G. , O. ). , and (. , v 1 ) have the same distribution on G * 2. reversibility: (G, o, v 1 ) and (G, v 1 , o) have the same distribution on G * *

, ) = P{ x 1 , x 2 ? ? } in terms of the capacity functional of ?

, Let ? be a general coverage model with capacity functional T ? , generated by a set process ? f with Laplace transform L ? f . Prove the following relation for

, Prove the expressions (14.1.13) and (14.1.14) for the volume fraction and the covariance function of a homogeneous germ-grain Boolean model

, Let ? f = i ? F i be a hard core set-process on R d , (i.e. the set-atoms F i do not overlap) with F i being balls

F. P{-k-?-?-}-=-e, Argue that for any connected K the full coverage functional is equal to the mean measure of ? f on

, Hint: K ? ? iff K is entirely contained in of the spherical set-atoms F i

, Let ? be a homogeneous germ-grain Boolean coverage model driven by a homogeneous Poisson process with intensity ?, 0 < ? < ? on R d and generic grain Z 0, p.1

, Let? = X i ,Z i be a stationary germ-grain model on R d as in (14.1.8), considered on some stationary framework (see the corresponding lesson) (?, A, {? x } x?R d , P)

, Using the flow of the stationary framework this means Z i = Z ? ? X i for some random closed set Z defined on the considered probability space. Stationary Voronoi tessellation and the Johnson-Mehl coverage model are examples of such process. Any stationary germ-grain process with

, Denote by ? f the corresponding set process (14.1.9) and its set-atoms by F i := X i + Z i

, Examples of stationary, ergodic random closed sets are homogeneous Boolean germ-grain models on R d . The case of deterministic grains follows immediately form the mixing property (implying ergodicity) of the homogeneous Poisson point process. The general case of (i.i.d.) compact grains requires some additional argument; cf, 1996.

, Consider the following two subsets of the plane, vol.2

, ? Let F 1 = {(x, y) ? R 2 : y = sin(1/x), x = 0} sup{(0, y) : y ?

, ? Let F 2 = {(x, y) ? R 2 : y = 1/|x|, x = 0}. Prove that F 2 is a closed and disconnected set

, ? F 2 ? {?} is closed and connected in the one-point (Alexandroff) compactifiation R 2 ? {?} of R 2 where the extra point ? is considered to be the limit of all sequences x n on R 2

, Hint: Denote by I i the indicator of the event that the grain X i + B X i (R i ) belongs to an unbounded component. Use the Campbell-Mecke-Matthes theorem

, Prove for a one-dimensional homogeneous Boolean model that E [R] < ? implies ? c = ? and E [R] = ? implies ? c = 0. (Strictly speaking in this latter case it is no longer a Boolean model

, Computer exercise. Estimate the normalized critical intensity ? * c for planar Boolean model with fixed spherical grains. (Recall, ? * c ? 1.1281). Use R with spatstat and igraph and SGCS

, Use the command clusters to obtain the connected components of the graph. For a given realization of germs, plot the fraction of nodes in the largest and second largest cluster in function of the grain radius. Estimate the critical radius as the radius when second largest component attains its maximal value, Simulate Poisson process with unit intensity in a reasonably large window. Construct the matrix of distances between the simulated points

, 188LESSON 15. CONNECTEDNESS OF RANDOM SETS AND CONTINUUM PERCOLATION Bibliography D. Aldous and R. Lyons. Processes on unimodular random networks, vol.12, pp.1454-1508, 2007.

D. Aldous and J. M. Steele, The objective method: Probabilistic combinatorial optimization and local weak convergence, Probability on discrete structures, pp.1-72, 2004.

K. B. Athreya and P. Ney, Branching processes, vol.9780486434742, 1972.

F. Baccelli and B. Laszczyszyn, Stochastic Geometry and Wireless Networks, Volume I -Theory, vol.3, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00403039

F. Baccelli, B. , and M. Karray, Random Measures, Point Processes, and Stochastic Geometry. Inria-Hal, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02460214

A. Baddeley, R. Turner, and E. Rubak, Datasets provided for spatstat

I. Benjamini, Coarse geometry and randomness-École d'Été de Probabilités de Saint-Flour, Lecture Notes in Mathematics, vol.2100, 2011.

N. Berglund, La probabilité d'extinction d'une espèce menacée, 2013.

I. Bienaymé, De la loi de multiplication et de la durée des familles, Soc. Philomat. Paris Extraits, Sér, vol.5, pp.37-39, 1845.

B. Laszczyszyn and D. Yogeshwaran, Clustering comparison of point processes with applications to random geometric models, Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms, vol.2120, pp.31-71, 2014.

B. Laszczyszyn, M. Haenggi, P. Keeler, and S. Mukherjee, Stochastic Geometry Analysis of Cellular Networks, 2018.

O. Bobrowski and M. Kahle, Topology of random geometric complexes: a survey, 2014.

B. Bollobás, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, European Journal of Combinatorics, vol.1, issue.4, pp.311-316, 1980.

S. R. Broadbent and J. M. Hammersley, Percolation processes: I. crystals and mazes, Mathematical Proceedings of the Cambridge Philosophical Society, vol.53, pp.629-641, 1957.

S. N. Chiu, D. Stoyan, W. Kendall, and J. Mecke, Stochastic geometry and its applications, 2013.

D. J. Daley and D. Vere-jones, An Introduction to the Theory of Point Processes, vol. I. Probability and Its Applications, 2003.

D. J. Daley and D. Vere-jones, An introduction to the theory of point processes: volume II: general theory and structure, 2007.

