Infinite Paths on a Random Environment of \({\mathbb {Z}}^2\) with Bounded and Recurrent Sums

  • Emilio De SantisEmail author
  • Mauro Piccioni


This paper considers a random structure on the lattice \({\mathbb {Z}}^2\) of the following kind. To each edge e a random variable \(X_e\) is assigned, together with a random sign \(Y_e \in \{-1,+1\}\). For an infinite self-avoiding path on \({\mathbb {Z}}^2\) starting at the origin consider the sequence of partial sums along the path. These are computed by summing the \(X_e\)’s for the edges e crossed by the path, with a sign depending on the direction of the crossing. If the edge is crossed rightward or upward the sign is given by \(Y_e\), otherwise by \(-Y_e\). We assume that the sequence of \(X_e\)’s is i.i.d., drawn from an arbitrary common law and that the sequence of signs \(Y_e\) is independent, with independent components drawn from a law which is allowed to change from horizontal to vertical edges. First we show that, with positive probability, there exists an infinite self-avoiding path starting from the origin with bounded partial sums. Moreover the process of partial sums either returns to zero or at least it returns to any neighborhood of zero infinitely often. These results are somewhat surprising at the light of the fact that, under rather mild conditions, there exists with probability 1 two sites with all the paths joining them having the partial sums exceeding in absolute value any prescribed constant.


Oriented percolation Random environment Recurrence Graph algorithms Optimization 

Mathematics Subject Classification

60K35 82B44 



The Authors thank their own University that funded in 2018 the project “Simmetrie e Disuguaglianze in Modelli Stocastici” of which this paper is a part.


  1. 1.
    Auffinger, A., Damron, M., Hanson, J.: 50 Years of First-Passage Percolation. University Lecture Series, vol. 68. American Mathematical Society, Providence, RI (2017)CrossRefzbMATHGoogle Scholar
  2. 2.
    Balister, P., Bollobás, B., Stacey, A.: Improved upper bounds for the critical probability of oriented percolation in two dimensions. Random Struct. Algorithms 5(4), 573–589 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berry, A.C.: The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49, 122–136 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bhamidi, S., van der Hofstad, R., Hooghiemstra, G.: First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab. 20(5), 1907–1965 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Booth, L., Meester, R.: Infinite paths with bounded or recurrent partial sums. Probab. Theory Relat. Fields 120(1), 118–142 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Damron, M., Lam, W.-K., Wang, X.: Asymptotics for \(2D\) critical first passage percolation. Ann. Probab. 45(5), 2941–2970 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Davis, B., McDonald, D.: An elementary proof of the local central limit theorem. J. Theor. Probab. 8(3), 693–701 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    den Hollander, F.: Large Deviations. Fields Institute Monographs, vol. 14. American Mathematical Society, Providence, RI (2000)zbMATHGoogle Scholar
  9. 9.
    Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12(4), 999–1040 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Esseen, C.-G.: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77, 1–125 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grimmett, G.: Percolation. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 2nd edn. Springer-Verlag, Berlin (1999)Google Scholar
  12. 12.
    Häggström, O., Hirscher, T.: Water transport on infinite graphs. Random Struct. Algorithm 54, 515–527 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems: Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Knight, F.B.: On the absolute difference chains. Z. Wahrsch. Verw. Gebiete 43(1), 57–63 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lang, S.: Introduction to Diophantine Approximations, second edn. Springer-Verlag, New York (1995)CrossRefzbMATHGoogle Scholar
  16. 16.
    Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25(1), 71–95 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lindvall, T.: Lectures on the coupling method. Dover Publications Inc, Mineola, NY (2002)zbMATHGoogle Scholar
  18. 18.
    Peigné, M., Woess, W.: Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity. Colloq. Math. 125(1), 31–54 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Williams, D.: Probability with Martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics.University of Rome SapienzaRomeItaly

Personalised recommendations