Combinatoire du polyn\^ome de Tutte et des cartes planaires

TL;DR

This thesis explores the Tutte polynomial through combinatorial enumeration of planar maps with spanning forests, revealing new asymptotic behaviors and proposing a unifying framework for activities related to the polynomial.

Contribution

It introduces a combinatorial characterization of the generating function for forested maps, establishes its differential algebraicity, and proposes a unifying notion of activity called Δ-activity for the Tutte polynomial.

Findings

Identified a phase transition at u=0 with a novel asymptotic regime.

Proved the generating function is differentially algebraic in z.

Unified various notions of activity into the Δ-activity framework.

Abstract

This thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight $z$ per face and a weight $u$ per non-root component of the forest. Equivalently, we count (with respect to the number of faces) the planar maps $C$ weighted by $T_C(u+1,1)$, where $T_C$ is the Tutte polynomial of $C$. We begin by a purely combinatorial characterization of the corresponding generating function, denoted by $F(z,u)$. We deduce from this that $F(z,u)$ is differentially algebraic in $z$, that is, satisfies a polynomial differential equation in $z$. Finally, for $u \geq -1$, we study the asymptotic behaviour of the $n$th coefficient of $F (z,u)$. We observe a phase transition at $0$, with a very unusual regime in $n^{-3}\ln^{-2} (n)$ for $u \in [-1,0[$, which testifies a new universality class for planar maps. In the second part, we propose a framework unifying the notions of activity used in the literature to describe the Tutte polynomial. The new notion of activity thereby defined is called $\Delta$-activity. It gathers all the notions of activities that were already known and has nice properties, as Crapo's property that defines a partition of the lattice of the spanning subgraphs into intervals with respect to the activity. Lastly we conjecture that every activity that describes the Tutte polynomial and that satisfies Crapo's property can be defined in terms of $\Delta$-activity.