The Annals of Probability

Greedy lattice animals: negative values and unconstrained maxima

Amir Dembo, Alberto Gandolfi, and Harry Kesten

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Abstract

Let $\{X_v, v \in \mathbb{Z}^d\}$ be i.i.d. random variables, and $S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice animal $\xi$. Let $N_n = \max\{S(\xi) : |\xi| = n$ \text{and $\xi$ contains the origin}\}$ and $G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$ . We show that, regardless of the negative tail of the distribution of $X_v$ , if $\mathbf{E}( X_v^+)^d (\log^+ X_v^+))^{d+a} < + \infty$ for some $a>0$, then first, $\lim_n n^{-1} N_n = N exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of $G_n$ depending on the sign of $N$: if $N > 0$ then $G_n \approx n^d$, and if $N < 0$ then $G_n \le cn$, for some $c > 0$. The exact behavior of $G_n$ in this last case depends on the positive tail of the distribution of $X_v$; we show that if it is nontrivial and has exponential moments, then $G_n \approx \log n$, with a transition from $G_n \approx n^d$ occurring in general not as predicted by large deviations estimates. Finally, if $x^d(1 - F(x)) \to \infty$as $x \to \infty$, then no transition takes place.

Article information

Source
Ann. Probab., Volume 29, Number 1 (2001), 205-241.

Dates
First available in Project Euclid: 21 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aop/1008956328

Digital Object Identifier
doi:10.1214/aop/1008956328

Mathematical Reviews number (MathSciNet)
MR1825148

Zentralblatt MATH identifier
1016.60048

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F10: Large deviations 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Optimization lattice animals percolation

Citation

Dembo, Amir; Gandolfi, Alberto; Kesten, Harry. Greedy lattice animals: negative values and unconstrained maxima. Ann. Probab. 29 (2001), no. 1, 205--241. doi:10.1214/aop/1008956328. https://projecteuclid.org/euclid.aop/1008956328


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References

  • Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036-1048.
  • Arratia, R. and Waterman, M. S. (1994). A phase transition for the score in matching random sequences allowing deletions. Ann. Appl. Probab. 4 200-225.
  • Cox, J. T., Gandolfi, A., Griffin, P. and Kesten, H. (1993). Greedy lattice animals I: Upper bounds. Ann. Appl. Probab. 3 1151-1169.
  • Cox, J. T. and Kesten, H. (1981). On the continuity of the time constant of first-passage percolation. J. Appl. Probab. 18 809-819.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, Berlin.
  • Deuschel, J.-D. and Pisztora, A. (1996). Surface order deviations for high-density percolation. Probab. Theory Related Fields 104 467-482.
  • Dunford, N. and Schwartz, J. T. (1958). Linear Operators, Part I. Interscience, New York.
  • Fisher, M. E. and Sykes, M. F. (1959). Excluded-volume problem and the Ising model of ferromagnetism. Phys. Rev. (2) 114 45-58.
  • Fontes, L. and Newman, C. M. (1993). First-passage percolation for random colorings of d. Ann. Appl. Probab. 3 746-762.
  • Gandolfi, A. and Kesten, H. (1994). Greedy lattice animals II: Linear growth. Ann. Appl. Probab. 4 76-107.
  • Grimmett, G. (1999). Percolation, 2nd ed. Springer, Berlin.
  • Howard, C. D. and Newman, C. M. (1999). From greedy lattice animals to Euclidean first-passage percolation. In Perplexing Problems in Probability (M. Bramson and R. Durrett, eds.) 107-119. Birkh¨auser, Boston.
  • Jiang, T. (1999). Maxima of partial sums indexed by geometric structures. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
  • Karlin, S., Dembo, A. and Kawabata, T. (1990). Statistical composition of high scoring segments from molecular sequences. Ann. Statist. 18 571-581.
  • Kesten, H. (1986). Aspects of first passage percolation. Lecture Notes in Math. 1180 125-264. Springer, Berlin.
  • Lee, S. (1993). An inequality for greedy lattice animals. Ann. Appl. Probab. 3 1170-1188.
  • Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71-95.
  • Martin, J. (2000). Linear growth for greedy lattice animals. Preprint.
  • Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. de l'I.H.E.S. 81 73-205.