Large NN Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension d≥2d\geq 2

We review an approach which aims at studying discrete (pseudo-)manifolds in dimension $d\geq 2$ and called random tensor models. More specifically, we insist on generalizing the two-dimensional notion of $p$-angulations to higher dimensions. To do so, we consider families of triangulations built out of simplices with colored faces. Those simplices can be glued to form new building blocks, called bubbles which are pseudo-manifolds with boundaries. Bubbles can in turn be glued together to form triangulations. The main challenge is to classify the triangulations built from a given set of bubbles with respect to their numbers of bubbles and simplices of codimension two. While the colored triangulations which maximize the number of simplices of codimension two at fixed number of simplices are series-parallel objects called melonic triangulations, this is not always true anymore when restricting attention to colored triangulations built from specific bubbles. This opens up the possibility of new universality classes of colored triangulations. We present three existing strategies to find those universality classes. The first two strategies consist in building new bubbles from old ones for which the problem can be solved. The third strategy is a bijection between those colored triangulations and stuffed, edge-colored maps, which are some sort of hypermaps whose hyperedges are replaced with edge-colored maps. We then show that the present approach can lead to enumeration results and identification of universality classes, by working out the example of quartic tensor models. They feature a tree-like phase, a planar phase similar to two-dimensional quantum gravity and a phase transition between them which is interpreted as a proliferation of baby universes

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Unimodular Random Trees

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also prove that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.Comment: 19 pages, 4 figure

Glauber Dynamics on Trees and Hyperbolic Graphs

We study continuous time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with $n$ vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap $|\lambda_1-\lambda_2|$) for the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time $\tau_2$ satisfies $\tau_2=O(1)$, then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp.Comment: To appear in Probability Theory and Related Field

Random graphs from a block-stable class

A class of graphs is called block-stable when a graph is in the class if and only if each of its blocks is. We show that, as for trees, for most $n$-vertex graphs in such a class, each vertex is in at most $(1+o(1)) \log n / \log\log n$ blocks, and each path passes through at most $5 (n \log n)^{1/2}$ blocks. These results extend to `weakly block-stable' classes of graphs

Uniform random sampling of planar graphs in linear time

This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Gim\'enez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $O(n^7)$.Comment: 55 page