Spatial tightness at the edge of Gibbsian line ensembles
Abstract
Consider a sequence of Gibbsian line ensembles, whose lowest labeled curves (i.e., the edge) have tight onepoint marginals. Then, given certain technical assumptions on the nature of the Gibbs property and underlying random walk measure, we prove that the entire spatial process of the edge is tight. We then apply this blackbox theory to the loggamma polymer Gibbsian line ensemble which we construct. The edge of this line ensemble is the transversal free energy process for the polymer, and our theorem implies tightness with the ubiquitous KPZ class $2/3$ exponent, as well as Brownian absolute continuity of all the subsequential limits. A key technical innovation which fuels our general result is the construction of a continuous grand monotone coupling of Gibbsian line ensembles with respect to their boundary data (entrance and exit values, and bounding curves). {\em Continuous} means that the Gibbs measure varies continuously with respect to varying the boundary data, {\em grand} means that all uncountably many boundary data measures are coupled to the same probability space, and {\em monotone} means that raising the values of the boundary data likewise raises the associated measure. This result applies to a general class of Gibbsian line ensembles where the underlying random walk measure is discrete time, continuous valued and logconvex, and the interaction Hamiltonian is nearest neighbor and convex.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.03045
 Bibcode:
 2021arXiv210103045B
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60J65;
 60K37;
 82B23
 EPrint:
 72 pages, 4 figures. We corrected some typos and a mistake in the proof of Lemma 2.13