TY - JOUR

T1 - Rate of convergence of the mean for sub-additive ergodic sequences

AU - Auffinger, Antonio

AU - Damron, Michael

AU - Hanson, Jack

N1 - Publisher Copyright:
© 2015 Elsevier Inc.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2015/11/5

Y1 - 2015/11/5

N2 - For sub-additive ergodic processes {Xm,n} with weak dependence, we analyze the rate of convergence of EX0,n/n to its limit g. We define an exponent γ given roughly by EX0,n~ng+nγ, and, assuming existence of a fluctuation exponent χ that gives VarX0,n~n2χ, we provide a lower bound for γ of the form γ≥χ. The main requirement is that χ≠1/2. In the case χ=1/2 and under the assumption VarX0,n=O(n/(log n)β) for some β>0, we prove γ≥χ-c(β) for a β-dependent constant c(β). These results show in particular that non-diffusive fluctuations are associated to non-trivial γ. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming χ exists. In the case of first-passage percolation in Zd, we provide a version of γ≥-1/2 without assuming existence of χ.

AB - For sub-additive ergodic processes {Xm,n} with weak dependence, we analyze the rate of convergence of EX0,n/n to its limit g. We define an exponent γ given roughly by EX0,n~ng+nγ, and, assuming existence of a fluctuation exponent χ that gives VarX0,n~n2χ, we provide a lower bound for γ of the form γ≥χ. The main requirement is that χ≠1/2. In the case χ=1/2 and under the assumption VarX0,n=O(n/(log n)β) for some β>0, we prove γ≥χ-c(β) for a β-dependent constant c(β). These results show in particular that non-diffusive fluctuations are associated to non-trivial γ. Various models, including first-passage percolation, directed polymers, the minimum of a branching random walk and bin packing, fall into our general framework, and the results apply assuming χ exists. In the case of first-passage percolation in Zd, we provide a version of γ≥-1/2 without assuming existence of χ.

KW - First-passage percolation

KW - Rate of convergence

KW - Sub-additive ergodic theory

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U2 - 10.1016/j.aim.2015.07.028

DO - 10.1016/j.aim.2015.07.028

M3 - Article

AN - SCOPUS:84939789303

VL - 285

SP - 138

EP - 181

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -