Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 5 (2008) open journal systems 


Existence and spatial limit theorems for lattice and continuum particle systems

Mathew D. Penrose, University of Bath


Abstract
We give a general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site. We allow for jump events at a site that affect the state of its neighbours. We give a law of large numbers and functional central limit theorem for additive set functions taken over an increasing family of subcubes of Zd. We discuss application to marked spatial point processes with births, deaths and jumps of particles, in particular examples such as continuum and lattice ballistic deposition and a sequential model for random loose sphere packing.

AMS 2000 subject classifications: 60K35, 60F17.

Keywords: Interacting particle system, functional central limit theorem, point process.

Creative Common LOGO

Full Text: PDF


Penrose, Mathew D., Existence and spatial limit theorems for lattice and continuum particle systems, Probability Surveys, 5, (2008), 1-36 (electronic). DOI: 10.1214/07-PS112.

References

[1]     Andjel, E. D. (1982) Invariant measures for the zero range processes. Ann. Probab. 10, 525–547. MR0659526

[2]     R. Atar, S. Athreya and M. Kang (2001) Ballistic deposition on a planar strip. Elec. Comm. Probab. 6, 31–38. MR1826165

[3]     Baddeley, A. and Møller, J. (1989) Nearest-Neighbour Markov Point processes and random sets. Internat. Statist. Rev. 57, 89–121.

[4]     Balázs, M., Rassoul-Agha, F., Seppäläinen, T. and Sethuraman, S. (2007) Existence of the zero range process and a deposition model with superlinear growth rates. Ann. Probab. 35 1201–1249. MR2330972

[5]     A.-L. Barabási and H. E. Stanley (1995) Fractal Concepts in Surface Growth. Cambridge University Press. MR1600794

[6]     Baryshnikov, Y. and Yukich, J. E. Gaussian fields and random packing. J. Stat. Physics, 111, 443–463. MR1964280

[7]     Basis, V. Ya. Infinite-dimensional Markov processes with almost local interaction of the components. Theory Probab. Appl. 21, 706–720. MR0426218

[8]     Basis, V. Ya. (1980) Stationarity and ergodicity of Markov interacting processes. Multicomponent random systems, pp. 37–58, Adv. Probab. Related Topics 6 Dekker, New York, 1980. MR0599532

[9]     J. D. Bernal (1959) A geometrical approach to the structure of liquids. Nature, 183 141–147.

[10]     J. D. Bernal and J. Mason (1960) Co-ordination of randomly packed spheres. Nature, 188, 910–911.

[11]     Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York. MR0233396

[12]     Chen, M. F. (1992) From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore. MR2091955

[13]     H. S. Choi, J. Talbot, G. Tarjus and P. Viot (1993) First-layer formation in ballistic deposition of spherical particles: Kinetics and structure. J. Chem. Phys. 99, 9296–9303.

[14]     D. J. Cumberland and R.J. Crawford (1987) The Packing of Particles. Handbook of Powder Technology, 6, Elsevier, AMsterdam.

[15]     Davis, M. H. A. (1993) Markov Models and Optimization. Chapman and Hall, London. MR1283589

[16]     De Masi, A. and Presutti, E. (1992) Mathematical Methods for Hydrodynamic Limits. Springer, Berlin.

[17]     Dhar, D. (1999) The Abelian sandpiles and related models. Phys. A 263, 4–25.

[18]     Doukhan, P., Lang, G., Louhichi, S. and Ycart, B. (2005). A functional central limit theorem for interacting particle systems on transitive graphs. Markov Proc. Rel. Fields, to appear. Preprint available at ArXiv.math-ph/0509041.

[19]     R. Durrett (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Pacific Grove, CA. MR0940469

[20]     E. B. Dynkin (1965). Markov Processes, Volume I. Springer, Berlin.

[21]     J. W. Evans (1993) Random and cooperative adsorption. Rev. Modern Phys. 65, 1281–1329.

[22]     Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York. MR0838085

[23]     Ferrari, P. A., Fernández, R., and Garcia, N. L. (2002) Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Applns. 102, 63–88. MR1934155

[24]     Fristedt, B. and Gray, L. (1997) A Modern Approach to Probability Theory. Birkhäuser, Boston. MR1422917

[25]     Garcia, N. L. and Kurtz, T. G. (2006) Spatial birth and death processes as solutions of stochastic equations. Alea 1, 288–303. MR2249658

[26]     Glötzl, E. (1981). Time-reversible and Gibbsian point processes I. Markovian and spatial birth and death processes on a general phase space. Math. Nachr. 102, 217–222. MR0642153

[27]     Glötzl, E. (1982). Time-reversible and Gibbsian point processes II. Markovian particle jump processes on a general phase space. Math. Nachr. 106, 63–71. MR0675745

[28]     Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture notes in Math. 724, Springer, Berlin. MR0538077

[29]     A. Gervois, M. Lichtenberg, L. Oger, and E. Guyon (1989) Coordination number of disordered packings of identical spheres. J. Phys. A: Math, Gen. 22 2119–2131. MR1004917

[30]     K. Gotoh and J. L. Finney (1974) Statistical geometrical approach to random packing density of equal spheres. Nature, 252, 202–205.

