Spectral Properties of Unimodular Lattice Triangulations
Abstract
Random unimodular lattice triangulations have been recently used as an embedded random graph model, which exhibit a crossover behavior between an ordered, largeworld and a disordered, smallworld behavior. Using the ergodic Pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance, Markov chain Monte Carlo simulations can be applied to study these graphs. Here, we consider the spectra of the adjacency and the Laplacian matrix as well as the algebraic connectivity and the spectral radius. Power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states. For random triangulations we find a qualitative agreement of the spectral properties with wellknown random graph models. In the microcanonical ensemble analytical approximations agree with numerical simulations. In the canonical ensemble a crossover behavior can be found for the algebraic connectivity and the spectral radius, thus combining largeworld and smallworld behavior in one model. The considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behavior allows a tuning of important transport quantities.
Keywords
Triangulations Random graphs Networks Spectral graph theory1 Introduction
Many real world systems consist of a set of equivalent objects and the pairwise interaction among themselves, so that they can be described approximately in terms of graph theory [16, 58]. Excluding all quantitative aspects of the interactions and taking only into account whether two objects interact or not, the considered objects are interpreted as the node of a graph, where edges exist between objects or nodes that do interact. Examples for such systems are neural networks in biology, where dendrites are modelled as nodes and axons between two dendrites are modelled as edges; or the worldwideweb, where an edge between two website nodes exist if there is a link from one website to the other.
A major tool in studying the physical properties of such graphs is spectral graph theory, which examines the spectra of the adjacency and the Laplacian matrices associated with the graph and some special eigenvalues of those, see e.g., [17]. The spectra of graphs are studied e.g., for quantum percolation [30] and Anderson transition on Bethe lattices [31, 32, 51] in terms of random matrices, as well as in biology [9] and chemistry [28, 70], for a review see [53] and the references therein. Special eigenvalues as the algebraic connectivity, the smallest nonzero eigenvalue of the Laplacian matrix, are important for characterizing the topology of graphs [41], for transport and dynamics on networks [3] and for optimization problems [55, 72], see [18, 54] for reviews. Additionally, spectral graph theory is wellconnected with the common mathematical theory of random matrices used in quantum physics [38]. An important application of the spectrum of the Laplacian matrix are (quantum) random walks on networks [13, 56], where the Laplacian matrix corresponds to the discretization of the operator used in Poisson and Schrödinger equations. In both cases the spectrum can be used for calculating return probabilities and inverse participation ratios.
For understanding and modelling the important features of real world networks artificially constructed graphs can be used, mainly random graphs [1]. Common examples are the Erdös–Rényi random graph [25, 26, 27], the Watts–Strogatz random graph [75] (and a slightly altered version denoted as Newman–Watts random graph [59]) and the Barabási–Albert random graph [10]. These random graphs were mainly constructed and optimised for creating models for real smallworld and scalefree networks, but also their spectral properties arise much interest [3, 57].
Recently, a new type of (embedded) random graph model was proposed by using canonical ensembles of unimodular lattice triangulations [46]. Random lattice triangulations are comparable with the usual random graph models; in the canonical ensemble a crossover behavior from ordered triangulations showing largeworld to disordered triangulations exhibiting smallworld and scalefree behavior can be found. The triangulations used are tessellations of the convex hull of a given point set into triangles that do not intersect [19]. Triangulations have been used as randomgraphs before [2, 5, 6, 45, 66, 81], but only the topological degrees of freedom were used there. In contrast to these models the actual coordinates of the nodes become important in lattice triangulations [46], which adds some (numerical) difficulties, but can be a concept useful in real networks where spatial coordinates of nodes and the induced length of the edges become important. The considered lattice triangulations are maximal planar graphs in the sense that no interior edge can be inserted without violating the planarity. Furthermore, one can always find an embedding of a planar graph with rational coordinates, by choosing first an arbitrary embedding with real coordinates and then wiggling the nonrational ones, since the rational numbers are dense in the reals. By scaling one can find an embedding with integer coordinates, so lattice triangulations can be considered as the supergraphs of all planar graphs.
Triangulations in general are used in quantum geometry for describing curved spacetimes in the approaches of causal dynamical triangulations [4] and in spin foams [64]. In topology and geometry one is interested in the number of triangulations of manifolds [68]. They can also be used as a tool for studying foams [7, 23, 60, 69], since the Voronoi tessellation is the dual of the Delaunay triangulation, which is a special triangulation of a point set.
