Open Access
2023 The geometry of near ground states in Gaussian polymer models
Shirshendu Ganguly, Alan Hammond
Author Affiliations +
Electron. J. Probab. 28: 1-80 (2023). DOI: 10.1214/22-EJP895


The energy and geometry of maximizing paths in integrable last passage percolation models are governed by the characteristic KPZ scaling exponents of one-third and two-thirds. When represented in scaled coordinates that respect these exponents, this random field of paths may be viewed as a complex energy landscape. We investigate the structure of valleys and connecting pathways in this landscape. The routed weight profile RRassociates to xR the maximum scaled energy obtainable by a path whose scaled journey from (0,0) to (0,1) passes through the point (x,12). Developing tools of Brownian Gibbs analysis from [Ham22] and [CHH19], we prove an assertion of strong similarity of this profile for Brownian last passage percolation to Brownian motion of rate two on the unit-order scale. A sharp estimate on the rarity that two macroscopically different routes in the energy landscape offer energies close to the global maximum results. We prove robust assertions concerning modulus of continuity for the energy and geometry of scaled maximizing paths, that develop the results and approach of [HS20], delivering estimates valid on all scales above the microscopic. The geometry of excursions of near ground states about the maximizing path is investigated: indeed, we estimate the energetic shortfall of scaled paths forced to closely mimic the geometry of the maximizing route while remaining disjoint from it. We also provide bounds on the approximate gradient of the maximizing path, viewed as a function, ruling out sharp steep movement down to the microscopic scale. Our results find application in a companion study [GH20a] of the stability, and fragility, of last passage percolation under a dynamical perturbation.

Funding Statement

The first author is partially supported by NSF grant DMS-1855688, NSF CAREER Award DMS-1945172, and a Sloan Research Fellowship. The second author is supported by the NSF under DMS grants 1855550 and 2153359 and by a Miller Professorship from the Miller Institute for Basic Research in Science at U.C. Berkeley.


The authors thank the referee for a detailed report. They also thank Milind Hegde for several useful discussions.


Download Citation

Shirshendu Ganguly. Alan Hammond. "The geometry of near ground states in Gaussian polymer models." Electron. J. Probab. 28 1 - 80, 2023.


Received: 29 January 2022; Accepted: 20 December 2022; Published: 2023
First available in Project Euclid: 2 May 2023

Digital Object Identifier: 10.1214/22-EJP895

Keywords: Brownian last passage percolation , dynamical models , geodesic excursions , geodesic modulus of continuity , stability and chaos

Vol.28 • 2023
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