Directional decay of the Green’s function for a random nonnegative potential on \(\mathbb{Z}^d\). (English) Zbl 0938.60098

Zerner, Martin P. W.

Ann. Appl. Probab. 8, No. 1, 246-280 (1998).

The author studies a simple random walk \((S_n)_{n\in\mathbb N_0}\) on \(\mathbb Z^d\) (in continuous or discrete time), moving in a random i.i.d.field of nonnegative random variables \(\omega(x), x\in\mathbb Z^d\), which is assumed to be independent of the walk. A transformed path measure is introduced via the density \(\exp(-\sum_{n=0}^{m-1} \omega(S_n))\) (suitably normalized). This random path measure can be interpreted in terms of first-passage site percolation and is a model for random electric networks or random walk with random killing. The author derives a shape theorem for the decay rate of the Green’s function for the transformed walk as the end-to-end distance goes to infinity and proves that the variance of its negative logarithm is linear in this distance. Furthermore, he proves a large-deviation principle for the endpoint of the path at a fixed time that goes to infinity, almost surely w.r.t.the environment. The author brings together methods used in earlier work by Sznitman on Brownian motion in a Poissonian potential with ideas from first-passage percolation. Most of the results can be seen as spatially discrete analogs of that work, but the methods of the present paper also cover the case that the starting point grows with the end-to-end distance (“uniform shape theorem”).

Reviewer: W.König (Berlin)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82B43	Percolation

Keywords:

random walk; random potential; Green’s function; asymptotic shape; first passage percolation; Lyapunov exponent; large deviations

References:

[1]	Boivin, D. (1990). First passage percolation: the stationary case. Probab. Theory Related Fields 86 491-499. · Zbl 0685.60103
[2]	Carmona, R. and Lacroix, J. (1990). Spectral Theory of Random Schrödinger Operators. Birkhäuser, Basel. · Zbl 0717.60074
[3]	Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques. Jones and Bartlett, Boston. · Zbl 0793.60030
[4]	Doy le, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. MAA, Washington, D.C. · Zbl 0583.60065
[5]	Durrett, R. (1988). Lecture Notes on Particle Sy stems and Percolation. Wadsworth, Belmont, CA. · Zbl 0659.60129
[6]	Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Ann. Math. Stud. 109. Princeton University Press. · Zbl 0568.60057
[7]	Hughes, B. D. (1995). Random Walks and Random Environments 1. Clarendon Press, Oxford. · Zbl 0820.60053
[8]	Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Nordhoff, Groningen. · Zbl 0219.60027
[9]	Kesten, H. (1986). Aspects of first passage percolation. Ecole d’ Été de Probabilités de St. Flour. Lecture Notes in Math. 1180 125-264. Springer, Berlin. · Zbl 0602.60098
[10]	Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338. · Zbl 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. Birkhäuser, Boston. · Zbl 0814.60063
[12]	Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston. · Zbl 1228.60004
[13]	Liggett, T. (1985). Interacting Particle Sy stems. Springer, New York.
[14]	Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston. · Zbl 0780.60103
[15]	Molchanov, A. (1994). Lectures on random media. Ecole d’ Été de Probabilités de St. Flour. Lecture Notes in Math. 1581 242-411. Springer, Berlin. · Zbl 0814.60093
[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. · Zbl 0826.60001
[18]	Ross, S. M. (1983). Stochastic Processes. Wiley, New York. · Zbl 0555.60002
[19]	Stoy an, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester. · Zbl 0536.60085
[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. · Zbl 0814.60022
[21]	Sznitman, A. S. (1995). Crossing velocities and random lattice animals. Ann. Probab. 23 1006-1023. · Zbl 0926.60083
[22]	Sznitman, A. S. (1996). Distance fluctuations and Ly apounov exponents. Ann. Probab. 24 1507-1530. · Zbl 0871.60088
[23]	van den Berg, J. and Kesten, H. (1993). Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 3 56-80. · Zbl 0771.60092