Euclidean Dynamical Triangulations and Their Relation to CDT

Euclidean dynamical triangulations (EDT) sum equilateral Euclidean triangulations without CDT’s preferred time foliation or causal gluing restrictions. The shared simplex building blocks do not imply a shared universality class: configuration-space entropy and phase order differ, and the original four-dimensional EDT model is dominated by crumpled and branched-polymer regimes.

Required background. Causal Dynamical Triangulations: Phases and Continuum Evidence defines the causal construction. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions supplies the topology contract.

Helpful background. Lattice Regulators and Target Continuum Theories and Complete Lattice Error Budgets supply comparison criteria.

Euclidean triangulation sum

For fixed four-dimensional topology and equilateral edge length $a$, the Regge action reduces to counts,

$$Z_{\mathrm{EDT}}(\kappa_2,\kappa_4) =\sum_T\frac{1}{C_T} \exp\!\left[\kappa_2N_2(T)-\kappa_4N_4(T)\right].$$

$N_2$ counts triangles and encodes integrated curvature; $N_4$ is volume. At fixed approximate $N_4$, $\kappa_2$ is scanned. No causal orientation or distinguished spatial slice is part of the configuration.

Original simulations found a crumpled phase with very large effective dimension and a branched-polymer phase with Hausdorff dimension near two, separated in a way not yielding an established smooth four-dimensional continuum limit Bialas et al. 1996.

Application: matched finite-size comparison

For EDT and CDT at the same nominal dimension, topology, and mean $N_4$, measure:

$$\chi_{N_2}=\frac{1}{N_4} \left(\langle N_2^2\rangle-\langle N_2\rangle^2\right),$$

the volume–volume correlator, graph geodesic radius, Hausdorff and spectral dimensions, Binder cumulants, and hysteresis. Reweight only within a verified overlap range and include autocorrelation errors.

In CDT also condition on time slices and simplex asymmetry; in EDT no corresponding observable exists. Quantities lacking a cross-regulator definition must be marked rather than forced into a numerical table. Compare transition order and finite-size exponents, not bare coupling locations.

Why causal restrictions matter

CDT permits a transfer matrix and a prescribed Wick rotation of Lorentzian triangulations. EDT samples a larger Euclidean configuration space whose entropy can overwhelm extended geometries. Removing configurations is a relevant change to the microscopic definition, not a gauge choice. Conversely, CDT’s foliation must be shown irrelevant or compatible with recovered covariance in the continuum.

Adversarial universality test

Add a local measure term to EDT or vary CDT asymmetry while tuning lines of constant physics. If universal observables converge to the same values with diverging correlation lengths, a shared universality class becomes plausible. Matching only spectral dimension over one window or a volume profile at one size is insufficient.

Scope

EDT is a precise statistical model and an important negative and comparative result. Modified measures may change its phase structure, but each modification requires a new continuum analysis. The next discrete route, Tensor Models and Random Geometry, generates related complexes from Feynman graphs.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Agishtein, M. E., and A. A. Migdal. “Simulations of Four-Dimensional Simplicial Quantum Gravity.” Modern Physics Letters A 7 (1992): 1039–1061. DOI.
• Bialas, P., Z. Burda, A. Krzywicki, and B. Petersson. “Focusing on the Fixed Point of 4D Simplicial Gravity.” Nuclear Physics B 472 (1996): 293–308. DOI.