Valentin Féray
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Valentin Féray has a total of 29 co-authors. Here are the top co-authors by number of shared publications:
- Adeline Pierrot
Université Paris-Saclay CNRS Laboratoire Interdisciplinaire des Sciences du Numérique 91400 Orsay France
4 shared publications - Lucas Gerin
CMAP Ecole Polytechnique France
4 shared publications - Mathilde Bouvel
Université de Lorraine, CNRS, Inria, LORIA, Nancy, France
4 shared publications - Colin Defant
Department of Mathematics, Harvard University, Cambridge, USA
2 shared publications - Frédérique Bassino
Université Paris, France
2 shared publications - Frédérique Bassino
Universit\'e Sorbonne Paris Nord, LIPN
2 shared publications
Latest Publications
- Open Access
Up-down chains and scaling limits: application to permuton- and graphon-valued diffusions
An up-down chain is a Markov chain in which each transition is a two-step process that moves up to a larger object and then back down to an object of the original size. The first goal of this paper is to present a general framework for analyzing these chains and computing their scaling limits. This approach unifies much of the existing literature while extending it in several directions. These include explicit conditions for constructing integrable up-down chains and convergence results for families of intertwined processes. The latter contribute to the method of intertwiners of Borodin and Olshanski. The second goal is to highlight a notable application of this framework to the settings of permutations and graphs. Here, we identify some integrable up-down chains and construct their scaling limits, a family of permuton- and graphon-valued Feller diffusions. Both the up-down chains and the limiting diffusions exhibit ergodicity, diagonalizable semigroups, and explicit expressions for the maximal separation distance to stationarity. For the diffusions, the stationary measures are the recursive separable permutons and recursive cographons recently introduced by the authors, and the separation distances turn out to be related to the Dedekind eta function.
Open AccessUp-down chains and scaling limits: application to permuton- and graphon-valued diffusions
An up-down chain is a Markov chain in which each transition is a two-step process that moves up to a larger object and then back down to an object of the original size. The first goal of this paper is to present a general framework for analyzing these chains and computing their scaling limits. This approach unifies much of the existing literature while extending it in several directions. These include explicit conditions for constructing integrable up-down chains and convergence results for families of intertwined processes. The latter contribute to the method of intertwiners of Borodin and Olshanski. The second goal is to highlight a notable application of this framework to the settings of permutations and graphs. Here, we identify some integrable up-down chains and construct their scaling limits, a family of permuton- and graphon-valued Feller diffusions. Both the up-down chains and the limiting diffusions exhibit ergodicity, diagonalizable semigroups, and explicit expressions for the maximal separation distance to stationarity. For the diffusions, the stationary measures are the recursive separable permutons and recursive cographons recently introduced by the authors, and the separation distances turn out to be related to the Dedekind eta function.
- Open Access
Tree-indexed sums of Catalan numbers
We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in $1/π$ with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane.Open AccessTree-indexed sums of Catalan numbers
We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in $1/π$ with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane. - Open Access
- 11 September 2024
- Published by Elsevier in Advances in Applied Mathematics
- Vol. 162
Binary search trees of permuton samples
- 11 September 2024
- Published by Elsevier in Advances in Applied Mathematics
- Vol. 162
Open AccessBinary search trees of permuton samples
- Open Access
A determinantal point process approach to scaling and local limits of random Young tableaux
We obtain scaling and local limit results for large random Young tableaux of fixed shape $λ^0$ via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). More precisely, we prove: (1) an explicit description of the limiting surface of a uniform random Young tableau of shape $λ^0$, based on solving a complex-valued polynomial equation; (2) a simple criteria to determine if the limiting surface is continuous in the whole domain; (3) and a local limit result in the bulk of a random Poissonized Young tableau of shape $λ^0$. Our results have several consequences, for instance: they lead to explicit formulas for the limiting surface of $L$-shaped tableaux, generalizing the results of Pittel and Romik (2007) for rectangular shapes; they imply that the limiting surface for $L$-shaped tableaux is discontinuous for almost-every $L$-shape; and they give a new one-parameter family of infinite random Young tableaux, constructed from the so-called random infinite bead process.Open AccessA determinantal point process approach to scaling and local limits of random Young tableaux
We obtain scaling and local limit results for large random Young tableaux of fixed shape $λ^0$ via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). More precisely, we prove: (1) an explicit description of the limiting surface of a uniform random Young tableau of shape $λ^0$, based on solving a complex-valued polynomial equation; (2) a simple criteria to determine if the limiting surface is continuous in the whole domain; (3) and a local limit result in the bulk of a random Poissonized Young tableau of shape $λ^0$. Our results have several consequences, for instance: they lead to explicit formulas for the limiting surface of $L$-shaped tableaux, generalizing the results of Pittel and Romik (2007) for rectangular shapes; they imply that the limiting surface for $L$-shaped tableaux is discontinuous for almost-every $L$-shape; and they give a new one-parameter family of infinite random Young tableaux, constructed from the so-called random infinite bead process. - Open Access
Asymptotic normality of pattern counts in conjugacy classes
We prove, under mild conditions on fixed points and two cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class.Additionally, we prove that the limiting variance is always non-degenerate for classical pattern counts. The proof uses weighted dependency graphs.
