The Annals of Applied Probability

Entropy-controlled Last-Passage Percolation

Quentin Berger and Niccolò Torri

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Abstract

We introduce a natural generalization of Hammersley’s Last-Passage Percolation (LPP) called Entropy-controlled Last-Passage Percolation (E-LPP), where points can be collected by paths with a global (path-entropy) constraint which takes into account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley’s LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in (Berger and Torri (2018)). We give applications in this context, which are essentials tools in (Berger and Torri (2018)): we show that the limiting variational problem conjectured in (Ann. Probab. 44 (2016) 4006–4048), Conjecture 1.7, is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in (Comm. Pure Appl. Math. 64 (2011) 183–204; Probab. Theory Related Fields 137 (2007) 227–275).

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1878-1903.

Dates
Received: May 2018
Revised: October 2018
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566845

Digital Object Identifier
doi:10.1214/18-AAP1448

Mathematical Reviews number (MathSciNet)
MR3914559

Zentralblatt MATH identifier
07057469

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments 60F05: Central limit and other weak theorems

Keywords
Last-passage percolation heavy-tail distributions path entropy

Citation

Berger, Quentin; Torri, Niccolò. Entropy-controlled Last-Passage Percolation. Ann. Appl. Probab. 29 (2019), no. 3, 1878--1903. doi:10.1214/18-AAP1448. https://projecteuclid.org/euclid.aoap/1550566845


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