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Abstract
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We develop a new probabilistic method for deriving deviation estimates in directed
planar polymer and percolation models. The key estimates are for exit points of
geodesics as they cross transversal down-right boundaries. These bounds are of
optimal cubic-exponential order. We derive them in the context of last-passage
percolation with exponential weights for a class of boundary conditions including the
stationary case. As a result, the probabilistic coupling method is empowered to treat
a variety of problems optimally, which could previously be achieved only via inputs
from integrable probability. As applications in the bulk setting, we obtain upper
bounds of cubic-exponential order for transversal fluctuations of geodesics, and
cube-root upper bounds with a logarithmic correction for distributional Busemann
limits and competition interface limits. Several other applications are already in the
literature.
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Keywords
Busemann limit, competition interface, coupling method,
corner growth model, exit point, geodesic, last-passage
percolation, path fluctuations, transversal fluctuations,
wandering exponent
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Mathematical Subject Classification
Primary: 60K35, 60K37
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Milestones
Received: 13 February 2022
Revised: 7 December 2022
Accepted: 26 January 2023
Published: 29 July 2023
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