[31]     Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89. MR0307392

[32]     Holley, R. (1972) Markovian interaction processes with finite range interactions. Ann. Math. Stat. 43, 1961–1967. MR0373070

[33]     Holley, R. and Stroock, D. W. (1978) Nearest neighbour birth and death processes on the real line. Acta Math. 140, 103–154. MR0488380

[34]     Holley, R. ad Stroock, D. W. (1979) Central limit phenomena of various interacting systems. Ann. Math. 110, 333–393. MR0549491

[35]     Jacobsen, M. (2006) Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser. MR2189574

[36]     W. S. Jodrey and E. M. Torey (1985) Computer simulation of close random packing of equal spheres. Phys. Review A, 32, 2347–2351.

[37]     R. Jullien and P. Meakin (1987) Simple three-dimensional models for ballistic deposition and restructuring. Europhys. Lett. 4, 1385–1390.

[38]     Kallenberg, O. (2002) Foundations of Modern Probability, 2nd ed., Springer, New York. MR1876169

[39]     Liggett, T. M. (1972) Existence theorems for infinite particle systems. Transactions Amer. Math. Soc. 165, 471–481. MR0309218

[40]     Liggett, T. M. (1985) Interacting Particle Systems. Springer, Berlin. MR0776231

[41]     Liggett, T. M. and Spitzer, F. (1981) Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56 443–468. MR0621659

[42]     Maes, C. M., Redig, F. and Saada, E. (2002) The abelian sandpile model on an infinite tree. Ann. Probab. 30, 2081–2107. MR1944016

[43]     Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton. MR2004226

[44]     Onoda, G. and Liniger, E. (1990) Random loose packings of uniform spheres and the dilatancy onset. Phys. Rev. Letters, 64, 2727–2730.

[45]     Penrose, M. D. (2001) Random parking, sequential adsorption, and the jamming limit. Comm. Math. Phys. 218, 153–176. MR1824203

[46]     Penrose, M. D. (2001) Limit theory for monolayer ballistic deposition in the continuum. J. Statist. Phys. 105, 561–583. MR1871657

[47]     Penrose, M. D. (2002) Limit theory for monotonic particle systems and sequential deposition. Stoch. Proc. Appln. 98 175–197. MR1887532

[48]     Penrose, M. D. (2005) Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33, 1945–1991. MR2165584

[49]     Penrose, M.D. and Sudbury, A. (2005) Exact and approximate results for deposition and annihilation processes on graphs. Ann. Appl. Probab. 15, 853–889 (2005). MR2114992

[50]     Penrose, M. D. and Yukich, J. E. (2002) Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12, 232–301. MR1890065

[51]     Penrose, M. D. and Yukich, J. E. (2001) Mathematics of random growing interfaces. J. Phys. A: Math. Gen. 34, 6239–6247. MR1862710

[52]     Preston, C. J. (1975) Spatial birth-and-death processes. Bull. Int. Statist. Inst. 46 (2) 371–391. MR0474532

[53]     Privman, V. (2000) Adhesion of Submicron Particles on Solid Surfaces, Colloids and Surfaces A, 165. Special Issue, edited by V. Privman.

[54]     Qi, X. (2007) Functional central limit theorem for spatial birth and death processes. Preprint, University of Wisconsin-Madison.

[55]     Scott, G. D. (1960) Packing of spheres. Nature, 188, 908–909.

[56]     B. Senger, J.-C. Voegel, P. Schaaf (2000) Irreversible adsorption of colloidal particles on solid substrates. Colloids and Surfaces A 165, 255–285.

[57]     Seppäläinen, T. (2000) Strong law of large numbers for the interface in ballistic deposition. Ann. Inst. H. Poincaré Probab. Statist. 36, 691–736. MR1797390

[58]     Sidoravićius, V. and Vares, M. E. (1995) Ergodicity of Spitzer’s renewal model. Stochastic Process. Appl. 55 119–130. MR1312152

[59]     H. Solomon (1967) Random Packing Density. Proc. Fifth Berkeley Symp. on Prob. and Stat. 3, 119–134, Univ. California Press. MR0256531

[60]     Spohn, H. (1991) Large scale dynamics of interacting particles. Springer, Berlin

[61]     T. Vicsek (1989) Fractal Growth Phenomena. World Scientific, Singapore. MR1020762

[62]     Williams, D. (1991) Probability with Martingales. Cambridge University Press, Cambridge. MR1155402

[63]     Zong, C. (1999) Sphere Packings. Springer, New York. MR1707318




Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787