Longest increasing paths with gaps

TL;DR

This paper investigates the maximal length of increasing paths with minimal gaps in Poisson and Bernoulli point processes, using couplings with known models to derive explicit shapes and fluctuation distributions.

Contribution

It introduces a novel variant of the Ulam-Hammersley problem with gap restrictions and provides explicit limiting shapes and fluctuation results via couplings.

Findings

Explicit limiting shapes for the paths in both models

Fluctuations follow Tracy-Widom distribution

Connections established with classical models

Abstract

We consider a variant of the continuous and discrete Ulam-Hammersley problems: we study the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path.For both cases (continuous and discrete) our approach rely on couplings with well-studied models: respectively the classical Ulam-Hammersley problem and last-passage percolation with geometric weights. Thanks to these couplings we obtain explicit limiting shapes in both settings.We also establish that, as in the classical Ulam-Hammersley problem, the fluctuations around the mean are given by the Tracy-Widom distribution.