In this paper we extend the results of [46] by considering spectral properties of lattice triangulations in order to relate graphs to physical properties. We measure the spectra of the adjacency and the Laplacian matrix of random triangulations, which is the ensemble of all triangulations with constant weights, using Metropolis Monte Carlo simulations, and compare the results with common models of random graphs. Defining an energy of a triangulation, which is well known in the literature, and that corresponds to the variance of the degree distribution (the histogram of the node degrees, which is the number of incident edges), we calculate the microcanonical and canonical expectation values of the algebraic connectivity and the spectral radius with Metropolis and Wang–Landau simulations. For the maximal ordered triangulations, which are the ground state of the energy functional used, analytical calculations are possible, as well as approximations for triangulations with low energies near the ground state. As an application of our results, we calculate the inverse participation ratio of certain eigenstates and show that random lattice triangulations on average exhibit stronger localization than comparable random graphs.
2 Triangulations and Spectral Graph Theory
2.1 Unimodular Lattice Triangulations
One special triangulation used throughout this paper is the maximal ordered triangulation. It is defined as the full triangulation of the \(M\times N\) integer lattice where the vertex with coordinates (i, j) is connected with at most six vertices with coordinates \((i, j \pm 1)\), \((i \pm 1, j)\) and \((i \pm 1, j \pm 1)\) whenever these coordinates are in \(\mathbb {Z}_M\) or \(\mathbb {Z}_N\) respectively. Note that postulating connections to vertices with coordinates \((i \pm 1, j \pm 1)\) is a convention, one could also use connections to \((i \pm 1, j \mp 1)\) instead. The maximal ordered lattice is often denoted as triangular lattice in the literature.
To determine the number of possible triangulations of an integer lattice is a nontrivial question. There are exact enumeration results and analytical bounds [42] as well as numerical approximations of this number [44] using Monte Carlo simulations, both show that there are exponentially many triangulations in terms of the system size \(M\times N\). So lattice triangulations are extensive and can be treated as a welldefined statistical system. Additionally the convergence of Glauber dynamics on lattice triangulations was examined in [14, 67] for different parameter sets.
A graph can be constructed from a triangulation by using the 0simplices (vertices) as graph nodes and the 1simplices (edges) as graph edges. In the notation of graph theory triangulations are planar, which is trivial since the graphs are defined in the actual embedding into an euclidean plane, they are even maximal planar with respect to the convex hull of the vertices, i.e., that by including an arbitrary other edge between two nodes that does not leave the convex hull of the vertices the planarity of the triangulation graph is violated.
2.2 Spectral Graph Theory
An undirected simple graph \(\mathscr {G} := (\mathbf A, E)\) is a pair consisting of a set \(\mathbf A\) (called vertices) with \(n = \mathbf A\) elements and a set E (called edges) of twoelement subsets of \(\mathbf A\). A triangulation can be interpreted as a graph using the point set \(\mathbf A\) as vertices and the 1simplices of the triangulation as edges E.
 The adjacency matrix \(A(\mathscr {G})\) withis a traceless, symmetric matrix that indicates whether two different vertices are connected. The matrix elements \(A(\mathscr {G})_{ij}^k\) equal the number of paths from \(v_i\) to \(v_j\) containing exactly k edges. We will denote its sorted eigenvalues by \(\alpha _0 \le \alpha _1 \le \dots \le \alpha _{n  1}\).$$\begin{aligned} A(\mathscr {G})_{ij} := {\left\{ \begin{array}{ll} 1 &{} \{v_i, v_j\} \in E \\ 0 &{} \{v_i, v_j\} \notin E \end{array}\right. } \end{aligned}$$
 The Laplacian matrix \(L(\mathscr {G}) := D(\mathscr {G})  A(\mathscr {G})\) with degree matrixis a discretization of the usual Laplace operator \(\mathbf {\nabla }^2\). The Laplacian matrix is symmetric and positivesemidefinite, the smallest eigenvalue is always 0. We denote the sorted eigenvalues of the Laplacian matrix by \(\lambda _0 \le \lambda _1 \le \dots \le \lambda _{n  1}\). The multiplicity of the eigenvalue 0 is the number of connected components of the graph.$$\begin{aligned} D(\mathscr {G})_{ij} := \delta _{ij} k_{v_i} \end{aligned}$$
 The normalized Laplacian matrixis useful for describing random walks on arbitrary geometries. This matrix is not considered in this paper, but our calculations can be extended simply to the normalized Laplacian.$$\begin{aligned} \mathscr {L} (\mathscr {G}) := \mathbbm {1}  D(\mathscr {G})^{1/2} A (\mathscr {G}) D(\mathscr {G})^{1/2} \end{aligned}$$
One can show that the algebraic connectivity is proportional to the inverse of the synchronization time in consensus dynamics on networks [3]. Additionally, for the return probability it governs the largetime in classical and the small frequency behavior in quantum random walks on networks, whereas the spectral radius governs shorttime, respectively, the large frequency behavior [57]. It is also possible to find a bound for the sum of the j largest eigenvalues in terms of the klargest vertex degrees [37]. The sum of the exponentiated eigenvalues of the adjacency and the Laplacian matrix is known as the (Laplacian) Estrada index and has many applications in the study of chemical molecules [22, 28, 29].
