Papers by Alessandra Caraceni
Springer proceedings in mathematics & statistics, 2019
We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two... more We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with the estimates of [15], shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is 4 (as is the case for the standard UIPQ); nevertheless, using our previous work [9] we show its law to be singular with respect to that of the standard UIPQ, that is-in the language of statistical physicsthe fact that disorder holds.
arXiv (Cornell University), Oct 27, 2021
We provide "growth schemes" for inductively generating uniform random 2p-angulations of the spher... more We provide "growth schemes" for inductively generating uniform random 2p-angulations of the sphere with n faces, as well as uniform random simple triangulations of the sphere with 2n faces. In the case of 2p-angulations, we provide a way to insert a new face at a random location in a uniform 2p-angulation with n faces in such a way that the new map is precisely a uniform 2p-angulation with n + 1 faces. Similarly, given a uniform simple triangulation of the sphere with 2n faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with 2n + 2 faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer.
arXiv (Cornell University), May 21, 2021
The class of ranked tree-child networks, tree-child networks arising from an evolution process wi... more The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper.
Random Structures and Algorithms, Dec 15, 2017
We give a new construction of the uniform infinite half-planar quadrangulation with a general bou... more We give a new construction of the uniform infinite half-planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus [9], which allows us to perform a detailed study of its geometry. We show that the process of distances to the root vertex read along the boundary contour of the UIHPQ evolves as a particularly simple Markov chain and converges to a pair of independent Bessel processes of dimension 5 in the scaling limit. We study the "pencil" of infinite geodesics issued from the root vertex as in [14], and prove that it induces a decomposition of the UIHPQ into three independent submaps. We are also able to prove that balls of large radius around the root are on average 7/9 times as large as those in the UIPQ, both in the UIHPQ and in the UIHPQ with a simple boundary; this fact we use in a companion paper to study self-avoiding walks on large quadrangulations.
Annales de l'I.H.P, Nov 1, 2016
A planar map is outerplanar if all its vertices belong to the same face. We show that random unif... more A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with n vertices suitably rescaled by a factor 1/ √ n converge in the Gromov-Hausdorff sense to 7 √ 2/9 times Aldous' Brownian tree. The proof uses the bijection of Bonichon, Gavoille and Hanusse [7].
Electronic Journal of Probability, 2020
We consider a natural local dynamic on the set of all rooted planar maps with n edges that is in ... more We consider a natural local dynamic on the set of all rooted planar maps with n edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the n-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times n −11/2. In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations as defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flip chain related to edge rotations via Tutte's bijection.
Probability Theory and Related Fields, Apr 22, 2019
We establish the first polynomial upper bound for the mixing time of random edge flips on rooted ... more We establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with n faces admits, up to constants, an upper bound of n −5/4 and a lower bound of n −11/2. In order to obtain the lower bound, we also consider a very natural Markov chain on plane trees-or, equivalently, on Dyck paths-and improve the previous lower bound for its spectral gap by Shor and Movassagh.
The Lagrangian of a hypergraph is a function that in a sense seems to measure how ‘tightly packed... more The Lagrangian of a hypergraph is a function that in a sense seems to measure how ‘tightly packed’ a subset of the hypergraph one can find. Lagrangians were first used by Motzkin and Straus to obtain a new proof of a classic theorem of Turán, and subsequently found a number of very valuable applications in Extremal Hypergraph Theory; one remarkable result they yield is the disproof of a famous "jumping conjecture" of Erdos, which we reprove entirely; we will also introduce a very recent method based on Razborov's flag algebras to show that, though the jumping conjecture is false in general, hypergraphs "do jump" in some cases
Discrete Mathematics
The class of ranked tree-child networks, tree-child networks arising from an evolution process wi... more The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper.
Springer Proceedings in Mathematics & Statistics, 2019
We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two... more We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the displacement of the SAW which, combined with the estimates of [15], shows that the self-avoiding walk is diffusive. As a byproduct this implies that the volume growth exponent of the lattice in question is 4 (as is the case for the standard UIPQ); nevertheless, using our previous work [9] we show its law to be singular with respect to that of the standard UIPQ, that is-in the language of statistical physicsthe fact that disorder holds.
Electronic Journal of Probability, 2020
We consider a natural local dynamic on the set of all rooted planar maps with n edges that is in ... more We consider a natural local dynamic on the set of all rooted planar maps with n edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the n-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times n −11/2. In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations as defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flip chain related to edge rotations via Tutte's bijection.
Probability Theory and Related Fields, 2019
We establish the first polynomial upper bound for the mixing time of random edge flips on rooted ... more We establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with n faces admits, up to constants, an upper bound of n −5/4 and a lower bound of n −11/2. In order to obtain the lower bound, we also consider a very natural Markov chain on plane trees-or, equivalently, on Dyck paths-and improve the previous lower bound for its spectral gap by Shor and Movassagh.
Random Structures & Algorithms, 2017
We give a new construction of the uniform infinite half‐planar quadrangulation with a general bou... more We give a new construction of the uniform infinite half‐planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus, which allows us to perform a detailed study of its geometry. We show that the process of distances to the root vertex read along the boundary contour of the UIHPQ evolves as a particularly simple Markov chain and converges to a pair of independent Bessel processes of dimension 5 in the scaling limit. We study the “pencil” of infinite geodesics issued from the root vertex as reported by Curien, Ménard, and Miermont and prove that it induces a decomposition of the UIHPQ into 3 independent submaps. We are also able to prove that balls of large radius around the root are on average 7/9 times as large as those in the UIPQ, both in the UIHPQ and in the UIHPQ with a simple boundary; this fact we use in a companion paper to study self‐avoiding walks on large quadrangulations.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2016
A planar map is outerplanar if all its vertices belong to the same face. We show that random unif... more A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with n vertices suitably rescaled by a factor 1/ √ n converge in the Gromov-Hausdorff sense to 7 √ 2/9 times Aldous' Brownian tree. The proof uses the bijection of Bonichon, Gavoille and Hanusse (J. Graph Algorithms Appl. 9 (2005) 185-204). Résumé. Une carte planaire est dite outerplanaire si tous ses sommets appartiennent à la même face. Nous montrons que les cartes outerplanaires aléatoires uniformes à n sommets, multipliées par le facteur d'échelle 1/ √ n, convergent au sens de Gromov-Hausdorff vers 7 √ 2/9 fois l'arbre Brownien d'Aldous. La preuve utilise la bijection de Bonichon, Gavoille et Hanusse (J. Graph Algorithms Appl. 9 (2005) 185-204).
Random Structures & Algorithms
We provide “growth schemes” for inductively generating uniform random ‐angulations of the sphere ... more We provide “growth schemes” for inductively generating uniform random ‐angulations of the sphere with faces, as well as uniform random simple triangulations of the sphere with faces. In the case of ‐angulations, we provide a way to insert a new face at a random location in a uniform ‐angulation with faces in such a way that the new map is precisely a uniform ‐angulation with faces. Similarly, given a uniform simple triangulation of the sphere with faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer.
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Papers by Alessandra Caraceni