht (1991)CrossRefzbMATHGoogle Scholar
  83. 83.
    Schladitz, K., Ohse, J., Nagel, W.: Measurement of intrinsic volumes of sets observed on lattices. Discrete Geom. Comput Imag. 37, 247–258 (2006)Google Scholar
  84. 84.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  85. 85.
    Schonmann, R.H.: Theorems and conjectures on the droplet-driven relaxation of stochastic Ising mode. In: Grimmett, G. (ed.) Probability and Phase Transition, pp. 265–301. Springer, Berlin (1994)CrossRefGoogle Scholar
  86. 86.
    Spanier, E.H.: Algebraic Topology. McGaw-Hill Book Co., New York (1966)zbMATHGoogle Scholar
  87. 87.
    Sunklodas, J.: Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal Distribution, pp. 113–165. Springer, Berlin (1991)zbMATHGoogle Scholar
  88. 88.
    Svane, A.M.: Valuations in Image Analysis, pp. 435–454. Springer International Publishing, Cham (2017)zbMATHGoogle Scholar
  89. 89.
    Velenik, Y.: Localization and delocalization of random interfaces. Probab. Surv 3, 112–169 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Werman, M., Wright, M.: Intrinsic volumes of random cubical complexes. Discrete Comput. Geom. 56, 93–113 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    Yukich, J.: Limit theorems in discrete stochastic geometry. In: Bandyopadhyay, B., et al. (eds.) Stochastic Geometry, Spatial Statistics and Random Fields, pp. 239–275. Springe, Heidelberg (2013)CrossRefGoogle Scholar

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Authors and Affiliations

  • Tulasi Ram Reddy
    • 1
  • Sreekar Vadlamani
    • 2
    • 3
  • D. Yogeshwaran
    • 4
  1. 1.Division of SciencesNew York University Abu DhabiAbu DhabiUAE
  2. 2.TIFR Center for Applicable MathematicsBangaloreIndia
  3. 3.Department of StatisticsLund UniversityLundSweden
  4. 4.Theoretical Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia

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