Geometry and Topology of the Boolean Model on a Stationary Point Processes : A Brief Survey
Keywords:
Boolean model, stationary point processes, topological phase transitions, central limit theorem, intrinsic volumes, Betti numbers.
Abstract
We shall review some of the author's recent results concerning geometric and topological features of the boolean model on a stationary point processes. While study of geometric features of the Poisson boolean model span a very rich literature, the literature
for topological features of the Poisson boolean model is very nascent and that for a general stationary point process is very little. In particular, the focus will be on asymptotics of geometric or topological statistics of the boolean model on a stationary point process. We
shall mainly give details about topological phase transitions and central limit theorem for geometric and topological statistics.
References
Adler R J (2014-2015) TOPOS. http://bulletin.imstat.org/
category/robert-adler/
Agarwala A and Shenoy V B (2017) Topological insulators in
amorphous systems Phys Rev Lett 118 236-402
Blaszczyszyn B, Merzbach E and Schmidt V (1997) A note on
expansion for functionals of spatial marked point
processes Stat Probab Letters 36 299-306
Baccelli F and Blaszczyszyn B (2009) Stochastic Geometry and
Wireless Networks: Volume I Theory, volume 3. Now
Publishers, Inc
Baccelli F and Blaszczyszyn B (2010) Stochastic geometry and
wireless networks: Volume II Applications, volume 4. Now
Publishers, Inc
Baryshnikov Y and Yukich J E (2005) Gaussian limits for random
measures in geometric probability Ann Appl Prob 15 213-
253
Ben-Houag J, Krishnapur M, Peres Y and Virág B (2009) Zeros
of Gaussian analytic functions and determinantal point
processes, volume 51 American Mathematical Society Provi
dence, RI
Blaszczyszyn B (1995) Factorial moment expansion for
stochastic systems Stoch Proc Appl 56 321-335
Blaszczyszyn B and Yogeshwaran D (2015) Clustering Comparison
of Point Processes, with Applications to Random
Geometric Models, pages 31-71. Springer International
Publishing, Cham
Blaszczyszyn B, Yogeshwaran D and Yukich J E (2016) Limit
theory for geometric statistics of clustering point processes
arXiv:1606.03988.
Bobrowski O and Kahle M (2014) Topology of random geometric
complexes: A survey. arXiv:1409.4734.
Bobrowski O, Kahle M and Skraba P (2016) Maximally Persistent
Cycles in Random Geometric Complexes. arXiv:1509.
04347, to appear in Ann Appl Prob
Bobrowski O and Mukherjee S (2015) The topology of
probability distributions on manifolds Prob Th Rel Fields
161 651-686
Bobrowski O and Oliveira G (2017) Random Cech Complexes
on Riemannian Manifolds. arXiv:1704.07204
Bobrowski O and Weinberger S (2017) On the vanishing of
homology in random Cech complexes Rand Struct Alg 51
14-51
Carlsson G (2014) Topological pattern recognition for point cloud
data Acta Numerica 23 289-368
Chatterjee S and Sen S (2017) Minimal spanning trees and steins
method Ann Appl Probab 27 1588-1645
Duy T K (2017) A remark on the convergence of betti numbers in
the thermodynamic regime Pac J Math for Industry
Duy T K, Hiraoka Y and Shirai T (2016) Limit theorems for
persistence diagrams. arXiv:1612.08371
Edelsbrunner H and Harer J (2010) Computational Topology, An
Introduction American Mathematical Society, Providence,
RI
Bj rner A (1995) Topological methods In Handbook of
Combinatorics. Elsevier, Amsterdam
o
Geometry and Topology of the Boolean Model on a Stationary Point Processes : A Brief Survey 557
Ghrist R (2014) Elementary Applied Topology. Createspace
Minkowski functionals-source detection via structure
quantification Astronomy & Astrophysics 555 A38
Grote J and Thäle C (2016) Gaussian polytopes: A cumulantbased
approach arXiv:1602.06148
Haenggi M (2012) Stochastic geometry for wireless networks.
Cambridge University Press
Heinrich L and Schmidt V (1985) Normal convergence of
multidimensional shot noise and rates of this convergence
Adv Appl Prob 17 709-730
Hiraoka Y, Nakamura T, Hirata A, Escolar E G, Matsue K and
Nishiura Y (2016) Hierarchical structures of amorphous
solids characterized by persistent homology Proceedings
of the National Academy of Sciences 113 7035-7040
Hiraoka Y and Shirai T (2015) Minimum spanning acycle and
lifetime of persistent homology in the Linial-Meshulam
process. arXiv:1503.05669
Kahle M (2011) Random geometric complexesDiscrete Comput.
