arXiv subjects

Emmanuel Guitter

Publications and source records attributed to Emmanuel Guitter.

At least 19 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↗

arXiv2026

Liouville Quantum Duality and Random Planar Maps II

This is Part II of our project on block-weighted planar maps and Liouville quantum duality. Focusing on the scaling properties at the dual critical point, we derive the conditional distribution of the root block size given the total size, as well as, conversely, the distribution of the total size for a fixed root block size. We show that these laws are in perfect agreement with the results of Liouville quantum gravity (LQG), obtained by modifying the standard Liouville random measure with additional atomic contributions representing localized quantum areas. The ratio of dual and direct partition functions with punctures is shown to be universal, its explicit LQG expression exactly matching its combinatorial analogue. We also investigate the block distance profile for doubly rooted maps, which is here rigorously related to the distance profile of maps consisting of a single block. Finally, we analyze the multifractal properties of the usual and dual Liouville measures, predicting the associated spectra, from both quantum and Euclidean perpectives. We illustrate our results through specific realizations of block-weighted planar maps, i.e., quadrangulations decomposed into simple blocks, tree-like structures formed by attaching quartic maps, and bicubic maps decomposed into 3-connected blocks. For each model, we give the single non-universal constant which uniquely determines the strength of the corresponding atomic Liouville measure.

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arXiv2025

Liouville Quantum Duality and Random Planar Maps

We consider models of block-weighted random planar maps in which possibly decorated maps are decomposed canonically into blocks, each block receiving the weight $u$. These maps present a transition at some critical value $u=u_{cr}$ above which the maps degenerate into Brownian trees. We show that the enumerative properties and critical exponents of the maps at $u=u_{cr}$ and those for $u<u_{cr}$ are connected by duality relations which are precisely those expected in the context of the Liouville quantum gravity description of random surfaces. We illustrate this result by various instances of block-weighted maps: random planar quadrangulations decomposed into simple blocks, Hamiltonian cycles on cubic or bicubic planar maps decomposed into irreducible blocks, and meandric systems.

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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↗

arXiv2024

Enumeration of maps with tight boundaries and the Zhukovsky transformation

We consider maps with tight boundaries, i.e. maps whose boundaries have minimal length in their homotopy class, and discuss the properties of their generating functions $T^{(g)}_{\ell_1,\ldots,\ell_n}$ for fixed genus $g$ and prescribed boundary lengths $\ell_1,\ldots,\ell_n$, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of $\omega^{(g)}_n(z_1,\ldots,z_n)$, a fundamental quantity in the Eynard-Orantin theory of topological recursion, thereby providing a combinatorial interpretation of the Zhukovsky transformation used in this context. This interpretation results from the so-called trumpet decomposition of maps with arbitrary boundaries. In the planar bipartite case, we obtain a fully explicit formula for $T^{(0)}_{2\ell_1,\ldots,2\ell_n}$ from the Collet-Fusy formula. We also find recursion relations satisfied by $T^{(g)}_{\ell_1,\ldots,\ell_n}$, which consist in adding an extra tight boundary, keeping the genus $g$ fixed. Building on a result of Norbury and Scott, we show that $T^{(g)}_{\ell_1,\ldots,\ell_n}$ is equal to a parity-dependent quasi-polynomial in $\ell_1^2,\ldots,\ell_n^2$ times a simple power of the basic generating function $R$. In passing, we provide a bijective derivation in the case $(g,n)=(0,3)$, generalizing a recent construction of ours to the non bipartite case.

math.CO↗

arXiv2023

Counting mobiles by integrable systems

Mobiles are a particular class of decorated plane trees which serve as codings for planar maps. Here we address the question of enumerating mobiles in their most general flavor, in correspondence with planar Eulerian (i.e., bicolored) maps. We show that the generating functions for such mobiles satisfy a number of recursive equations which lie in the field of integrable systems, leading us to explicit expressions for these generating functions as ratios of particular determinants. In particular we recover known results for mobiles associated with uncolored maps and prove some conjectured formulas for the generating functions of mobiles associated with $p$-constellations.

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arXiv2023

Hamiltonian cycles on bicolored random planar maps

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges $c=-1$ and $c=-2$. The first group includes generic $p$-regular maps with vertices of fixed valency $p\geq 3$, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of $2q$-regular maps ($q\geq 2$) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent $\gamma$, as well as a new universal critical exponent $\nu$ characterizing the number of long-distance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of $\nu$, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for $p$-regular maps with $p=3,4,\ldots,7$, and for maps with mixed valencies $(2,3)$, $(2,4)$ and $(3,4)$.

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arXiv2022

Exponents for Hamiltonian paths on random bicubic maps and KPZ

We evaluate the configuration exponents of various ensembles of Hamiltonian paths drawn on random planar bicubic maps. These exponents are estimated from the extrapolations of exact enumeration results for finite sizes and compared with their theoretical predictions based on the KPZ relations, as applied to their regular counterpart on the honeycomb lattice. We show that a naive use of these relations does not reproduce the measured exponents but that a simple modification in their application may possibly correct the observed discrepancy. We show that a similar modification is required to reproduce via the KPZ formulas some exactly known exponents for the problem of unweighted fully packed loops on random planar bicubic maps.

