Electronic Journal of Probability

Finitely dependent insertion processes

Avi Levy

Full-text: Open access

Abstract

A $q$-coloring of $\mathbb Z$ is a random process assigning one of $q$ colors to each integer in such a way that consecutive integers receive distinct colors. A process is $k$-dependent if any two sets of integers separated by a distance greater than $k$ receive independent colorings. Holroyd and Liggett constructed the first stationary $k$-dependent $q$-colorings by introducing an insertion algorithm on the complete graph $K_q$. We extend their construction from complete graphs to weighted directed graphs. We show that complete multipartite analogues of $K_3$ and $K_4$ are the only graphs whose insertion process is finitely dependent and whose insertion algorithm is consistent. In particular, there are no other such graphs among all unweighted graphs and among all loopless complete weighted directed graphs. Similar results hold if the consistency condition is weakened to eventual consistency. Finally we show that the directed de Bruijn graphs of shifts of finite type do not yield $k$-dependent insertion processes, assuming eventual consistency.

Article information

Source
Electron. J. Probab. Volume 22 (2017), paper no. 32, 19 pp.

Dates
Received: 3 April 2016
Accepted: 1 February 2017
First available in Project Euclid: 12 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1491962643

Digital Object Identifier
doi:10.1214/17-EJP35

Subjects
Primary: 60G10: Stationary processes 05C15: Coloring of graphs and hypergraphs 60C05: Combinatorial probability

Keywords
proper coloring directed graph weighted graph m-dependence stationary process shift of finite type

Rights
Creative Commons Attribution 4.0 International License.

Citation

Levy, Avi. Finitely dependent insertion processes. Electron. J. Probab. 22 (2017), paper no. 32, 19 pp. doi:10.1214/17-EJP35. https://projecteuclid.org/euclid.ejp/1491962643


Export citation

References

  • [1] J. Aaronson, D. Gilat, and M. Keane. On the structure of $1$-dependent Markov chains. J. Theoret. Probab., 5(3):545–561, 1992.
  • [2] J. Aaronson, D. Gilat, M. Keane, and V. de Valk. An algebraic construction of a class of one-dependent processes. Ann. Probab., 17(1):128–143, 1989.
  • [3] N. Alon and O. N. Feldheim. A note on general sliding window processes. Electron. Commun. Probab., 19(66), 7, 2014.
  • [4] B. Bollobás. Modern graph theory, volume 184 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998.
  • [5] C. Borgs, J. Chayes, L. Lovász, V. T. Sós, and K. Vesztergombi. Counting graph homomorphisms. In Topics in discrete mathematics, volume 26 of Algorithms Combin., pages 315–371. Springer, Berlin, 2006.
  • [6] A. Borodin, P. Diaconis, and J. Fulman. On adding a list of numbers (and other one-dependent determinantal processes). Bull. Amer. Math. Soc. (N.S.), 47(4):639–670, 2010.
  • [7] R. M. Burton, M. Goulet, and R. Meester. On $1$-dependent processes and $k$-block factors. Ann. Probab., 21(4):2157–2168, 1993.
  • [8] N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch., Proc., 49:758–764 = Indagationes Math., 8:461–467, 1946.
  • [9] V. de Valk. One-dependent processes: two-block factors and non-two-block factors, volume 85 of CWI Tract. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1994.
  • [10] R. Durrett. Probability: theory and examples, volume 31 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fourth edition, 2010.
  • [11] A. E. Holroyd. One-dependent coloring by finitary factors, arXiv:1411.1463
  • [12] A. E. Holroyd and T. M. Liggett. Symmetric 1-dependent colorings of the integers. Electron. Commun. Probab., 20(31), 8, 2015.
  • [13] A. E. Holroyd and T. M. Liggett. Finitely dependent coloring. Forum Math. Pi, 4:e9, 43, 2016.
  • [14] A. E. Holroyd, O. Schramm, and D. B. Wilson. Finitary coloring, arXiv:1412.2725
  • [15] I. A. Ibragimov and Y. V. Linnik. Independent and stationarily connected variables. Izdat. “Nauka”, Moscow, 1965.
  • [16] I. A. Ibragimov and Y. V. Linnik. Independent and stationary sequences of random variables. Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman.
  • [17] S. Janson. Runs in $m$-dependent sequences. Ann. Probab., 12(3):805–818, 1984.
  • [18] O. Kallenberg. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002.
  • [19] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995.