2.3 Random Graphs
In this paper we compare the graph interpretation of unimodular lattice triangulations with some common random graph models. Similar to [46], where graph measures as the clustering coefficient (which is the average ratio of the numbers of actual present and possible edges between the neighbors of a node) and the average shortest path length (the shortest path length between two nodes is the minimal number of edges that have to be run through to go from the one to the other node within the graph) were compared to these random graph models, we choose the respective parameters of the random graph models such that the average numbers of nodes and edges are equal to the number MN of triangulation vertices and to the number \(3MN  2(M+N) + 1\) of triangulation edges for comparing the spectral properties. In the following we describe shortly the different models of random graphs we are comparing the lattice triangulation graphs to, as well as our choice of parameters for these random graphs.
Erdös and Rényi [25] categorised the random graphs \(\mathscr {G}_{n,p}\) according to the asymptotics of the average number np of edges incident with a vertex. For our choice of parameters (6) with \(np \rightarrow 6\) they found that the graph has one giant component with \(n\cdot (1  2.5\cdot 10^{3})\) vertices on average, and \(n\cdot 0.25\cdot 10^{3}\) connected components for \(np=6\) and \(n\rightarrow \infty \). The latter result implies that the average Erdös–Rényi random graph is not connected for a sufficiently high number of vertices for our choice of parameters, in contrast to the considered lattice triangulations which are always connected.
Watts–Strogatz and Newman–Watts Random Graph Another common random graph model was proposed by Newmann and Watts [59]. It consists of a regular graph on a periodic lattice of n vertices (a ring in one dimension), where each vertex is connected with its \(L = 2 k \cdot d\) nearest neighbors (with d being the dimension of the lattice and integer k), superimposed with an Erdös–Rényi random graph with connection probability p. Numerical calculations for a similar model [75] showed that there is a crossover region in the parameter p where this model shows a smallworld (but not scalefree) behavior.
Barabási–Albert Random Graph Barabási and Albert [10] proposed a preferential attachment random graph model with the following construction in order to achieve a powerlaw degree distribution: Start from a graph of m vertices without edges. Then execute t of the following steps: Insert one vertex and edges from this vertex to m existing vertices so that the probability \(p_i\) for an edge between vertex \(v_i\) and the new vertex is proportional to the node degree \(k_i\). Hence vertices with high vertex degree have a high probability to be linked to the new vertex (preferential attachment). The resulting graph consists of \(n = m + t\) vertices and \(m\cdot t\) edges.
To compare the Barabási–Albert random graph to a random triangulation on an \(M\times N\) integer lattice, we choose \(m = 3\) and \(t = MN  3\) so that the corresponding Barabási–Albert random graph has 3MN vertices and \(MN  9\) edges. The number of vertices matches the number of triangulation vertices, for \(M,N \gg 1\) also the number of edges match.
3 Analytical Solution
The perturbation theory unfortunately fails if one considers the algebraic connectivity. For the case \(M = N\) the corresponding eigenvalue is sixfold degenerated and the perturbation matrix has to be diagonalized numerically. This leads for all M, N to a decrease in the algebraic connectivity, which is consistent with a direct diagonalisation of the new Laplacian \(L + V\) for the periodic case. In contrast to that, if one diagonalises \(L + V\) directly for the nonperiodic case, the algebraic connectivity increases. So for the small eigenvalues the difference between a regular and a periodic triangulation leads to a qualitative change in the behavior of the algebraic connectivity, which is quite comprehensible since the algebraic connectivity is dominated by the vertices with low degree (two for nonperiodic and six for periodic triangulations), which can be seen already in the bounds (3). The spectral radius is determined by the vertices with high degree, which is six for both periodic and nonperiodic triangulations, also apparent in the bounds (4).
