Freezing of triangulations

Abstract

Zero temperature dynamics of two dimensional triangulations of a torus with curvature energy is described. Numerical simulations strongly suggest that the model get frozen in metastable states, made of topological defects on flat surfaces, that group into clusters of same topological charge. It is conjectured that freezing is related to high temperature structure of baby universes.

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

“…Similar results for the canonical averaged triangulation degree distribution can be found in [34] for positive temperatures.…”

“…Similar energy functions were already applied to graphs in [40]; in [37] the total length of all edges was used as energy function in lattice triangulations, which qualitatively agrees with our choice since high energy leads to long edges in the triangulations. Previous works [29,33,34] considering topological triangulations as graphs also use the energy function (1)…”

“…For the numerical calculation of the expectation values we use the Wang-Landau algorithm for calculating the density of states (DOS). This makes it possible to calculate equilibrium properties for all temperatures, in contrast to Metropolis or Glauber dynamics used in [29,33,34], where it is hard to access negative temperatures and low temperatures, but we lose the ability to consider dynamical behaviour like quenches or glass-transitions previously studied in the literature. In the canonical ensemble we find in all considered observables a transition from an ordered large-world behaviour for positive temperatures to small-world and scale-free behaviour for negative temperatures.…”

Random and frozen states in complex triangulations

“…At very low temperatures (β → ∞), the probability allows only moves that lower the energy or leave it unchanged, driving therefore the system towards the ground state where all vertices have degree equal to k (if it is an integer number). However, it was observed that, for low genus embeddings, a 'freezing' transition takes place and the system cannot reach its ground state in any finite time [9,22,23]. The general cases with arbitrary genus g have not been studied yet, and this is indeed the topic of the present paper.…”

“…The appealing aspect of such a topological glass is that the typical features of structural glasses are observed in a simple triangulation of a surface [9]. It has been shown that this model has some of the typical properties of a strong glass with aging behaviour, but without breakdown of the fluctuation-dissipation relation [9,[20][21][22][23] . In this paper, we extend the topological glass model, originally developed in [9], to the general case of triangulations embedded on complex surfaces with different genus.…”