arXiv subjects

Guillaume Chapuy

Publications and source records attributed to Guillaume Chapuy.

At least 19 recordsLinked to original sources

arXiv2026

Composition of random functions and word reconstruction

Given two functions $\mathbf{a}\!:\! [n] \rightarrow [n]$ and $\mathbf{b}\!:\! [n] \rightarrow [n]$ chosen uniformly at random, any word $w=w_1w_2\dots w_k\in \{a,b\}^k$ induces a random function $\mathbf{w}\!:\! [n] \rightarrow [n]$ by composition, i.e. $\mathbf{w}=ϕ_{w_k}\circ \dots \circ ϕ_{w_1}$ with $ϕ_a=\mathbf{a}$ and $ϕ_b=\mathbf{b}$. We study the following question: assuming $w$ is fixed but unknown, and $n$ goes to infinity, does the random function $\mathbf{w}$ carry enough information to (partially) recover the word $w$ with good enough probability, in one or several i.i.d. samples? We prove that the random functions stemming from two non-isomorphic words can be discriminated with probability arbitrarily close to $1$ with a bounded number of samples, when $n$ goes to infinity. Equivalently, the total variation distance between their distributions is bounded away from zero. The proof relies on the study of certain auto-correlation functions appearing in the variance of the weighted number of quasi-leaves. Whether the total variation distance goes to 1, or equivalently, whether one sample is enough to distinguish the words with high probability, is the major question we leave open. We show that this is the case when the two words have different lengths or different exponents.

math.PR↗

arXiv2024

On the scaling of random Tamari intervals and Schnyder woods of random triangulations (with an asymptotic D-finite trick)

We consider a Tamari interval of size $n$ (i.e., a pair of Dyck paths which are comparable for the Tamari relation) chosen uniformly at random. We show that the height of a uniformly chosen vertex on the upper or lower path scales as $n^{3/4}$, and has an explicit limit law. By the Bernardi-Bonichon bijection, this result also describes the height of points in the canonical Schnyder trees of a uniform random plane triangulation of size $n$. The exact solution of the model is based on polynomial equations with one and two catalytic variables. To prove the convergence from the exact solution, we use a version of moment pumping based on D-finiteness, which is essentially automatic and should apply to many other models. We are not sure to have seen this simple trick used before. It would be interesting to study the universality of this convergence for decomposition trees associated to positive Bousquet-Mélou--Jehanne equations.

math.CO↗

arXiv2023

$b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral

We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We give an evolution equation for this model and derive from it Virasoro constraints, thereby proving a conjecture of Féray on Jack characters. A combinatorial model of non-oriented monotone Hurwitz maps which generalizes monotone transposition factorizations is provided. In the case $b=1$ we obtain an explicit Schur expansion of the model and show that it obeys the BKP integrable hierarchy. This Schur expansion also proves a conjecture of Oliveira--Novaes relating zonal polynomials with irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Brézin--Gross--Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.

math.CO↗

arXiv2023

Distances and isoperimetric inequalities in random triangulations of high genus

We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by at most an additive constant. Our main tool to prove those results is an isoperimetric inequality of independent interest: any part of the triangulation whose size is large compared to $\log n$ has a perimeter proportional to its volume.

math.PR↗

arXiv2023

The Discrete Gaussian model, I. Renormalisation group flow at high temperature

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature.

math.PR↗

arXiv2023

Short Synchronizing Words for Random Automata

We prove that a uniformly random automaton with $n$ states on a 2-letter alphabet has a synchronizing word of length $O(n^{1/2}\log n)$ with high probability (w.h.p.). That is to say, w.h.p. there exists a word $ω$ of such length, and a state $v_0$, such that $ω$ sends all states to $v_0$. Prior to this work, the best upper bound was the quasilinear bound $O(n\log^3n)$ due to Nicaud (2016). The correct scaling exponent had been subject to various estimates by other authors between $0.5$ and $0.56$ based on numerical simulations, and our result confirms that the smallest one indeed gives a valid upper bound (with a log factor). Our proof introduces the concept of $w$-trees, for a word $w$, that is, automata in which the $w$-transitions induce a (loop-rooted) tree. We prove a strong structure result that says that, w.h.p., a random automaton on $n$ states is a $w$-tree for some word $w$ of length at most $(1+ε)\log_2(n)$, for any $ε>0$. The existence of the (random) word $w$ is proved by the probabilistic method. This structure result is key to proving that a short synchronizing word exists.

cs.FL↗

arXiv2023

A telescopic proof of Cayley's formula

We give a short proof of the fact that the number of labelled trees on $n$ vertices is $n^{n-2}$. Although many short proofs are known, we have not seen this one before.

math.CO↗

arXiv2023

On the edge-density of the Brownian co-graphon and common ancestors of pairs in the CRT

