T16 — Dimer Mobility Scaling

How does the fraction of mobile punctures scale with horizon size?

Objective

Compute the fraction of “mobile” punctures (those participating in flippable plaquettes) as the horizon area grows. This tests whether the RVB picture survives in the thermodynamic limit.

Key Results

N_p Graph Matchings f_dimer Frozen Full
6 random 7 0.905 0.0% 71.4%
12 random 103 0.702 2.9% 16.5%
18 random 1,030 0.669 0.4% 4.4%
24 random 10,104 0.629 0.2% 1.0%

ℹ️ Scaling Behavior

  • f_dimer decreases with N_p: Larger graphs have lower average mobility fraction per matching (0.905 → 0.629).
  • Frozen matchings decrease: The fraction of fully frozen coverings drops as N_p grows (0% → 0.2%).
  • Thermodynamic limit: Extrapolation suggests f_dimer → 0.537 as N_p → ∞.
  • RVB state: In the thermodynamic limit, roughly half the punctures are mobile in a typical matching.

Scaling Analysis

Extrapolation to N_p → ∞

Using the fit f = f_∞ + c/N_p:

\[ f_{dimer}(\infty) \approx 0.537 \]

⚠️ Partial Mobility

The QHE/BHE analogy predicts that in a macroscopic black hole, about 54% of punctures are mobile in a typical dimer covering, while 46% are locally frozen. This is a weaker form of topological order than the QHE Laughlin state (which has perfect mobility).

Distribution of Matchings

N_p % Frozen % Partial % Full
6 0.0% 28.6% 71.4%
12 2.9% 80.6% 16.5%
18 0.4% 95.2% 4.4%
24 0.2% 98.8% 1.0%

✅ Pattern Emerging

As N_p grows, both frozen and full-mobility matchings vanish. The vast majority of matchings are partially mobile, with ~54% of punctures flippable on average.

Visualization

Dimer mobility scaling

Interpretation

The dimer mobility fraction f_dimer measures what fraction of punctures participate in at least one flippable plaquette in a typical matching. As the horizon grows:

  • The number of dimer coverings grows exponentially (10^4 at N_p=24)
  • The fraction of frozen matchings vanishes (0.2% at N_p=24)
  • The fraction of full-mobility matchings also vanishes (1.0% at N_p=24)
  • Most matchings are partially mobile, with ~54% of punctures flippable

This suggests the QHE/BHE horizon is in a partially liquid state: not fully rigid (solid-like) but not fully resonant (liquid-like) either. The punctures form a mixed phase where some regions are mobile and others are frozen.

Comparison to QHE

In the quantum Hall effect, the Laughlin state is a true liquid with perfect mobility (all electrons participate in collective motion). The BHE analogy predicts a weaker form of topological order: a partially resonating state where only ~54% of punctures are mobile.

This is a testable prediction: if the BHE analogy is correct, numerical simulations of LQG horizons should show that the RVB state is only partially mobile, not fully resonant.

Code

The Python code used to generate these results:

  • t16_dimer_scaling.py — Exact dimer coverings for random spherical triangulations, mobility statistics, and scaling analysis

All code is MIT-licensed and part of the qhe_bhe repository.

Status

🟢 Completed - Exact results for N_p = 6, 12, 18, 24 ✅ - Extrapolation to thermodynamic limit ✅ - Scaling analysis and visualization ✅

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