Directional decay of the Green's function for a random nonnegative potential on ${\bf Z}\sp d$
Martin P. W. Zerner
Source: Ann. Appl. Probab. Volume 8, Number 1 (1998), 246-280.
Abstract
We derive a shape theorem type result for the almost sure exponential decay of the Green's function of $-\Delta + V$, where the potentials $V(x), x \epsilon \mathbb{Z}^d$ are i.i.d. nonnegative random variables. This result implies a large deviation principle governing the position of a d-dimensional random walk moving in the same potential.
Primary Subjects: 60K35, 82D30
Keywords: Random walk; random
potential; Green's function; asymptotic
shape; first passage percolation; Lyapounov
exponent; large deviations
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1027961043
Mathematical Reviews number (MathSciNet):
MR1620370
Digital Object Identifier: doi:10.1214/aoap/1027961043
Zentralblatt MATH identifier:
0938.60098
References
[1] Boivin, D. (1990). First passage percolation: the stationary case. Probab. Theory Related Fields 86 491-499.
Mathematical Reviews (MathSciNet):
MR92f:60177
[2] Carmona, R. and Lacroix, J. (1990). Spectral Theory of Random Schr¨odinger Operators. Birkh¨auser, Basel.
Mathematical Reviews (MathSciNet):
MR92k:47143
[3] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques. Jones and Bartlett, Boston.
Mathematical Reviews (MathSciNet):
MR95a:60034
[4] Doy le, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. MAA, Washington, D.C.
Mathematical Reviews (MathSciNet):
MR89a:94023
[5] Durrett, R. (1988). Lecture Notes on Particle Sy stems and Percolation. Wadsworth, Belmont, CA.
Mathematical Reviews (MathSciNet):
MR89k:60157
[6] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Ann. Math. Stud. 109. Princeton University Press.
Mathematical Reviews (MathSciNet):
MR87g:60066
[7] Hughes, B. D. (1995). Random Walks and Random Environments 1. Clarendon Press, Oxford.
[8] Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Nordhoff, Groningen.
[9] Kesten, H. (1986). Aspects of first passage percolation. Ecole d' ´Et´e de Probabilit´es de St. Flour. Lecture Notes in Math. 1180 125-264. Springer, Berlin.
[10] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338.
Mathematical Reviews (MathSciNet):
MR94m:60205
Zentralblatt MATH:
0783.60103
[11] Khanin, K. M., Mazel, A. E., Shlosman, S. B. and Sinai, Ya. G. (1994). Loop condensation effects in the behavior of random walks. In The Dy nkin Festschrift (M. Freidlin, ed.) 167-184. Birkh¨auser, Boston.
Mathematical Reviews (MathSciNet):
MR95m:60104
[12] Lawler, G. F. (1991). Intersections of Random Walks. Birkh¨auser, Boston.
[13] Liggett, T. (1985). Interacting Particle Sy stems. Springer, New York.
Mathematical Reviews (MathSciNet):
MR86e:60089
[14] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkh¨auser, Boston.
Mathematical Reviews (MathSciNet):
MR94f:82002
[15] Molchanov, A. (1994). Lectures on random media. Ecole d' ´Et´e de Probabilit´es de St. Flour. Lecture Notes in Math. 1581 242-411. Springer, Berlin.
[16] Newman, C. M. (1996). Private communication.
[17] Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford.
[18] Ross, S. M. (1983). Stochastic Processes. Wiley, New York.
[19] Stoy an, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.
Mathematical Reviews (MathSciNet):
MR85f:60147
[20] Sznitman, A. S. (1994). Shape theorem, Ly apounov exponents, and large deviations for Brownian motion in a Poissonian potential. Comm. Pure Appl. Math. 47 1655-1688.
Mathematical Reviews (MathSciNet):
MR96b:60217
[21] Sznitman, A. S. (1995). Crossing velocities and random lattice animals. Ann. Probab. 23 1006-1023.
Mathematical Reviews (MathSciNet):
MR96j:60066
Zentralblatt MATH:
0926.60083
[22] Sznitman, A. S. (1996). Distance fluctuations and Ly apounov exponents. Ann. Probab. 24 1507-1530.
Mathematical Reviews (MathSciNet):
MR98b:60175
[23] van den Berg, J. and Kesten, H. (1993). Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 3 56-80.
Mathematical Reviews (MathSciNet):
MR94a:60134
Zentralblatt MATH:
0771.60092
The Annals of Applied Probability
