Emergent spectral geometry in the Coherence--Curvature Model

TL;DR

The paper addresses how coherent diffusion and curvature shape emergent geometry in ensembles of discrete graphs. It introduces the Coherence--Curvature Model (CCM) with the Hamiltonian $H(G) = -\alpha\lambda_2(G) + \beta|E| + \gamma\sum_v R_O(v)$ and samples low-energy graphs via connected simulated annealing, measuring $d_s$, $d_h$, and $\langle d\rangle$. It finds a finite-dimensional, fractal-like geometry with $d_s > d_h$, where $d_s$ grows with system size and can approach four, while $d_h$ remains near three, and large-scale distances grow only slowly, indicating small-world behavior; crucially, increasing the curvature coupling $\gamma$ tunes $d_s$ largely independently of $d_h$. The study demonstrates decoupling of spectral and volumetric dimensions and identifies a rich parameter space via $(\gamma,\beta)$ that shapes locality and diffusion, providing a controlled numerical laboratory for geometrogenesis in discrete gravity-inspired models. While limited by modest system sizes and slow convergence of spectral observables, the CCM offers a concrete framework to compare emergent geometries with other discrete approaches (e.g., causal dynamical triangulations, group field theories) and to map how microscopic dynamics translate to finite-dimensional, fractal-like manifolds.

Abstract

We investigate the Coherence--Curvature Model (CCM), a dynamical ensemble of connected graphs governed by a Hamiltonian that couples algebraic connectivity, Ollivier-Ricci curvature, and an edge-density penalty. Using connected simulated annealing we generate low-energy graph configurations and characterize their emergent geometry through the spectral dimension (ds), the Hausdorff dimension (dh), and the average distance. Finite-size scaling shows a clear growth of ds with system size, while dh increases more mildly. At the largest volumes explored the data are compatible with ds ~ 4 and dh ~ 3, implying ds > dh and a nontrivial hierarchy between spectral and volumetric notions of dimension. We also map the dependence on the curvature coupling gamma and the locality coupling beta, and we find a slow power-law growth of typical distances with a small exponent eta. The CCM therefore provides a controlled numerical laboratory in which the interplay of coherence, curvature, and locality yields finite-dimensional, fractal-like geometries.

Figures (5)

• Figure 1: Finite-size scaling of the CCM ensemble at fixed couplings $(\alpha,\beta,\gamma) = (1.0, 0.0091, 0.15)$ and annealing depth $S_{\max} = 16000$ steps. The panels show (a) the spectral dimension $d_s$, (b) the Hausdorff dimension $d_h$, and (c) the ratio $d_s/d_h$ as functions of the number of vertices $N \in \{128,256,512,1024\}$. Error bars denote one standard deviation across seeds, including fitting uncertainties where applicable. The data indicate a growth of $d_s$ with $N$ towards values compatible with four within current errors, a milder increase of $d_h$ towards values close to three, and a slow growth of typical distances consistent with the scaling form in Eq. \ref{['eq:distance-scaling']}.
• Figure 2: Annealing-depth study at fixed size $N=512$ and couplings $(\alpha,\beta,\gamma)=(1.0,0.0091,0.15)$. The panels show the dependence of energy density, algebraic connectivity $\lambda_2$, mean Ollivier--Ricci curvature, spectral dimension $d_s$, Hausdorff dimension $d_h$, and average distance $\langle d \rangle$ on the maximum number of Monte Carlo steps $S_{\max}\in\{4000,16000,32000\}$. Error bars denote one standard deviation across seeds. The results indicate a slow convergence of the spectral properties, with $d_s$ increasing for deeper annealing, while $d_h$ and $\langle d \rangle$ exhibit milder variations and stabilize more rapidly.
• Figure 3: Dependence of the CCM ensemble on the curvature coupling $\gamma$ at fixed size $N=256$, locality coupling $\beta=0.0091$, and annealing depth $S_{\max}=16000$ steps. The panels show the mean energy, algebraic connectivity $\lambda_2$, mean Ollivier--Ricci curvature, spectral dimension $d_s$, Hausdorff dimension $d_h$, and average distance $\langle d \rangle$ as functions of $\gamma\in\{0.05,0.15,0.30,0.60,1.20\}$. Error bars denote one standard deviation across seeds. Increasing $\gamma$ drives the system towards configurations with higher $\lambda_2$, less negative curvature, and larger $d_s$, while $d_h$ and $\langle d \rangle$ vary more slowly.
• Figure 4: Dependence of the emergent geometry on the locality coupling $\beta$ at fixed size $N=256$, curvature coupling $\gamma=0.15$, and annealing depth $S_{\max}=16000$ steps. Panel (a) shows the spectral dimension $d_s$ versus $\beta$, panel (b) the Hausdorff dimension $d_h$ versus $\beta$, and panel (c) the average distance $\langle d \rangle$ versus $\beta$, for $\beta\in\{0.0050,0.0091,0.0150,0.0250,0.0400\}$. Error bars denote one standard deviation across seeds. Increasing $\beta$ strengthens the edge-density penalty, leading to sparser graphs with larger typical distances and modest variations in $d_h$, while $d_s$ exhibits a more intricate, non-monotonic response.
• Figure 5: Consistency checks for the CCM ensemble at fixed couplings $(\alpha,\beta,\gamma)=(1.0,0.0091,0.15)$ and annealing depth $S_{\max}=16000$ steps. The panels illustrate the scaling of the average distance $\langle d \rangle(N)$ across the FSS volumes $N=128$--$1024$, together with a power-law fit of the form $\langle d \rangle(N)\sim N^\eta$ and the associated residuals. The effective exponent is $\eta=0.126$, indicating a slow growth of typical distances with $N$ consistent with an effectively small-world, yet geometrically nontrivial, regime.