TY - JOUR

T1 - Lyapunov Exponents of Random Walks in Small Random Potential

T2 - The Lower Bound

AU - Mountford, Thomas

AU - Mourrat, Jean Christophe

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/11

Y1 - 2013/11

N2 - We consider the simple random walk on ℤd, d ≥ 3, evolving in a potential of the form β V, where (V(x))x∈ℤd are i.i.d. random variables taking values in [0, + ∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -Δ+ β V.

AB - We consider the simple random walk on ℤd, d ≥ 3, evolving in a potential of the form β V, where (V(x))x∈ℤd are i.i.d. random variables taking values in [0, + ∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -Δ+ β V.

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U2 - 10.1007/s00220-013-1781-3

DO - 10.1007/s00220-013-1781-3

M3 - Article

AN - SCOPUS:84884816407

VL - 323

SP - 1071

EP - 1120

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -