Critical condition of the water-retention model

TL;DR

This paper investigates the water retention capacity of a random landscape modeled on a 2D lattice, identifying a phase transition point where the system retains a large volume of water, and relates it to percolation theory.

Contribution

It introduces a phase transition analysis for water retention in a random height landscape and connects the criticality to the two-dimensional percolation universality class.

Findings

Identifies the critical point for water retention in the model.

Shows the criticality belongs to the 2D percolation universality class.

Provides a universal upper bound for water-covered area in large systems.

Abstract

We study how much water can be retained without leaking through boundaries when each unit square of a two-dimensional lattice is randomly assigned a block of unit bottom area but with different heights from zero to $n-1$. As more blocks are put into the system, there exists a phase transition beyond which the system retains a macroscopic volume of water. We locate the critical points and verify that the criticality belongs to the two-dimensional percolation universality class. If the height distribution can be approximated as continuous for large $n$, the system is always close to a critical point and the fraction of the area below the resulting water level is given by the percolation threshold. This provides a universal upper bound of areas that can be covered by water in a random landscape.