O. Dousse, M. Franceschetti, N. Macris, R. Meester, and P. Thiran, Percolation in the signal to interference ratio graph, Journal of Applied Probability, vol.43, issue.2, pp.552-562, 2006.

M. Draief and L. Massoulié, Epidemics and Rumours in Complex Networks, ISBN 0521734436, 9780521734431, 2010.

H. Duminil-copin and V. Tassion, A new proof of the sharpness of the phase transition for bernoulli percolation on Z d, 2015.

P. Erd?s and A. Rényi, On random graphs I, Publicationes Mathematicae, vol.6, pp.290-297, 1959.

B. Forghani and K. Mallahi-karai, Amenability of trees. Groups, Graphs and Random Walks, vol.436, p.176, 2017.

M. Franceschetti and R. Meester, Random networks for communication: from statistical physics to information systems, vol.24, 2008.

E. N. Gilbert, Random graphs, The Annals of Mathematical Statistics, vol.30, issue.4, pp.1141-1144, 1959.

G. Grimmett, Random labelled trees and their branching networks, Journal of the Australian Mathematical Society, vol.30, issue.2, pp.229-237, 1980.

G. Grimmett, Percolation. Grundlehren der mathematischen Wissenschaften, 2013.

P. Hall, Introduction to the Theory of Coverage Processes, 1988.

H. Hermann and A. Elsner, Geometric models for isotropic random porous media: a review, Advances in Materials Science and Engineering, 2014.

M. Heveling and G. Last, Characterization of palm measures via bijective point-shifts. The Annals of Probability, vol.33, pp.1698-1715, 2005.

C. Hirsch and G. Last, On maximal hard-core thinnings of stationary particle processes, 2017.

P. W. Holland, K. B. Laskey, and S. Leinhardt, Stochastic blockmodels: First steps, Social networks, vol.5, issue.2, pp.109-137, 1983.

P. Jacquet and W. Szpankowski, Analytical depoissonization and its applications, Theoretical Computer Science, vol.201, issue.1, pp.167-176, 1998.

S. Janson, Random coverings in several dimensions, Acta Math, vol.156, pp.83-118, 1986.

S. Janson and M. J. Luczak, A new approach to the giant component problem, Random Structures and Algorithms, vol.34, issue.2, pp.197-216, 2009.

O. Kallenberg, Random measures. Academic Pr, 1983.

O. Kallenberg, Foundations of modern probability, 2002.

J. L. Kelley, General Topology, 1955.

M. Kerscher, K. Mecke, P. Schücker, H. Bohringer, L. Guzzo et al., Non-gaussian morphology on large scales: Minkowski functionals of the reflex cluster catalogue, Astronomy & Astrophysics, vol.377, issue.1, pp.1-16, 2001.

G. Last and M. Penrose, Lectures on the Poisson Process. Institute of Mathematical Statistics Textbooks, 2017.

L. Gadar, Generate (random) graphs with igraph, pp.2017-2027

J. Boudec, Understanding the simulation of mobility models with palm calculus. Performance Evaluation, vol.64, pp.126-147, 2007.

G. Matheron, Random sets and integral geometry, 1975.

J. Mecke, Stationäre zufällige maße auf lokalkompakten abelschen gruppen, Probability Theory and Related Fields, vol.9, pp.36-58, 1967.

K. R. Mecke, Morphology of spatial patterns-porous media, spinodal decomposition and dissipative structures, Acta Physica Polonica. Series B, vol.28, issue.8, pp.1747-1782, 1997.

R. Meester and R. Roy, Continuum percolation, vol.119, 1996.

I. Molchanov, Theory of random sets, 2005.

M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence. Random structures & algorithms, vol.6, pp.161-180, 1995.

M. Molloy and B. Reed, The size of the giant component of a random graph with a given degree sequence, Combinatorics, probability and computing, vol.7, issue.3, pp.295-305, 1998.

R. A. Neher, K. Mecke, and H. Wagner, Topological estimation of percolation thresholds, Journal of Statistical Mechanics: Theory and Experiment, issue.01, p.1011, 2008.

B. L. Okun, Euler characteristic in percolation theory, Journal of Statistical Physics, vol.59, issue.1, pp.523-527, 1990.

T. S. Package, Spatstat quick reference guide, 2017.

M. Penrose, The longest edge of the random minimal spanning tree, The Annals of Applied Probability, vol.7, issue.2, pp.340-361, 1997.

E. Roubin and J. Colliat, Critical probability of percolation over bounded region in N-dimensional Euclidean space, Journal of Statistical Mechanics: Theory and Experiment, vol.2016, issue.3, p.33306, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01305745

Y. A. Rozanov, Markov Random Fields, 1982.

R. Schneider and W. Weil, Stochastic and integral geometry, 2008.

C. Song, P. Wang, and H. A. Makse, A phase diagram for jammed matter, Nature, vol.453, issue.7195, pp.629-632, 2008.

D. Stauffer and A. Aharony, Introduction To Percolation Theory, 2003.

J. M. Stoyanov, Counterexamples in probability. Courier Corporation, 2013.

S. Torquato, O. Uche, and F. Stillinger, Random sequential addition of hard spheres in high euclidean dimensions, Physical Review E, vol.74, issue.6, p.61308, 2006.

B. Tsirelson, Measurability and Continuity, Lecture at the School of Mathematical Sciences, 2013.

R. Turner, Package deldir, 2017.

R. Van-der and . Hofstad, preparation, vol.2, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00901082

R. Van-der and . Hofstad, of Cambridge Series in Statistical and probabilistic Mathematics, vol.1, 2017.

H. W. Watson and F. Galton, On the probability of the extinction of families, The Journal of the Anthropological Institute of Great Britain and Ireland, vol.4, pp.138-144, 1875.