Bassino et al. (arXiv:1907.08517) have shown that uniform random co-graphs (graphs without induced $P_4$) of size $n$ converge to a certain non-deterministic graphon. The edge-density of this graphon is a random variable $Λ\in [0,1]$ whose first moments have been computed by these authors. The first purpose of this note is to observe that, in fact, these moments can be computed by a simple recurrence relation. The problem leads us to the following question of independent interest: given $k$ i.i.d. uniform pairs of points $(X_1,Y_1),$ $\dots$, $(X_k,Y_k)$ in the Brownian CRT, what is the size $S_k$ of the set $\{X_i\wedge Y_i, i=1\dots k\}$ formed by their pairwise last common ancestors? We show that ${S_k}\sim c{\sqrt{k}\log k}$ in probability, with $c=(\sqrt{2π})^{-1}$. The method to establish the recurrence relation is reminiscent of Janson's computation of moments of the Wiener index of (large) random trees. The logarithm factor in the convergence result comes from the estimation of Riemann sums in which summands are weighted by the integer divisor function -- such sums naturally occur in the problem. We have not been able to analyse the asymptotics of moments of $Λ$ directly from the recurrence relation, and in fact our study of $S_k$ is independent from it. Several things remain to be done, in particular we only scratch the question of large deviations of $S_k$, and the precise asymptotics of moments of $Λ$ is left open.

math.CO↗

arXiv2022

Note on the density of ISE and a related diffusion

The integrated super-Brownian excursion (ISE) is the occupation measure of the spatial component of the head of the Brownian snake with lifetime process the normalized Brownian excursion. It is a random probability measure on $\mathbb{R}$, and it is known to describe the continuum limit of the distribution of labels in various models of random discrete labelled trees. We show that $f_{ISE}$, its (random) density has a.s. a derivative $f'_{ISE}$ which is continuous and $\left(\frac{1}{2}-a\right)$-Hölder for any $a >0$ but for no $a<0$ (proving a conjecture of Bousquet-Mélou and Janson). We conjecture that $f_{ISE}$ can be represented as a second-order diffusion of the form $$df'_{ISE}(t) = \sqrt{2f_{ISE}(t)}\, dB_t + g\left(f'_{ISE}(t), f_{ISE}(t),\int_{-\infty}^t f_{ISE}(s)ds\right)dt,$$ for some continuous function $g$, for $t>0$, and we give a number of remarks and questions in that direction. The proof of regularity is based on a moment estimate coming from a discrete model of trees, while the heuristic of the diffusion comes from an analogous statement in the discrete setting, which is a reformulation of explicit product formulas of Bousquet-Mélou and the first author (2012).

math.PR↗

arXiv2022

Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the "$b$-conjecture" of Goulden and Jackson from 1996, expansions of the $β$-ensemble matrix model, deformations of the HCIZ integral, and $b$-Hurwitz numbers that we introduce here and that are $b$-deformations of classical (single or double) Hurwitz numbers obtained for $b=0$. A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable $b$-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.

math.CO↗

arXiv2022

Topological recursion for Orlov-Scherbin tau functions, and constellations with internal faces

We study the correlators $W_{g,n}$ arising from Orlov-Scherbin 2-Toda tau functions with rational content-weight $G(z)$, at arbitrary values of the two sets of time parameters. Combinatorially, they correspond to generating functions of weighted Hurwitz numbers and $(m,r)$-factorisations of permutations. When the weight function is polynomial, they are generating functions of constellations on surfaces in which two full sets of degrees (black/white) are entirely controlled, and in which internal faces are allowed in addition to boundaries. We give the spectral curve (the "disk" function $W_{0,1}$, and the "cylinder" function $W_{0,2}$) for this model, generalising Eynard's solution of the 2-matrix model which corresponds to $G(z)=1+z$, by the addition of arbitrarily many free parameters. Our method relies both on the Albenque-Bouttier combinatorial proof of Eynard's result by slice decompositions, which is strong enough to handle the polynomial case, and on algebraic arguments. Building on this, we establish the topological recursion (TR) for the model. Our proof relies on the fact that TR is already known at time zero (or, combinatorially, when the underlying graphs have only boundaries, and no internal faces) by work of Bychkov-Dunin-Barkowski-Kazarian-Shadrin (or Alexandrov-Chapuy-Eynard-Harnad for the polynomial case), and on the general idea of deformation of spectral curves due to Eynard and Orantin, which we make explicit in this case. As a result of TR, we obtain strong structure results for all fixed-genus generating functions.