Open AccessAsymptotic normality of pattern counts in conjugacy classes
We prove, under mild conditions on fixed points and two cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class.Additionally, we prove that the limiting variance is always non-degenerate for classical pattern counts. The proof uses weighted dependency graphs.
- Open Access
Dense and nondense limits for uniform random intersection graphs
We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval $[0,1]$ endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size $k$ ($k$ fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique.Open AccessDense and nondense limits for uniform random intersection graphs
We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval $[0,1]$ endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size $k$ ($k$ fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique. - Open Access
The permuton limit of random recursive separable permutations
We introduce and study a simple Markovian model of random separable permutations. Our first main result is the almost sure convergence of these permutations towards a random limiting object in the sense of permutons, which we call the recursive separable permuton. We then prove several results on this new limiting object: a characterization of its distribution via a fixed-point equation, a combinatorial formula for its expected pattern densities, an explicit integral formula for its intensity measure, and lastly, we prove that its distribution is absolutely singular with respect to that of the Brownian separable permuton, which is the large size limit of uniform random separable permutations.
Open AccessThe permuton limit of random recursive separable permutations
We introduce and study a simple Markovian model of random separable permutations. Our first main result is the almost sure convergence of these permutations towards a random limiting object in the sense of permutons, which we call the recursive separable permuton. We then prove several results on this new limiting object: a characterization of its distribution via a fixed-point equation, a combinatorial formula for its expected pattern densities, an explicit integral formula for its intensity measure, and lastly, we prove that its distribution is absolutely singular with respect to that of the Brownian separable permuton, which is the large size limit of uniform random separable permutations.
- Open Access
- 12 January 2024
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 31 (1) , P1.41
Wiener Indices of Minuscule Lattices
The Wiener index of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.- 12 January 2024
- Published by The Electronic Journal of Combinatorics in The Electronic Journal of Combinatorics
- Vol. 31 (1) , P1.41
Open AccessWiener Indices of Minuscule Lattices
The Wiener index of a finite graph $G$ is the sum over all pairs $(p,q)$ of vertices of $G$ of the distance between $p$ and $q$. When $P$ is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals. - Open Access
Components in meandric systems and the infinite noodle
We investigate here the asymptotic behaviour of a large typical meandric system. More precisely, we show the quenched local convergence of a random uniform meandric system $M_n$ on $2n$ points, as $n \rightarrow \infty$, towards the infinite noodle introduced by Curien, Kozma, Sidoravicius and Tournier ({\em Ann. Inst. Henri Poincar\'e D}, {6}(2):221--238, 2019). As a consequence, denoting by $cc( M_n)$ the number of connected components of $ M_n$, we prove the convergence in probability of $cc(M_n)/n$ to some constant $\kappa$, answering a question raised independently by Goulden--Nica--Puder ({\em Int. Math. Res. Not.}, 2020(4):983--1034, 2020) and Kargin ({\em Journal of Statistical Physics}, 181(6):2322--2345, 2020). This result also provides information on the asymptotic geometry of the Hasse diagram of the lattice of non-crossing partitions. Finally, we obtain expressions of the constant $\kappa$ as infinite sums over meanders, which allows us to compute upper and lower approximations of $\kappa$.Open AccessComponents in meandric systems and the infinite noodle
We investigate here the asymptotic behaviour of a large typical meandric system. More precisely, we show the quenched local convergence of a random uniform meandric system $M_n$ on $2n$ points, as $n \rightarrow \infty$, towards the infinite noodle introduced by Curien, Kozma, Sidoravicius and Tournier ({\em Ann. Inst. Henri Poincar\'e D}, {6}(2):221--238, 2019). As a consequence, denoting by $cc( M_n)$ the number of connected components of $ M_n$, we prove the convergence in probability of $cc(M_n)/n$ to some constant $\kappa$, answering a question raised independently by Goulden--Nica--Puder ({\em Int. Math. Res. Not.}, 2020(4):983--1034, 2020) and Kargin ({\em Journal of Statistical Physics}, 181(6):2322--2345, 2020). This result also provides information on the asymptotic geometry of the Hasse diagram of the lattice of non-crossing partitions. Finally, we obtain expressions of the constant $\kappa$ as infinite sums over meanders, which allows us to compute upper and lower approximations of $\kappa$. - Open Access
Wiener Indices of Minuscule Lattices
The Wiener index of a finite graph G is the sum over all pairs (p, q) of vertices of G of the distance between p and q. When P is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.
Open AccessWiener Indices of Minuscule Lattices
The Wiener index of a finite graph G is the sum over all pairs (p, q) of vertices of G of the distance between p and q. When P is a finite poset, we define its Wiener index as the Wiener index of the graph of its Hasse diagram. In this paper, we find exact expressions for the Wiener indices of the distributive lattices of order ideals in minuscule posets. For infinite families of such posets, we also provide results on the asymptotic distribution of the distance between two random order ideals.