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 -