# Local convergence of random planar graphs

## TL;DR

This paper investigates the local structure of large random planar graphs, establishing a new infinite graph limit and connecting graph decomposition techniques with probabilistic methods to understand their asymptotic behavior.

## Contribution

It introduces a novel uniform infinite planar graph limit and applies probabilistic decomposition methods to analyze random planar graphs and maps.

## Key findings

- Established the uniform infinite planar graph as a local limit.
- Derived asymptotic formulas for the number of planar graphs.
- Connected graph decomposition with probabilistic techniques.

## Abstract

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with n labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the local topology as n tends to infinity. We also establish such limits for random 2-connected planar graphs and maps as their number of edges tends to infinity. Our approach encompasses a new probabilistic view on the Tutte decomposition. This allows us to follow the path along the decomposition of connectivity from planar maps to planar graphs in a uniformed way, basing each step on condensation phenomena for random walks under subexponentiality and Gibbs partitions. Using large deviation results, we recover the asymptotic formula by Gim\'enez and Noy (2009) for the number of planar graphs.