Unimodular lattice triangulations as small-world and scale-free random graphs

Real-world networks, e.g. the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte-Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real world graphs, for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees $k \geq 5$. Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.Comment: 17 pages, 6 figures, will appear in New J. Phy

Uniform Infinite Planar Triangulations

The existence of the weak limit as n --> infinity of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio

Dimension and Dimensional Reduction in Quantum Gravity

If gravity is asymptotically safe, operators will exhibit anomalous scaling at the ultraviolet fixed point in a way that makes the theory effectively two-dimensional. A number of independent lines of evidence, based on different approaches to quantization, indicate a similar short-distance dimensional reduction. I will review the evidence for this behavior, emphasizing the physical question of what one means by `dimension' in a quantum spacetime, and will discuss possible mechanisms that could explain the universality of this phenomenon.Comment: For proceedings of the conference in honor of Martin Reuter: "Quantum Fields---From Fundamental Concepts to Phenomenological Questions"; 14 pages; based in part on my review article arXiv:1705.0541

The Harris-Luck criterion for random lattices

The Harris-Luck criterion judges the relevance of (potentially) spatially correlated, quenched disorder induced by, e.g., random bonds, randomly diluted sites or a quasi-periodicity of the lattice, for altering the critical behavior of a coupled matter system. We investigate the applicability of this type of criterion to the case of spin variables coupled to random lattices. Their aptitude to alter critical behavior depends on the degree of spatial correlations present, which is quantified by a wandering exponent. We consider the cases of Poissonian random graphs resulting from the Voronoi-Delaunay construction and of planar, ``fat'' $\phi^3$ Feynman diagrams and precisely determine their wandering exponents. The resulting predictions are compared to various exact and numerical results for the Potts model coupled to these quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one figure added for clarification, minor re-wordings and typo cleanu

Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $\sigma$ has weight $\lambda^{|\sigma|}$, where $\lambda$ is a positive real parameter, and $|\sigma|$ is the total length of the edges in $\sigma$. Empirically, this model exhibits a "phase transition" at $\lambda=1$ (corresponding to the uniform distribution): for $\lambda<1$ distant edges behave essentially independently, while for $\lambda>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $\lambda<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $\lambda>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org