# Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition

## TL;DR

This study introduces a lattice filling model of percolation with tunable growth probability, revealing a crossover from continuous to discontinuous phase transitions and identifying a tricritical region with changing critical exponents.

## Contribution

It presents a novel percolation model with a touch and stop growth rule that demonstrates a crossover from continuous to discontinuous transitions depending on growth probability.

## Key findings

- For g ≤ 0.5, the model shows continuous percolation transitions.
- For g ≥ 0.8, the model exhibits discontinuous percolation transitions.
- A tricritical region exists between g = 0.5 and g = 0.8 with evolving critical exponents.

## Abstract

A random growth lattice filling model of percolation with touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty sites and the clusters are grown from these nucleation centers with a tunable growth probability g. As the growth probability g is varied from 0 to 1 two distinct regimes are found to occur. For g\le 0.5, the model exhibits continuous percolation transitions as ordinary percolation whereas for g\ge 0.8 the model exhibits discontinuous percolation transitions. The discontinuous transition is characterized by discontinuous jump in the order parameter, compact spanning cluster and absence of power law scaling of cluster size distribution. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously till the crossover from continuous to discontinuous transition is completed.