4 Triangulation Ensembles as Random Graph Models

The random ensemble assigns to each triangulation the constant probability \(P = 1/\Omega _{\mathbf A}\) and can be viewed as a democratic sum since each triangulation contributes in equal measure. This ensemble is independent of the definition of a triangulation energy. Ensemble averages of the observable \(\mathscr {O}\) in the random ensemble will be denoted by \(\langle \mathscr {O} \rangle _{\mathrm {rnd}}\) or simply \(\langle \mathscr {O} \rangle \)

The microcanonical ensemble at constant energy E assigns every triangulation with energy E a constant probability, and every triangulation with a different energy the probability 0. This ensemble is a random ensemble in the subset of all triangulations \(\Omega _{\mathbf A}_E := \{ \mathscr {T} \in \Omega _{\mathbf A} \mid E(\mathscr {T}) = E \}\) with energy E. Microcanonical ensemble averages of an observable \(\mathscr {O}\) at energy E will be denoted by \(\langle \mathscr {O} \rangle _{\mathrm {mc}}(E)\)

The canonical ensemble at constant inverse temperature \(\beta \) assigns every triangulation \(\mathscr {T}\) with energy E the Boltzmann probability \(P(\mathscr {T}) = \exp (\beta E) / Z\) with \(Z = \sum _{\mathscr {T} \in \Omega _{\mathbf A}} \exp (\beta E(\mathscr {T}))\). Formally one can define the canonical ensemble also for negative temperatures \(\beta < 0\). In the case of triangulations with the considered energy function (2) negative temperatures correspond to the usual positive temperatures with negative coupling constant \(J < 0\). The canonical ensemble at infinite temperature \((\beta = 0)\) equals the ensemble of random triangulations. Canonical ensemble averages of an observable \(\mathscr {O}\) at inverses temperature \(\beta \) will be denoted by \(\langle \mathscr {O} \rangle _{\mathrm {c}}(\beta )\)
4.1 Random Triangulation
In this section we consider spectral graph observables of random lattice triangulations and their scaling behavior in terms of the size of the integer lattice. The results of these calculations are especially interesting because they do not depend on the choice of the energy function, since every possible triangulation contributes with constant weight.
The spectral observables of the random ensemble are calculated using the Metropolis algorithm [50] averaging over more than 1000 different random triangulations. Due to the fact that every triangulation occurs in the random ensemble with constant probability, the acceptance probability (15) used in the Metropolis algorithm is always 1, so every possible and suggested Pachner move will be executed. To obtain a stationary distribution of random triangulations, we start with the maximal ordered triangulation and propose \(10^5 M\cdot N\) Pachner moves, which is well above the diameter \(o(MN(M+N))\) of the flip graph [14] for the considered system sizes. In order to avoid an influence of the autocorrelation of successive triangulations in the Markov chain, we ensure that the number of steps between two measurements is larger than 100 autocorrelation times, which we measured to be smaller than \(10 M\cdot N\) for the considered spectral observables and system sizes, and which was measured for other graph observables before in Ref. [46].
For the calculations in this paper we use only quadratic lattice sizes \(N\times N\); considering rectangular lattices is possible but does not change the results much, for large lattices the only relevant parameter is the number of vertices \(M\cdot N\) [46]. The accessible system sizes are rather limited compared to topological triangulations [6, 45] for two reasons: First, one has to check whether a proposed flip is executable (compare Sect. 2.1), which can be done for twodimensional lattice triangulations by checking whether the two diagonals have the same mids, or by testing for convexity in more general cases. These checks consume more computation time for a flip than in the topological case, where only incidences have to be looked up. Second, every time encountering a nonexecutable flip the triangulation remains unchanged, which increases e.g., the autocorrelation time of the algorithm. Note that proposing only executable flips is not suitable, because this makes it difficult to fulfill the detailed balance. We were able to calculate the spectral properties of random triangulations for system sizes between \(4 \times 4\) and \(64 \times 64\).
