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 $\epsilon$.

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

References in corpus (4)

• Fluctuation properties of the TASEP with periodic initial configuration
• The directed landscape
• Competition interfaces and second class particles
• Invariant Measures for TASEP with a Slow Bond

Cited by in corpus (14)

• The directed landscape
• Bulk properties of the Airy line ensemble
• Three-halves variation of geodesics in the directed landscape
• Local stationarity of exponential last passage percolation
• Optimal Exponent for Coalescence of Finite Geodesics in Exponential Last Passage Percolation
• The stationary horizon and semi-infinite geodesics in the directed landscape
• Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation
• Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison
• Tightness of discrete Gibbsian line ensembles with exponential interaction Hamiltonians
• Tightness of Bernoulli Gibbsian line ensembles
• Brownian regularity for the KPZ line ensemble
• Short- and long-time path tightness of the continuum directed random polymer
• Optimal tail estimates in $\beta$-ensembles and applications to last passage percolation
• When the geodesic becomes rigid in the directed landscape