arXiv subjects

Hugo Manet

Publications and source records attributed to Hugo Manet.

3 recordsLinked to original sources

arXiv2026

Distance statistics of block-weighted planar quadrangulations

We study random planar quadrangulations in which each block -- i.e., each simple (without multiple edges) component -- is assigned a weight $u$. We derive an explicit expression for the distance-dependent two-point function, defined as the generating function of such block-weighted maps with two marked edges at a fixed graph distance. Using contour integral representations in two variables combined with a delicate saddle-point analysis, we compute the associated distance profile in the scaling limit of large quadrangulations. We recover the known phase transition at $u = 9/5$, characterized by distinct scaling exponents and different scaling functions, below, at, and above criticality. We also discuss the block distance profile, where the two marked edges are conditioned to lie within the same block.

math.CO↗

arXiv2024

Enumeration of planar bipartite tight irreducible maps

We consider planar bipartite maps which are both tight, i.e. without vertices of degree $1$, and $2b$-irreducible, i.e. such that each cycle has length at least $2b$ and such that any cycle of length exactly $2b$ is the contour of a face. It was shown by Budd that the number $\mathcal N_n^{(b)}$ of such maps made out of a fixed set of $n$ faces with prescribed even degrees is a polynomial in both $b$ and the face degrees. In this paper, we give an explicit expression for $\mathcal N_n^{(b)}$ by a direct bijective approach based on the so-called slice decomposition. More precisely, we decompose any of the maps at hand into a collection of $2b$-irreducible tight slices and a suitable two-face map. We show how to bijectively encode each $2b$-irreducible slice via a $b$-decorated tree drawn on its derived map, and how to enumerate collections thereof. We then discuss the polynomial counting of two-face maps, and show how to combine it with the former enumeration to obtain $\mathcal N_n^{(b)}$.

math.CO↗

arXiv2019

Topology-Preserving Terrain Simplification

We give necessary and sufficient criteria for elementary operations in a two-dimensional terrain to preserve the persistent homology induced by the height function. These operations are edge flips and removals of interior vertices, re-triangulating the link of the removed vertex. This problem is motivated by topological terrain simplification, which means removing as many critical vertices of a terrain as possible while maintaining geometric closeness to the original surface. Existing methods manage to reduce the maximal possible number of critical vertices, but increase thereby the number of regular vertices. Our method can be used to post-process a simplified terrain, drastically reducing its size and preserving its favorable properties.

cs.CG↗