We compare the ensemble averages for random triangulations as graphs with the Erdös–Rényi, the Newman–Watts and the Barabási–Albert random graphs with parameters chosen according to Sect. 2.3, the generation of these random graphs was done using the NetworkX framework [39].
Spectrum of the Adjacency Matrix For the Erdös–Rényi random graph there are some analytical results for the spectrum of the adjacency matrix. The random graph adjacency spectra are strongly related with the spectra of random matrices, which follow in most cases a semicircle distribution [40, 76, 77].
Numerical investigations showed that for constant edge probability \(p \ne 0, 1\) the adjacency spectrum of the Erdös–Rényi model converges to the semicircle distribution up to the largest eigenvalue, but there are additional peaks for \(p \propto n^{1}\), which we actually choose to compare with random triangulations (see [31, 32] for investigations on the level of random matrices and [11, 34] for investigations on random graphs).
In Fig. 4 the probability density function (PDF) for the spectrum of the adjacency matrix of random triangulations is displayed in an indexresolved and an indexsummed way for different lattice sizes. The spectrum is compared with the spectra of the maximal ordered triangulations and the Erdös–Rényi random graph. Figure 5 displays cuts through the random triangulation and the Erdös–Rényi adjacency PDF for the eigenvalue index \(i = MN/2\) and the eigenvalue magnitude \(\alpha = 0\). The displayed PDFs are created using a kernel density estimation [62, 63] with Gaussian kernel and Silverman’s rule [65] for the width of the Gaussian.
The adjacency spectra of random and maximal ordered triangulations both have a main peak around the eigenvalue magnitude of 2. The only difference is that at \(\alpha \approx 0\) the random PDF has a higher value than the maximal ordered one, and vice versa at \(\alpha \approx 4.5\).
If one compares the Laplacian random spectrum with the spectrum of the maximal ordered triangulation two main things differ: For the maximal ordered triangulation the largest possible eigenvalue is 9 (as shown analytically for the periodic maximal ordered triangulation), whereas the largest eigenvalues of the random triangulations are much higher for increasing lattices sizes, which can be understood in terms of the lower bound (4) of the spectral radius given by the maximal vertex degree that is higher for random triangulations. The indexsummed PDF is peaked around an eigenvalue magnitude of 8 for the maximal ordered triangulation, for the random triangulation (of lattices with size bigger than \(10\times 10\)) there is a peak around an eigenvalue magnitude of 5 which is less dominant than the peak of the maximal ordered one.
For the considered parameter set of Erdös–Rényi random graphs there are approximative analytical calculations [20] for the spectrum of the Laplacian that coincide with earlier numerical calculations [12]. Ding and Jiang [21] showed that the spectrum of the adjacency matrix of a random (Erdös–Rényi) graph converges to the semicircle distribution, and the spectrum of the Laplacian matrix converges to a free convolution of the semicircle distribution and a normal distribution.
The Laplacian spectrum of the Erdös–Rényi random graph looks similar to the spectrum of the triangulations for eigenvalue magnitudes bigger than 6, both exhibit a linear density decrease for increasing eigenvalue magnitudes. For eigenvalue magnitudes between 2 and 6 there are several peaks in the spectrum of the random triangulation which also survive if one considers the limit \(MN \rightarrow \infty \), whereas the random graph spectrum in this interval is nearly linear. Between eigenvalue magnitudes 0 and 2 the Laplacian spectrum of the random graph shows two peaks and three dips, while the spectrum of the random triangulation is smooth at most for large lattice sizes.
Algebraic Connectivity and Spectral Radius In this section we examine the dependence of the smallest Laplacian eigenvalue \(\lambda _1\) (algebraic connectivity) and the biggest Laplacian eigenvalue \(\lambda _{MN1}\) (spectral radius) of random triangulations on the lattice size. The results of the MonteCarlo simulations for random triangulations can be found in Fig. 8, as well as the values of the algebraic connectivity and the spectral radius for the different considered models of random graphs.
