Fracturing ranked surfaces

TL;DR

This paper investigates the properties of ranked surfaces and bridges in discretized landscapes, revealing a new tricritical point, fractal dimensions, and universality classes across different models and dimensions.

Contribution

It uncovers a new tricritical point associated with bridges in ranked surfaces and demonstrates their fractal nature and universality across models and dimensions.

Findings

Identifies a tricritical point at the percolation threshold p_c.

Finds bridges have a fractal dimension of approximately 1.22 in 2D.

Reveals a self-similar fracture line as p approaches 1.

Abstract

Discretized landscapes can be mapped onto ranked surfaces, where every element (site or bond) has a unique rank associated with its corresponding relative height. By sequentially allocating these elements according to their ranks and systematically preventing the occupation of bridges, namely elements that, if occupied, would provide global connectivity, we disclose that bridges hide a new tricritical point at an occupation fraction $p=p_{c}$, where $p_{c}$ is the percolation threshold of random percolation. For any value of $p$ in the interval $p_{c}< p \leq 1$, our results show that the set of bridges has a fractal dimension $d_{BB} \approx 1.22$ in two dimensions. In the limit $p \rightarrow 1$, a self-similar fracture is revealed as a singly connected line that divides the system in two domains. We then unveil how several seemingly unrelated physical models tumble into the same universality class and also present results for higher dimensions.