Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
Abstract
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 $\epsilon$ of each other is bounded above by $\epsilon^{(k^2 - 1)/2 \, + \, o(1)}$. 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 [Ham17c] 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 $\epsilon^{2/3}$ on short scales $\epsilon$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2017
- DOI:
- 10.48550/arXiv.1709.04110
- arXiv:
- arXiv:1709.04110
- Bibcode:
- 2017arXiv170904110H
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics
- E-Print:
- 67 pages with eight figures. An ancillary file contains the latex source code for a version, available on the author's webpage, that contains an appendix in which explicit bounds on certain constants are derived. Some typographical errors corrected in this version