Scaling of algebraic connectivity, spectral radius and average inverse participation ratio with the system size
Quantity  Graph  Scaling behavior  Reduced \(\chi ^2\) 

\(\langle \lambda _1 \rangle \)  Per. triangular lat.  \(12 \pi (MN)^{1}\)  – 
rnd. triangs.  \(0 + (10.7 \pm 0.2)\cdot (MN)^{0.949 \pm 0.003}\)  \(2.5 \cdot 10^{8}\)  
Erdös–Rényi  \(\rightarrow 0\)  –  
Newman–Watts  \(0.541 \pm 0.003\)  –  
Barabási–Albert  \(1.235 \pm 0.001\)  –  
\(\langle \lambda _{MN1} \rangle \)  per. triangular lat.  9  – 
rnd. triangs.  \((27.4 \pm 0.5)  (27.6 \pm 0.3)\cdot (MN)^{0.147 \pm 0.006}\)  \(4.3 \cdot 10^{3}\)  
Erdös–Rényi  \((21.4 \pm 0.2)  (23.6 \pm 0.2)\cdot (MN)^{0.236 \pm 0.006}\)  \(3.5 \cdot 10^{3}\)  
Newman–Watts  \((16.01 \pm 0.08)  (20.8 \pm 0.2)\cdot (MN)^{0.301 \pm 0.006}\)  \(2.6 \cdot 10^{3}\)  
Barabási–Albert  \(\rightarrow \infty \)  –  
\(\langle \overline{\chi }\rangle \)  Per. triangular lat.  \((MN)^{1}\)  – 
rnd. triangs.  \((0.643 \pm 0.009)\cdot (MN)^{0.497 \pm 0.004} + (0.0522 \pm 0.0002)\)  \(2.4 \cdot 10^{7}\)  
Erdös–Rényi  \((1.19 \pm 0.03)\cdot (MN)^{0.601 \pm 0.007} + (0.0074 \pm 0.0007)\)  \(1.7 \cdot 10^{6}\)  
\((1.02 \pm 0.03)\cdot (MN)^{0.539 \pm 0.007}\)  \(8.8 \cdot 10^{6}\)  
Newman–Watts  \((0.59 \pm 0.05)\cdot (MN)^{0.58 \pm 0.02}\)  \(1.6 \cdot 10^{5}\)  
Barabási–Albert  \((0.820 \pm 0.006)\cdot (MN)^{0.383 \pm 0.002}\)  \(1.5 \cdot 10^{6}\) 
Inverse Participation Ratio and Localization In this section the IPR (5) of the Laplacian spectrum of random triangulations is examined in terms of the system size. In Fig. 9 the average IPR \(\langle \overline{\chi }\rangle \), as well as the IPR of the algebraic connectivity \(\langle \chi _1 \rangle \) and the spectral radius \(\langle \chi _{MN1} \rangle \) are displayed and compared with the common random graph models.
The IPR \(\langle \chi _1 \rangle \) of the algebraic connectivity random lattice triangulations is approximately equal to that of the maximal ordered triangulation and decreases with a power law \(\propto x^{0.896 \pm 0.006}\), since the algebraic connectivity is determined by the vertices with low degree (which are the vertices at the corner of the lattice), and the degree is likely to be unchanged for the random triangulations. One can find a similiar decrease for Barabási–Albert , but not for Erdös–Rényi or Newman–Watts random graphs (which converge towards a finite value for increasing system size). The IPR of the spectral radius converges to a value above 0.8, with a similiar functional dependency as the other random graph models.
The probability distribution function of the IPRs for random triangulations is comparable to the one of random graphs, there are differences only in the probability of the largest IPRs, which correspond to the strongest localization.
4.2 Microcanonical Ensemble
In this section we consider triangulations with a fixed energy which corresponds to a microcanonical ensemble and examine the Laplacian spectrum, the algebraic connectivity \(\lambda _1\) and the spectral radius \(\lambda _{MN1}\) in terms of the energy for different system sizes. For each lattice size we measure the energy in units of the average energy \(\langle E \rangle _{\mathrm {rnd}}\) of random triangulations on an equal sized lattice to make the results for different lattice sizes comparable. We will use \(\epsilon = E / E_{\mathrm {rnd}}\) to denote this rescaled energy.

Start with a random triangulation with arbitrary energy, generated as described in Sect. 4.1.
 Perform Metropolis Monte Carlo steps [50] with the acceptance probabilityand check after each step whether the obtained triangulation has the desired energy, then stop (the inverse temperature \(\beta \) can be tuned to find the desired energy more quickly). Note that the actual number of steps necessary to find a suitable triangulation depends on the given energy and cannot be predicted.$$\begin{aligned} A_{\mathrm {Metropolis}} (\mathscr {T}_1 \rightarrow \mathscr {T}_2) := \min \left( 1, \frac{\exp [\beta E(\mathscr {T}_2)]}{\exp [\beta E(\mathscr {T}_1)]} \right) \end{aligned}$$(18)

If the desired energy was reached, take the triangulation for measuring the observables and perform 1000MN steps at \(\beta = 0\) to randomize the triangulation and avoid autocorrelations between successive measurements. Note that the autocorrelation time is always below 10MN as explained in Sect. 4.1.
