Uniform random colored complexes

Carrance, Ariane

doi:10.1002/rsa.20845

Cited by 6 publications

(8 citation statements)

References 29 publications

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“…A motivation for the study of triangulations comes from the discretization of space in quantum gravity, see [3,17]. Recently there has been a renewed interest in this topic via coloured tensor models, which are a higher dimensional generalization of matrix integrals, see [14,8]. The objects that naturally arise in these models are coloured triangulations: roughly speaking, a triangulation is coloured if the vertices are coloured with colours from 1 to d + 1, and all colours appear in each d-simplex.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On the Number of Coloured Triangulations of d-Manifolds

Chapuy

Perarnau

2020

Discrete Comput Geom

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We give superexponential lower and upper bounds on the number of coloured d-dimensional triangulations whose underlying space is a manifold, when the number of simplices goes to infinity and d ≥ 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 Θ(n) n 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 ≥ 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. an invitation in Paris funded by the ERC grant CombiTop, during which this project was advanced.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Since vertices of X(G) are in bijection with connected components of 3-coloured subgraphs of G, there exist distinct colours i, j, k such that κ i,j,k ≥ Kn 4 log n . By (8) with I = {i, j, k} and up to relabelling i, j, k, we have κ i,j − κ i,j,k ≤ n 6 − 1 3 κ i,j,k ≤ n 6 − Kn 12 log n .…”

Section: Remaining Proofsmentioning

confidence: 99%

On the Number of Coloured Triangulations of d-Manifolds

Chapuy

Perarnau

2020

Discrete Comput Geom

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“…Given the last two lemmas, the proof of Corollary 3 is rather straightforward. Indeed, combining (8) and (9) we deduce that with probability at least 1 − 2ε all vertices of ConfigModel(U 2n ) are linked to V δn are linked to each other. Hence the diameter of the graph is less than 3 and we even have d gr (u, v) 2 as soon as u or v has degree larger than A.…”

Section: P All Vertices With Degreementioning

confidence: 99%

Universality for Random Surfaces in Unconstrained Genus

Budzinski¹,

Curien²,

Petri³

2019

Electron. J. Combin.

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Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

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“…• one distinguished white vertex (resulting from the contraction of the root-edge). 7 In [26], constellations were called stacked maps, as a central quantity in that work was the sum of faces over a certain set of submaps. Here we use the term constellations, as it is a common name in the combinatorics literature for (the dual) of these objects.…”

Section: Bijection With Constellationsmentioning

confidence: 99%

“…Still, fixing the order of the graph is responsible for the a.a.s. non-singular topology of the building block (in [7], it is shown that a uniform (q + 1)-colored graph with q > 2 and all of its residues are a.a.s. dual to pseudo-manifolds with singularities).…”

Section: Topological Structure Of Large Random Syk Graphs Of Fixed Ordermentioning

confidence: 99%

Combinatorial study of graphs arising from the Sachdev–Ye–Kitaev model

Fusy

Lionni

Tanasă

2020

European Journal of Combinatorics

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We consider the graphs involved in the theoretical physics model known as the colored Sachdev-Ye-Kitaev (SYK) model. We study in detail their combinatorial properties at any order in the so-called 1/N expansion, and we enumerate these graphs asymptotically.Because of the duality between colored graphs involving q + 1 colors and colored triangulations in dimension q, our results apply to the asymptotic enumeration of spaces that generalize unicellular maps -in the sense that they are obtained from a single building block -for which a higherdimensional generalization of the genus is kept fixed.

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Uniform random colored complexes

Cited by 6 publications

References 29 publications

On the Number of Coloured Triangulations of d-Manifolds

On the Number of Coloured Triangulations of d-Manifolds

Universality for Random Surfaces in Unconstrained Genus

Combinatorial study of graphs arising from the Sachdev–Ye–Kitaev model

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