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Exponents governing the rarity of disjoint polymers in Brownian last passage percolation. (English) Zbl 07194974
Summary: In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two-thirds power of the interpolating distance. This two-thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit-order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of \(k\) disjoint polymers crossing a unit-order region while beginning and ending within a short distance \(\varepsilon\) of each other is bounded above by \(\varepsilon^{( k^2 - 1) / 2 + o (1)}\). This result, which we conjecture to be sharp, yields understanding of the uniform nature of the coalescence structure of polymers, and plays a foundational role in Hammond (Forum Math. Pi 7 (2019) e2, 69) in proving comparison on unit-order scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an on-scale articulation of the two-thirds power law for polymer geometry: polymers fluctuate by \(\varepsilon^{2 / 3}\) on short scales \(\varepsilon\).

MSC:
82C43 Time-dependent percolation in statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
82B23 Exactly solvable models; Bethe ansatz
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
82D60 Statistical mechanical studies of polymers
60J65 Brownian motion
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