Laplacian spectrum of a microcanonical ensemble of \(16\times 16\) triangulations. The color code (a) on the left shows the indexresolved probability density function (PDF) in terms of the index of the eigenvalue and the magnitude of the eigenvalues for normalized energies \(E / \langle E \rangle _{\mathrm {rnd}} = 0.2, 0.6, 1.0, 3.0\), with \(\langle E \rangle _{\mathrm {rnd}} \approx 1000\). The indexsummed PDF (b) is also displayed for the same normalized energies (Color figure online)
4.3 Canonical Ensemble
The results for the microcanonical ensemble of triangulations as calculated in the previous section depend strongly on the choice of the energy functional. In this section we present some results for canonical averages of observables in lattice triangulations. These have in our opinion the advantage that the results should not change qualitatively if one changes the energy functional quantitatively. Especially the canonical average infinite temperature (\(\beta = 0\), which is equivalent to the ensemble of random triangulations) is independent of the choice of the energy functional, additionally the limits for \(\beta \rightarrow \pm \infty \) should agree qualitatively.
One can use the standard Metropolis Monte Carlo simulations [50] for calculating canonical ensemble averages for triangulations by using the acceptance probability (18), but there is a problem occurring in our setup if trying to access the regime of negative temperatures, which can also be identified with negative coupling and positive temperatures [44, 46]: For high energies there are triangulations that are local minima in the energy landscape, i.e., all triangulations that are connected with these triangulations have a higher energy, so for all possible steps the Metropolis acceptance probabilities (18) are small. For \(\beta < 0\) small enough the algorithm cannot leave these states in the available computation time, although of course there are other states that contribute to the canonical expectation values. So the Metropolis results are wrong and the system is not computationally ergodic anymore. Using a parallel tempering approach [24] also fails since there is a quasi phase transition for small negative \(\beta \) due to the large free energy barrier [46].
For negative temperatures there can also be a problem with increasing autocorrelation time, as considered numerically in [46]. A similar problem was considered analytically for lattice triangulations in Refs. [14, 67], where Glauber dynamics was used, which is basically the Metropolis algorithm with a slightly different choice of the acceptance probabilities. Therein, the authors used an energy function that measures the sum of the edge lengths that qualitatively agrees with the energy function (2). Instead of using the inverse temperature as control parameter, the coupling constant of the energy was modified and allows to create ordered and disordered lattice triangulations. It was shown in Ref. [14] that the mixing time, which is the time until the Markov chain is near to statistical equilibrium, scales exponentially with the system size for the analog of \(\beta < 0\), and polynomial for the analog of a sufficiently high \(\beta > \beta _0 > 0\), furthermore it was conjectured that the mixing time is polynomial for all \(\beta > 0\). For random triangulations (which is equal to \(\beta = 0\)) no result could be obtained. Although these results are for the mixing and not for the autocorrelation time, one can conjecture a strong relation between both.
Density of states (black filled curve) for \(8\times 8\)triangulations and the probability P(E) (colored lines) for the triangulation to have energy E in the canonical ensemble for different values of the inverse temperature \(\beta \) (Color figure online)
For twodimensional unimodular lattice triangulations the density of states can only be calculated for all possible energies up to \(11\times 11\) triangulations using the Wang–Landau algorithm, but it is possible to calculate the DOS for energies \(E \le E_c\) smaller than a cutoff \(E_c\) for up to \(25 \times 25\) triangulations [44]. This DOS with cutoff can only be used for calculating canonical expectation values for \(\beta > \beta _c\), where \(\beta _c\) is a cutoff in the inverse temperatures, which is negative for the DOS calculated in [44].
One has to keep in mind that the regime of negative couplings (\(\beta < 0\)) is not well defined in the thermodynamic limit (infinite system sizes) since the new specific groundstate energy (which is the negative maximal energy of the positivecoupling model) is not bounded from below [46]. So one can consider only finite system sizes for \(\beta < 0\).
