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Order of the variance in the discrete Hammersley process with boundaries

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Version 2 2023-06-12, 09:07
Version 1 2023-06-09, 17:54
journal contribution
posted on 2023-06-12, 09:07 authored by Federico Ciech, Nicos GeorgiouNicos Georgiou
We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope. We show that along characteristic directions the order of the variance of the last passage time is of order N^{2/3} in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N) in the boundary model and to O(1) for the non-boundary model. This behaviour is the result of the two flat edges of the shape function.

Funding

The flat edge in last passage percolation; G2031; EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCIL; EP/P021409/1

History

Publication status

  • Published

File Version

  • Published version

Journal

Journal of Statistical Physics

ISSN

0022-4715

Publisher

Springer

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2019-05-24

First Open Access (FOA) Date

2019-06-07

First Compliant Deposit (FCD) Date

2019-05-23

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