## The Annals of Applied Probability

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

### Entropy-controlled Last-Passage Percolation

Quentin Berger and Niccolò Torri

#### 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