# The geometry of a critical percolation cluster on the UIPT

## TL;DR

This paper investigates the geometric properties of critical percolation clusters on the uniform infinite planar triangulation, focusing on tail distributions of key metrics and highlighting differences from previous models.

## Contribution

It provides new tail distribution exponents for critical clusters on the UIPT, revealing differences from earlier half-plane triangulation results.

## Key findings

- Derived tail exponents for peeling time, perimeter, and volume.
- Identified a factor of 2 difference in exponents compared to previous work.
- Enhanced understanding of critical percolation geometry on UIPT.

## Abstract

We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical cluster. The exponents obtained here differs by a factor 2 from those computed previously by Angel and Curien (2015) in the case of critical site percolation on the uniform infinite half-plane triangulation.