Random geometric complexes in the thermodynamic regime
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Akcoglu, M.A., Krengel, U.: Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323, 53–67 (1981)
Björner, A.: Topological methods. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. 2, pp. 1819–1872. Elsevier, Amsterdam (1995)
Bobrowski, O., Adler, R.J.: Distance functions, critical points, and the topology of random Čech complexes. Homol. Homotopy Appl. 16(2), 311–344 (2014)
Bobrowski, O., Kahle, M.: Topology of random geometric complexes: a survey (2014). arXiv:1409.4734
Bobrowski, O., Mukherjee, S.: The topology of probability distributions on manifolds. Probab. Theory Relat. Fields 161(3–4), 651–686 (2015)
Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (N.S.) 46(2), 255–308 (2009)
Chalker, T.K., Godbole, A.P., Hitczenko, P., Radcliff, J., Ruehr, O.G.: On the size of a random sphere of influence graph. Adv. Appl. Probab. 31(3), 596–609 (1999)
Costa, A.E., Farber, M., Kappeler, T.: Topics of stochastic algebraic topology. In: Proceedings of the Workshop on Geometric and Topological Methods in Computer Science (GETCO). Electronic Notes in Theoretical Computer Science, vol. 283(0), pp. 53–70 (2012)
Decreusefond, L., Ferraz, E., Randriam, H., Vergne, A.: Simplicial homology of random configurations. Adv. Appl. Probab. 46, 1–20 (2014)
Delfinado, C.J.A., Edelsbrunner, H.: An incremental algorithm for Betti numbers of simplicial complexes. In: Proceedings of the Ninth Annual Symposium on Computational Geometry, SCG ’93, New York, NY, USA, pp. 232–239. ACM (1993)
Edelsbrunner, H., Harer, J.L.: Computational Topology, An Introduction. American Mathematical Society, Providence (2010)
Forman, R.: A user’s guide to discrete Morse theory. Sém. Lothar. Comb. 48, (2002)
Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. (N.S.) 45(1), 61–75 (2008)
Hall, P.: Introduction to the theory of coverage processes. In: Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1988)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hug, D., Last, G., Schulte, M.: Second order properties and central limit theorems for geometric functionals of Boolean models. Ann. Appl. Probab. (2013) (to appear). arXiv:1308.6519
Kahle, M.: Topology of random simplicial complexes: a survey. In: Tillmann, U., Galatius, S., Sinha, D. (eds.) Algebraic Topology: Applications and New Directions. Contemporary Mathematics, vol. 620, pp. 201–221. American Mathematical Society, Providence (2014)
Kahle, M., Meckes, E.: Limit theorems for Betti numbers of random simplicial complexes. Homol. Homotopy Appl. 15(1), 343–374 (2013)
Last, G., Penrose, M.D.: Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150(3–4), 663–690 (2011)
Last, G., Peccatti, G., Schulte, M.: Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequality and stabilization. Probab. Theory Relat. Fields 1–57 (2015). doi: 10.1007/s00440-015-0643-7
Linial, N., Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26(4), 475–487 (2006)
Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Reading (1984)
Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discret. Comput. Geom. 39(1–3), 419–441 (2008)
Niyogi, P., Smale, S., Weinberger, S.: A topological view of unsupervised learning from noisy data. SIAM J. Comput. 40(3), 646–663 (2011)
Nourdin, I., Peccati, G.: Normal approximations with Malliavin calculus. In: Cambridge Tracts in Mathematics, vol. 192. Cambridge University Press, Cambridge (2012). From Stein’s method to universality
Pemantle, R., Peres, Y.: Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures. Combin. Probab. Comput. 23, 140–160 (2013)
Penrose, M.D.: Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13(4), 1124–1150 (2007)
Penrose, M.D., Yukich, J.E.: Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11(4), 1005–1041 (2001)
Sarkar, A.: Co-existence of the occupied and vacant phase in Boolean models in three or more dimensions. Adv. Appl. Probab. 29(4), 878–889 (1997)
Schneider, R., Weil, W.: Stochastic and integral geometry. In: Probability and its Applications (New York). Springer, Berlin (2008)
Schulte, M.: Malliavin–Stein method in stochastic geometry. Ph.D. thesis (2013). http://repositorium.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2013031910717
Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Co., New York (1966)
Steele, J.M.: An Efron-Stein inequality for nonsymmetric statistics. Ann. Stat. 14(2), 753–758 (1986)
Steele, J.M.: Probability theory and combinatorial optimization. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69. SIAM, Philadelphia (1997)
Stoyan, D., Kendall, W., Mecke, J.: Stochastic Geometry and its Applications. Wiley, Chichester (1995)
Tanemura, H.: Critical behavior for a continuum percolation model. In: Probability Theory and Mathematical Statistics (Tokyo, 1995), pp. 485–495. World Scientific Publishing, River Edge (1996)
Yogeshwaran, D., Adler, R.J.: On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25(6), 3338–3380 (2015)
Yukich, J.E.: Probability theory of classical Euclidean optimization problems. Lecture Notes in Mathematics, vol. 1675. Springer, Berlin, Heidelberg (1998)
Zomorodian, A.: Topological data analysis. In: Advances in Applied and Computational Topology. Proceedings of Symposia in Applied Mathematics, vol. 70, pp. 1–39. American Mathematical Society, Providence (2012)
Zomorodian, A.J.: Topology for computing. Cambridge Monographs on Applied and Computational Mathematics, vol. 16. Cambridge University Press, Cambridge (2009)