Pattern occurrences in random planar maps

TL;DR

This paper studies how often specific patterns appear in large random planar maps, showing that their expected counts grow linearly with the size of the map, using an extension of a known combinatorial formula.

Contribution

It introduces an extension of Liskovets' formula to analyze pattern occurrences in random planar maps with a Boltzmann distribution.

Findings

Expected pattern occurrences grow linearly with the number of edges.

Extension of Liskovets' formula for planar maps.

Asymptotic analysis of pattern counts.

Abstract

We consider planar maps adjusted with a (regular critical) Boltzmann distribution and show that the expected number of pattern occurrences of a given map is asymptotically linear when the number n of edges goes to infinity. The main ingredient for the proof is an extension of a formula by Liskovets (1999).