2017/09/12 by Alan Hammond, Hammond, Alan
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1709.04115
openalex publication_date 2017/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In last passage percolation models lying in the KPZ universality class, the\nenergy of long energy-maximizing paths may be studied as a function of the\npaths' pair of endpoint locations. Scaled coordinates may be introduced, so\nthat these maximizing paths, or polymers, now cross unit distances with\nunit-order fluctuations, and have scaled energy, or weight, of unit order. In\nthis article, we consider Brownian last passage percolation in these scaled\ncoordinates. In the narrow wedge case, one endpoint of such polymers is fixed,\nsay at (0,0) \∈ \ℝ2, and the other is varied horizontally, over\n(z,1), z \∈ \ℝ, so that the polymer weight profile may be studied\nas a function of z \∈ \ℝ. This profile is known to manifest a\none-half power law, having 1/2--H "older continuity. The polymer weight\nprofile may be defined beginning from a much more general initial condition. In\nthis article, we present a more general assertion of this one-half power law,\nas well as a bound on the poly-logarithmic correction. For a very broad class\nof initial data, the polymer weight profile has a modulus of continuity of the\norder of x1/2 \( \log x-1 \)2/3, with a high degree of\nuniformity in the scaling parameter and the initial condition.\n