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 twothirds power of the interpolating distance. This twothirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unitorder 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 unitorder 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 unitorder scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an onscale articulation of the twothirds power law for polymer geometry: polymers fluctuate by $\epsilon^{2/3}$ on short scales $\epsilon$.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 DOI:
 10.48550/arXiv.1709.04110
 arXiv:
 arXiv:1709.04110
 Bibcode:
 2017arXiv170904110H
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 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