Related papers: Map enumeration from a dynamical perspective
Discrete dynamical systems defined by the iteration of a polynomial map of the unit simplex to itself appear in the context of population genetic systems evolving under mutation, recombination and weak selection. Although exceptional…
In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based…
In this article we describe an algorithm that can be applied for the generation of various classes of maps on orientable surfaces. It uses existing generators for abstract graphs and combines them with an efficient embedding and isomorphism…
We compute, for each genus $g\geq 0$, the generating function $L_g\equiv L_g(t;p_1,p_2,\dots)$ of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables…
We solve three enumerative problems concerning families of planar maps. More precisely, we establish algebraic equations for the generating function of non-separable triangulations in which all vertices have degree at least d, for a certain…
This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general…
A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by…
Given a graph embedded in an orientable surface, a process consisting of random excitations and random node and face balancing is constructed and analyzed. It is shown that given a priori bounds g' on the genus and n' on the number of…
Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…
Traditional map-making relies heavily on Geographic Information Systems (GIS), requiring domain expertise and being time-consuming, especially for repetitive tasks. Recent advances in generative AI (GenAI), particularly image diffusion…
We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
This paper summarizes our latest understanding and results about the application of the Mathematics Of Enumeration to Tanner Graphs that have a regular structure called Balanced Tanner Graphs. Some preliminaries of permutation groups have…
We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients of orthogonal polynomials associated with…
We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial…
This work studies certain aspects of graphs embedded on surfaces. Initially, a colored graph model for a map of a graph on a surface is developed. Then, a concept analogous to (and extending) planar graph is introduced in the same spirit as…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These…
Traditionally, autonomous reconnaissance applications have acted on explicit sets of historical observations. Aided by recent breakthroughs in generative technologies, this work enables robot teams to act beyond what is currently known…
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…