Geom 45 553-573
Kahle M (2014) Topology of random simplicial complexes: a
survey. In Algebraic topology: Applications and new
directions Contemp Math 620 201-221 Amer Math Soc
Providence, RI
Kesten H and Lee S (1996) The central limit theorem for weighted
minimal spanning trees on random pointsAnn Appl Probab
495-527
Klette R and Rosenfeld A (2004) Digital geometry: geometric
methods for digital picture analysis. Elsevier
Kong T Y and Rosenfeld A (1989) Digital topology: introduction
and survey Computer Vision, Graphics and Image
Processing 48 357-393
Last G, Peccati G and Schulte M (2016) Normal approximation
on poisson spaces: Mehler's formula, second order
poincare inequalities and stabilization Prob Th Rel Fields
165 667-723
Last G and Penrose M D (2017+) Lectures on the Poisson Process.
to be published as IMS Textbook by Cambridge University
Press. http://www.math.kit.edu/stoch/last/page/lectures_
on_the_poisson_process/en
Munkres J R (1996) Elements Of Algebraic Topology, 1st edition.
Westview Press
Owada T (2016) Limit Theorems for the Sum of Persistence
Barcodes. arXiv:1604.04058
Peccati G and Reitzner M, editors (2016) Stochastic analysis for
Poisson point processes. Mallavin calculus, Wiener-Ito
chaos expansions and stochastic geometry, volume 7 of
Bocconi & Springer Series. Bocconi University Press,
Springer, [Cham]
Penrose M (2003) Random Geometric Graphs. Oxford
University Press, New York
Penrose M and Yukich J E (2001) Central limit theorems for
some graphs in computational geometryAnn Appl Probab
11 1005-1041
Penrose M D (1997) The longest edge of the random minimal
spanning tree Ann Appl Prob 340-361
Penrose M D and Yukich J E (2013) Limit theory for point
processes in manifolds Ann Appl Probab 23 2161-2211
Ram Reddy T, Vadlamani S and Yogeshwaran D (2017) Central
limit theorem for quasi-local statistics of spin models on
Cayley graphs arXiv:1709.10424
Schneider R and Weil W (2008) Stochastic and Integral Geometry.
Probability and its Applications (New York). Springer-
Verlag, Berlin
Schreiber T and Yukich J (2008) Variance asymptotics and central
limit theorems for generalized growth processes with
applications to convex hulls and maximal points Ann
Probab 36 363-396
Skraba P, Thoppe G and Yogeshwaran D (2017) Randomly
Weighted dcomplexes: Minimal Spanning Acycles and
Persistence Diagrams. arXiv:1701.00239v1
Svane A (2017) Valuations in image analysis, pages 435-454
Springer International Pub- lishing, Cham
Yogeshwaran D and Adler R J (2015) On the topology of random
com- plexes built over stationary point processes Ann
Appl Probab 25 3338-3380
Yogeshwaran D, Subag E and Adler R J (2017) Random geometric
com- plexes in the thermodynamic regime Prob Th Rel
Fields 167 107-142
Yukich J E (2006) Probability theory of classical Euclidean
optimization problems. Springer
Yukich J E (2013) Limit theorems in discrete stochastic geometry.
In Stochastic geometry, spatial statistics and random fields,
Lecture Notes in Math 2068 239-275 Springer,
Heidelberg.
category/robert-adler/
Agarwala A and Shenoy V B (2017) Topological insulators in
amorphous systems Phys Rev Lett 118 236-402
Blaszczyszyn B, Merzbach E and Schmidt V (1997) A note on
expansion for functionals of spatial marked point
processes Stat Probab Letters 36 299-306
Baccelli F and Blaszczyszyn B (2009) Stochastic Geometry and
Wireless Networks: Volume I Theory, volume 3. Now
Publishers, Inc
Baccelli F and Blaszczyszyn B (2010) Stochastic geometry and
wireless networks: Volume II Applications, volume 4. Now
Publishers, Inc
Baryshnikov Y and Yukich J E (2005) Gaussian limits for random
measures in geometric probability Ann Appl Prob 15 213-
253
Ben-Houag J, Krishnapur M, Peres Y and Virág B (2009) Zeros
of Gaussian analytic functions and determinantal point
processes, volume 51 American Mathematical Society Provi
dence, RI
Blaszczyszyn B (1995) Factorial moment expansion for
stochastic systems Stoch Proc Appl 56 321-335
Blaszczyszyn B and Yogeshwaran D (2015) Clustering Comparison
of Point Processes, with Applications to Random
Geometric Models, pages 31-71. Springer International
Publishing, Cham
Blaszczyszyn B, Yogeshwaran D and Yukich J E (2016) Limit
theory for geometric statistics of clustering point processes
arXiv:1606.03988.
Bobrowski O and Kahle M (2014) Topology of random geometric
complexes: A survey. arXiv:1409.4734.
Bobrowski O, Kahle M and Skraba P (2016) Maximally Persistent
Cycles in Random Geometric Complexes. arXiv:1509.