math.CO↗

arXiv2022

Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group $W$, in terms of the spectrum of an associated operator, the $W$-Laplacian. This covers in particular all finite Coxeter groups. The results of this paper include generalizations of the Matrix Tree and Matrix Forest theorems to reflection groups, and cover reduced (shortest length) as well as arbitrary length factorizations. Our formulas are relative to a choice of weighting system that consists of $n$ free scalar parameters and is defined in terms of a tower of parabolic subgroups. To study such systems we introduce (a class of) variants of the Jucys-Murphy elements for every group, from which we define a new notion of `tower equivalence' of virtual characters. A main technical point is to prove the tower equivalence between virtual characters naturally appearing in the problem, and exterior products of the reflection representation of $W$. Finally we study how this $W$-Laplacian matrix we introduce can be used in other problems in Coxeter combinatorics. We explain how it defines analogues of trees for $W$ and how it relates them to Coxeter factorizations, we give new numerological identities between the Coxeter number of $W$ and those of its parabolic subgroups, and finally, when $W$ is a Weyl group, we produce a new, explicit formula for the volume of the corresponding root zonotope.

math.CO↗

arXiv2021

Enumeration of non-oriented maps via integrability

In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence formulas for their enumeration according to the genus and size parameters. Using techniques already known in the orientable case (elimination of variables via Virasoro constraints or Tutte equations), we naturally obtain recurrence formulas with non-polynomial coefficients. This non-polynomiality reflects the presence of shifts of the charge parameter in the BKP equation. Nevertheless, we show that it is possible to obtain non-shifted versions, meaning pure ODEs for the associated generating functions, from which recurrence relations with polynomial coefficients can be extracted. We treat the cases of triangulations, general maps, and bipartite maps. These recurrences with polynomial coefficients are conceptually interesting but bigger to write than those with non-polynomial coefficients. However they are relatively nice-looking in the case of one-face maps. In particular we show that Ledoux's recurrence for non-oriented one-face maps can be recovered in this way, and we obtain the analogous statement for the (bivariate) bipartite case.

math.CO↗

arXiv2021

Counting chains in the noncrossing partition lattice via the W-Laplacian

We give an elementary, case-free, Coxeter-theoretic derivation of the formula $h^nn!/|W|$ for the number of maximal chains in the noncrossing partition lattice $NC(W)$ of a real reflection group $W$. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the $W$-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

math.CO↗

arXiv2020

On the number of coloured triangulations of $d$-manifolds

We give superexponential lower and upper bounds on the number of coloured $d$-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $d\geq 3$ is fixed. In the special case of dimension $3$, the lower and upper bounds match up to exponential factors, and we show that there are $2^{O(n)} n^{\frac{n}{6}}$ coloured triangulations of $3$-manifolds with $n$ tetrahedra. Our results also imply that random coloured triangulations of $3$-manifolds have a sublinear number of vertices. Our upper bounds apply in particular to coloured $d$-spheres for which they seem to be the best known bounds in any dimension $d\geq 3$, even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edge-coloured graphs having the property that each $3$-coloured component is planar, which is of independent interest.

math.CO↗

arXiv2018

On tessellations of random maps and the $t_g$-recurrence

We study the masses of the two cells in a Voronoï tessellation of the Brownian surface of genus $g\geq 0$ centered on two uniform random points. Making use of classical bijections and asymptotic estimates for maps of fixed genus, we relate the second moment of these random variables to the Painlevé-I equation satisfied by the double scaling limit of the one-matrix model, or equivalently to the "$t_g$-recurrence" satisfied by the constants $t_g$ driving the asymptotic number of maps of genus $g\geq0$. This raises the question of giving an independent probabilistic or combinatorial derivation of this second moment, which would then lead to new proof of the $t_g$-recurrence. More generally we conjecture that for any $g\geq 0$ and $k\geq 2$, the masses of the cells in a Voronoï tessellation of the genus-$g$ Brownian surface by $k$ uniform points follows a Dirichlet$(1,1,\dots,1)$ distribution.

math.PR↗

arXiv2018

Perfect sampling algorithm for Schur processes

We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entropy optimal (i.e. it reads a minimal number of random bits as an input). The algorithm encompasses previous growth procedures for special Schur processes related to the primal and dual RSK algorithm, as well as the famous domino shuffling algorithm for domino tilings of the Aztec diamond. It can be easily adapted to deal with symmetric Schur processes and general Schur processes involving infinitely many parameters. It is more concrete and easier to implement than Borodin's algorithm, and it is entropy optimal. At a technical level, it relies on unified bijective proofs of the different types of Cauchy and Littlewood identities for Schur functions, and on an adaptation of Fomin's growth diagram description of the RSK algorithm to that setting. Simulations performed with this algorithm suggest interesting limit shape phenomena for the corresponding tiling models, some of which are new.

math.PR↗

arXiv2017

Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that if $\mathcal{G}_n$ is any bridge-addable class of graphs on $n$ vertices, and $G_n$ is taken uniformly at random from $\mathcal{G}_n$, then $G_n$ is connected with probability at least $e^{-\frac{1}{2}} + o(1)$, when $n$ tends to infinity. This lower bound is asymptotically best possible since it is reached for forests. Our proof uses a "local double counting" strategy that may be of independent interest, and that enables us to compare the size of two sets of combinatorial objects by solving a related multivariate optimization problem. In our case, the optimization problem deals with partition functions of trees relative to a supermultiplicative functional.

math.CO↗