Laplacian spectrum of a canonical ensemble of \(8\times 8\) triangulations. The color code (a) left shows the indexresolved probability density function (PDF) in terms of the index of the eigenvalue and the magnitude of the eigenvalues for inverse temperatures \(\beta = 0.1, 0.04, 0, 0.2\). The indexsummed PDF (b) is also displayed for the different inverse temperatures (Color figure online)
Algebraic Connectivity and Spectral Radius The results for the canonical expectation values of algebraic connectivity \(\lambda _1\) and the spectral radius \(\lambda _{MN  1}\) are displayed in Fig. 14 for lattice size \(8\times 8\).
Canonical averages of the algebraic connectivity (a) and spectral radius (b) in terms of the inverse temperature \(\beta \) for different lattice sizes, calculated using the Wang–Landau algorithm. The algebraic connectivity is independent of the system size for negative temperatures, whereas the spectral radius is independent of the system size for positive temperatures
Canonical averages of the inverse participation ratio for \(8 \times 8\) lattices. (a) Expectation values of the average IPR (black, solid line), the IPR of the algebraic radius (black, dashed line) and of the spectral radius (black, dashdotted line) in terms of the inverse temperature. (b) The same expectation values in terms of the (logarithmically plotted) distance from a quasicritical temperature \(\beta _{\mathrm {qc}}\), for \(\beta > \beta _{\mathrm {qc}}\) (\(\langle \overline{\chi }\rangle \), \(\langle \chi _{MN1} \rangle \)) and for \(\beta < \beta _{\mathrm {qc}}\) (\(\langle \chi _{1} \rangle \)). The thinner, red lines are linear fits with respect to \(\log \beta  \beta _{\mathrm {qc}}\). Note that the value of \(\langle \chi _1 \rangle \) is streched by a factor 10 (Color figure online)
Inverse Participation Ratio In Fig. 15 the average IPR and the IPRs of the algebraic connectivity and the spectral radius are plotted in terms of the inverse temperatures. As expected, all considered IPRs and therewith the localization increases for increasing disorder in the triangulations. Near the quasicritical inverse temperature \(\beta _{\mathrm {qc}}\) one observes that the IPRs scale with the logarithm \(\log \beta  \beta _{qc}\) of the reduced inverse temperature.
5 Conclusions
In this paper we examined the spectral properties of the graph interpretation of twodimensional unimodular lattice triangulations. These triangulations show a transition from ordered largeworld to disordered smallworld behavior.
For the random triangulations we calculated the spectrum of the adjacency and the Laplacian matrix numerically and compared them with common random graph models. The algebraic connectivity, which is the smallest nonvanishing eigenvalue of the Laplacian matrix, vanished with a power law in terms of the system size, which is similar to the Erdös–Rényi graph, but different to the Newman–Watts and Barabási–Albert random graph, which tend to a fixed value for increasing lattice size. Calculating the average inverse participation ratio we showed that random lattice triangulations on average exhibit stronger localization than comparable random graphs.
We introduced an energy function that corresponds to the order and the disorder of the lattice triangulation to calculate the dependence of the spectral observables on the order of the triangulation in the microcanonical and the canonical ensemble.
For the microcanonical ensemble we find for small energies a linear dependence of the algebraic connectivity and the spectral radius on the energy of the triangulations. In the case of the spectral radius the linear dependence can be understood analytically using the ground state values, that can be calculated analytically, and perturbation theory. For the algebraic connectivity perturbation theory fails because the fixed boundary conditions have stronger influence than the perturbation of the flip.
In the canonical ensemble, that was numerically calculated with the Wang–Landau algorithm, we find a crossover between ordered, largeworld behavior with a systemsize independent spectral radius and a disordered, smallworld behavior with a systemsize independent algebraic connectivity and stronger localization than in the ordered phase. This crossover behavior can also be found in other observables as the mean energy, the clustering coefficient and the average shortest path length [46].
Our results can be applied to planar realworld graphs where the actual coordinates of the vertices or the distance between vertices becomes important or must be optimised, e.g., in the power grid or in transport problems. Using also the inserting and removing Pachner moves that lead to nonfull triangulations the properties of a grandcanonical ensemble of triangulations can be studied in the future. In order to take into account arbitrary embedded planar networks, one can also examine triangulations of random point sets instead of a square lattice. It can also be promising to study the localization in greater detail to understand better the underlying mechanism, e.g., on which vertices the eigenvectors of the Laplacian matrix are localized.
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