04347, to appear in Ann Appl Prob
Bobrowski O and Mukherjee S (2015) The topology of
probability distributions on manifolds Prob Th Rel Fields
161 651-686
Bobrowski O and Oliveira G (2017) Random Cech Complexes
on Riemannian Manifolds. arXiv:1704.07204
Bobrowski O and Weinberger S (2017) On the vanishing of
homology in random Cech complexes Rand Struct Alg 51
14-51
Carlsson G (2014) Topological pattern recognition for point cloud
data Acta Numerica 23 289-368
Chatterjee S and Sen S (2017) Minimal spanning trees and steins
method Ann Appl Probab 27 1588-1645
Duy T K (2017) A remark on the convergence of betti numbers in
the thermodynamic regime Pac J Math for Industry
Duy T K, Hiraoka Y and Shirai T (2016) Limit theorems for
persistence diagrams. arXiv:1612.08371
Edelsbrunner H and Harer J (2010) Computational Topology, An
Introduction American Mathematical Society, Providence,
RI
Bj rner A (1995) Topological methods In Handbook of
Combinatorics. Elsevier, Amsterdam
o
Geometry and Topology of the Boolean Model on a Stationary Point Processes : A Brief Survey 557
Ghrist R (2014) Elementary Applied Topology. Createspace
Minkowski functionals-source detection via structure
quantification Astronomy & Astrophysics 555 A38
Grote J and Thäle C (2016) Gaussian polytopes: A cumulantbased
approach arXiv:1602.06148
Haenggi M (2012) Stochastic geometry for wireless networks.
Cambridge University Press
Heinrich L and Schmidt V (1985) Normal convergence of
multidimensional shot noise and rates of this convergence
Adv Appl Prob 17 709-730
Hiraoka Y, Nakamura T, Hirata A, Escolar E G, Matsue K and
Nishiura Y (2016) Hierarchical structures of amorphous
solids characterized by persistent homology Proceedings
of the National Academy of Sciences 113 7035-7040
Hiraoka Y and Shirai T (2015) Minimum spanning acycle and
lifetime of persistent homology in the Linial-Meshulam
process. arXiv:1503.05669
Kahle M (2011) Random geometric complexesDiscrete Comput.
Geom 45 553-573
Kahle M (2014) Topology of random simplicial complexes: a
survey. In Algebraic topology: Applications and new
directions Contemp Math 620 201-221 Amer Math Soc
Providence, RI
Kesten H and Lee S (1996) The central limit theorem for weighted
minimal spanning trees on random pointsAnn Appl Probab
495-527
Klette R and Rosenfeld A (2004) Digital geometry: geometric
methods for digital picture analysis. Elsevier
Kong T Y and Rosenfeld A (1989) Digital topology: introduction
and survey Computer Vision, Graphics and Image
Processing 48 357-393
Last G, Peccati G and Schulte M (2016) Normal approximation
on poisson spaces: Mehler's formula, second order
poincare inequalities and stabilization Prob Th Rel Fields
165 667-723
Last G and Penrose M D (2017+) Lectures on the Poisson Process.
to be published as IMS Textbook by Cambridge University
Press. http://www.math.kit.edu/stoch/last/page/lectures_
on_the_poisson_process/en
Munkres J R (1996) Elements Of Algebraic Topology, 1st edition.
Westview Press
Owada T (2016) Limit Theorems for the Sum of Persistence
Barcodes. arXiv:1604.04058
Peccati G and Reitzner M, editors (2016) Stochastic analysis for
Poisson point processes. Mallavin calculus, Wiener-Ito
chaos expansions and stochastic geometry, volume 7 of
Bocconi & Springer Series. Bocconi University Press,
Springer, [Cham]
Penrose M (2003) Random Geometric Graphs. Oxford
University Press, New York
Penrose M and Yukich J E (2001) Central limit theorems for
some graphs in computational geometryAnn Appl Probab
11 1005-1041
Penrose M D (1997) The longest edge of the random minimal
spanning tree Ann Appl Prob 340-361
Penrose M D and Yukich J E (2013) Limit theory for point
processes in manifolds Ann Appl Probab 23 2161-2211
Ram Reddy T, Vadlamani S and Yogeshwaran D (2017) Central
limit theorem for quasi-local statistics of spin models on
Cayley graphs arXiv:1709.10424
Schneider R and Weil W (2008) Stochastic and Integral Geometry.
Probability and its Applications (New York). Springer-
Verlag, Berlin
Schreiber T and Yukich J (2008) Variance asymptotics and central
limit theorems for generalized growth processes with
applications to convex hulls and maximal points Ann
Probab 36 363-396
Skraba P, Thoppe G and Yogeshwaran D (2017) Randomly
Weighted dcomplexes: Minimal Spanning Acycles and
Persistence Diagrams. arXiv:1701.00239v1
Svane A (2017) Valuations in image analysis, pages 435-454
Springer International Pub- lishing, Cham
Yogeshwaran D and Adler R J (2015) On the topology of random
com- plexes built over stationary point processes Ann
Appl Probab 25 3338-3380
Yogeshwaran D, Subag E and Adler R J (2017) Random geometric
com- plexes in the thermodynamic regime Prob Th Rel
Fields 167 107-142
Yukich J E (2006) Probability theory of classical Euclidean
optimization problems. Springer
Yukich J E (2013) Limit theorems in discrete stochastic geometry.
In Stochastic geometry, spatial statistics and random fields,
Lecture Notes in Math 2068 239-275 Springer,
Heidelberg.
Published
2018-07-09
Issue
Section
Review Articles
Copyright (c) 2018 D Yogeshwaran
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