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

Antonio Auffinger, Michael Damron*, Jack Hanson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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~n, 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.

Original languageEnglish (US)
Pages (from-to)138-181
Number of pages44
JournalAdvances in Mathematics
StatePublished - Nov 5 2015


  • First-passage percolation
  • Rate of convergence
  • Sub-additive ergodic theory

ASJC Scopus subject areas

  • Mathematics(all)

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