# Vanishing of the anchored isoperimetric profile in bond percolation at p c

## TL;DR

This paper investigates the behavior of the anchored isoperimetric profile in bond percolation at the critical probability, showing that if the profile's limit exists at criticality, it must be zero, extending previous definitions to finite boxes.

## Contribution

It extends the definition of the anchored isoperimetric profile to the critical point and provides a partial result indicating its vanishing at criticality.

## Key findings

- The anchored isoperimetric profile at p_c, if the limit exists, must be zero.
- Extension of the profile's definition to finite boxes at p_c.
- Partial proof linking the profile's limit to its vanishing at criticality.

## Abstract

We consider the anchored isoperimetric profile of the infinite open cluster, defined for $p > p\_c$, whose existence has been recently proved in [3]. We extend adequately the definition for $p = p\_c$, in finite boxes. We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at $p\_c$ exists, it has to vanish.