Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Concentration inequalities for a removal-driven thinning process

Authors: Joe Klobusicky and Govind Menon
Journal: Quart. Appl. Math. 75 (2017), 677-696
MSC (2010): Primary 35R60, 60K25, 82C23, 82C70
DOI: https://doi.org/10.1090/qam/1474
Published electronically: June 5, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening introduced by the first author (2014). The system consists of $ n$ particles in $ (0,\infty )$ that move at unit speed to the left. Each time a particle hits the boundary point 0, it is removed from the system along with a second particle chosen uniformly from the particles in $ (0,\infty )$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $ f_0(x) \in L^1_+(0,\infty )$, the empirical measure of the particle system at time $ t$ is shown to converge to the measure with density $ f(x,t)$, where $ f$ is the unique solution to the kinetic equation with nonlinear boundary coupling

$\displaystyle \partial _t f (x,t) - \partial _x f(x,t) = -\frac {f(0,t)}{\int _0^\infty f(y,t)\, dy} f(x,t), \quad 0<x < \infty ,$    

and initial condition $ f(x,0)=f_0(x)$.

The proof relies on a concentration inequality for an urn model studied by Pittel, and Maurey's concentration inequality for Lipschitz functions on the permutation group.

References [Enhancements On Off] (What's this?)

  • [1] K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp, and S. Taasan, Critical events, entropy, and the grain boundary character distribution, Physical Review B, 83 (2011), p. 134117.
  • [2] C. Beenakker, Evolution of two-dimensional soap-film networks, Physical Review Letters, 57 (1986), p. 2454.
  • [3] Albert Cohen, A probabilistic analysis of two dimensional grain growth, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)-Carnegie Mellon University. MR 2710785
  • [4] M. H. A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353-388. With discussion. MR 790622
  • [5] R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. MR 1932358
  • [6] A. Dvoretzky, J. Kiefer, and J. Wolfowitz, Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, Ann. Math. Statist. 27 (1956), 642-669. MR 0083864, https://doi.org/10.1214/aoms/1177728174
  • [7] Matt Elsey, Selim Esedoá¸lu, and Peter Smereka, Large-scale simulation of normal grain growth via diffusion-generated motion, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2126, 381-401. MR 2748098, https://doi.org/10.1098/rspa.2010.0194
  • [8] Philippe Flajolet, Joaquim Gabarró, and Helmut Pekari, Analytic urns, Ann. Probab. 33 (2005), no. 3, 1200-1233. MR 2135318, https://doi.org/10.1214/009117905000000026
  • [9] H. Flyvbjerg, Model for coarsening froths and foams, Physical Review E, 47 (1993), p. 4037.
  • [10] V. Fradkov, M. Glicksman, M. Palmer, J. Nordberg, and K. Rajan, Topological rearrangements during 2D normal grain growth, Physica D: Nonlinear Phenomena, 66 (1993), pp. 50-60.
  • [11] Reiner Henseler, Michael Herrmann, Barbara Niethammer, and Juan J. L. Velázquez, A kinetic model for grain growth, Kinet. Relat. Models 1 (2008), no. 4, 591-617. MR 2448609, https://doi.org/10.3934/krm.2008.1.591
  • [12] Donald E. Knuth, John McCarthy, Walter Stromquist, Daniel H. Wagner, and Tim Hesterberg, Problems and Solutions: Solutions: E3429, Amer. Math. Monthly 99 (1992), no. 7, 684. MR 1542179
  • [13] H.-K. Hwang, M. Kuba, and A. Panholzer, Analysis of some exactly solvable diminishing urn models, Formal Power Series and Algebraic Combinatorics (Tianjin, China, 2007).
  • [14] J. Klobusicky, Kinetic limits of piecewise deterministic Markov processes and grain boundary coarsening, Ph.D. thesis, Brown University, Providence, RI, 2014.
  • [15] D. Knuth and J. McCarthy, Problem e3429: Big pills and little pills, American Mathematical Monthly, 98 (1991), p. 264.
  • [16] A. N. Kolmogorov and V. M. Tihomirov, $ \varepsilon $-entropy and $ \varepsilon $-capacity of sets in function spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3-86 (Russian). MR 0112032
  • [17] M. Marder, Soap-bubble growth, Physical Review A, 36 (1987), p. 438.
  • [18] J. Mason, E. Lazar, R. MacPherson, and D. J. Srolovitz, Statistical topology of cellular networks in two and three dimensions, Physical Review E, 86 (2012), p. 051128.
  • [19] Bernard Maurey, Construction de suites symétriques, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 14, A679-A681 (French, with English summary). MR 533901
  • [20] Govind Menon, Barbara Niethammer, and Robert L. Pego, Dynamics and self-similarity in min-driven clustering, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6591-6618. MR 2678987, https://doi.org/10.1090/S0002-9947-2010-05085-8
  • [21] Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
  • [22] W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956), 900-904. MR 0078836
  • [23] B. Pittel, An urn model for cannibal behavior, J. Appl. Probab. 24 (1987), no. 2, 522-526. MR 889816
  • [24] Takumi Saegusa and Jon A. Wellner, Weighted likelihood estimation under two-phase sampling, Ann. Statist. 41 (2013), no. 1, 269-295. MR 3059418, https://doi.org/10.1214/12-AOS1073
  • [25] John von Neumann, Collected works. Vol. VI: Theory of games, astrophysics, hydrodynamics and meteorology, General editor: A. H. Taub. A Pergamon Press Book, The Macmillan Co., New York, 1963. MR 0157876

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35R60, 60K25, 82C23, 82C70

Retrieve articles in all journals with MSC (2010): 35R60, 60K25, 82C23, 82C70

Additional Information

Joe Klobusicky
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180
Email: klobuj@rpi.edu

Govind Menon
Affiliation: Division of Applied Mathematics, Box F, Brown University, Providence, Rhode Island 02912
Email: govind_menon@brown.edu

DOI: https://doi.org/10.1090/qam/1474
Keywords: Piecewise-deterministic Markov process, functional law of large numbers, diminishing urns
Received by editor(s): April 27, 2017
Published electronically: June 5, 2017
Article copyright: © Copyright 2017 Brown University

American Mathematical Society