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arXiv2022

On quasi-polynomials counting planar tight maps

A tight map is a map with some of its vertices marked, such that every vertex of degree $1$ is marked. We give an explicit formula for the number $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees $d_1,\ldots,d_n$, where a marked vertex is seen as a face of degree $0$. It is a quasi-polynomial in $(d_1,\ldots,d_n)$, as shown previously by Norbury. Our derivation is bijective and based on the slice decomposition of planar maps. In the non-bipartite case, we also rely on enumeration results for two-type forests. We discuss the connection with the enumeration of non necessarily tight maps. In particular, we provide a generalization of Tutte's classical slicings formula to all non-bipartite maps.

math.CO↗

arXiv2021

Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit.

math.CO↗

arXiv2020

Maps of unfixed genus and blossoming trees

We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex degrees. The mapping from blossoming trees to maps is a generalization to unfixed genus of Schaeffer's closing construction for planar Eulerian maps. The inverse mapping relies on the existence of canonical orientations which allow to equip the maps with canonical spanning trees, as proved by Bernardi. Our bijection gives in particular (here in the Eulerian case) a combinatorial explanation to the striking similarity between the (infinite) recursive system of equations which determines the partition function of maps with unfixed genus (as obtained via matrix models and orthogonal polynomials) and that determining the partition function of planar maps. All the functions in the recursive system get a combinatorial interpretation as generating functions for maps endowed with particular multiple markings of their edges. This allows us in particular to give a combinatorial proof of some differential identities satisfied by these functions. We also consider face-colored Eulerian maps with unfixed genus and derive some striking identities between their generating functions and those of properly weighted marked maps. The same methodology is then applied to deal with $m$-regular bipartite maps with unfixed genus, leading to similar results. The case of cubic maps is also briefly discussed.

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arXiv2019

Arctic curves of the twenty-vertex model with domain wall boundaries

We use the tangent method to compute the arctic curve of the Twenty-Vertex (20V) model with particular domain wall boundary conditions for a wide set of integrable weights. To this end, we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20V and 6V models via the Kagome lattice ice model. This allows to express refined partition functions of the 20V model in terms of their 6V counterparts, leading to explicit parametric expressions for the various portions of its arctic curve. The latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases. A number of numerical simulations are also presented, which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method. We finally compute the arctic curve of the Quarter-turn symmetric Holey Aztec Domino Tiling (QTHADT) model, a problem closely related to the 20V model and whose asymptotics may be analyzed via a similar tangent method approach. Again results for the QTHADT model are found to be in perfect agreement with our numerical simulations.

math-ph↗

arXiv2019

Twenty-Vertex model with domain wall boundaries and domino tilings

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries 0 and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed.

math.CO↗

arXiv2019

The Arctic curve for Aztec rectangles with defects via the Tangent Method

The Tangent Method of Colomo and Sportiello is applied to the study of the asymptotics of domino tilings of large Aztec rectangles, with some fixed distribution of defects along a boundary. The associated Non-Intersecting Lattice Path configurations are made of Schr\"oder paths whose weights involve two parameters $\gamma$ and $q$ keeping track respectively of one particular type of step and of the area below the paths. We derive the arctic curve for an arbitrary distribution of defects, and illustrate our result with a number of examples involving different classes of boundary defects.

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arXiv2018

A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points

We use a tangent method approach to obtain the arctic curve in a model of non-intersecting lattice paths within the first quadrant, including a q-dependent weight associated with the area delimited by the paths. Our model is characterized by an arbitrary sequence of starting points along the positive horizontal axis, whose distribution involves an arbitrary piecewise differentiable function. We give an explicit expression for the arctic curve in terms of this arbitrary function and of the parameter q. A particular emphasis is put on the deformation of the arctic curve upon varying q, and on its limiting shapes when q tends to 0 or infinity. Our analytic results are illustrated by a number of detailed examples.

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arXiv2018

Arctic curves for paths with arbitrary starting points: a tangent method approach

We use the tangent method to investigate the arctic curve in a model of non-intersecting lattice paths with arbitrary fixed starting points aligned along some boundary and whose distribution is characterized by some arbitrary piecewise differentiable function. We find that the arctic curve has a simple explicit parametric representation depending of this function, providing us with a simple transform that maps the arbitrary boundary condition to the arctic curve location. We discuss generic starting point distributions as well as particular freezing ones which create additional frozen domains adjacent to the boundary, hence new portions for the arctic curve. A number of examples are presented, corresponding to both generic and freezing distributions.

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arXiv2017

Statistics of the Voronoi cell perimeter in large bi-pointed maps

We study the statistics of the Voronoi cell perimeter in large bi-pointed planar quadrangulations. Such maps have two marked vertices at a fixed given distance $2s$ and their Voronoi cell perimeter is simply the length of the frontier which separates vertices closer to one marked vertex than to the other. We characterize the statistics of this perimeter as a function of $s$ for maps with a large given volume $N$ both in the scaling limit where $s$ scales as $N^{1/4}$, in which case the Voronoi cell perimeter scales as $N^{1/2}$, and in the local limit where $s$ remains finite, in which case the perimeter scales as $s^2$ for large $s$. The obtained laws are universal and are characteristics of the Brownian map and the Brownian plane respectively.

math.CO↗

arXiv2017

A universal law for Voronoi cell volumes in infinitely large maps

We discuss the volume of Voronoi cells defined by two marked vertices picked randomly at a fixed given mutual distance 2s in random planar quadrangulations. We consider the regime where the mutual distance 2s is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoi cells keeps a finite volume, which scales as s^4 for large s. We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.

math.CO↗