Random-environment dimers and spanning trees

Reproducible Julia experiments for height fluctuations in random tilings, spanning trees, and loop-erased random walks. The project tests whether variance grows like (\log L) or develops a super-rough ((\log L)^2) component, with particular attention to separating conditional sampling noise from fluctuations induced by a shared random environment.

The repository combines exact small-system checks, deterministic Monte Carlo campaigns, environment-blocked bootstrap inference, and restart-safe Slurm workflows. The active Julia package has no third-party runtime dependencies.

Research highlights

• Aztec diamonds: paired spatial-height increments show positive finite-size quadratic-log curvature in the disorder covariance for the original and stronger Gamma laws.
• Structured square-grid disorder: spanning-tree/Temperley experiments up to (L=6144) do not show a stable positive quadratic-log contribution.
• Direct weighted dimers: a 1,312-environment square-grid Glauber campaign separates conditional, disorder, and total central-height variance. An exact finite-volume Kasteleyn replay of all 960 Gamma environments confirms that no component has a robust positive quadratic-log coefficient over (L=2,\ldots,20), without relying on MCMC mixing.
• Negative controls: uniform or all-one environments do not create a spurious disorder component.

These are finite-size numerical findings, not asymptotic proofs. Exact estimates, uncertainty intervals, diagnostics, and limitations are recorded in the results ledger.

Quick start

Requirements: Julia 1.10 or newer.

cd research

# Run mathematical-reference and workflow tests.
julia --project=aztec -e 'using Pkg; Pkg.test()'
sh aztec/test/smoke_workflows.sh

Run a small deterministic Aztec-height campaign:

JULIA_NUM_THREADS=4 julia --project=aztec \
  aztec/scripts/run_height_campaign.jl \
  --config aztec/configs/gamma_height_smoke.csv \
  --output-dir aztec/output/height_smoke

Generated batches and scratch analyses belong under the ignored aztec/output/ directory. Retained observations and reviewed results live in aztec/data/ and aztec/results/.

Core estimators

For two conditionally independent replicas (H_1,H_2) in the same frozen environment (\omega),

[ \frac12\operatorname{Var}(H_1-H_2) =\mathbb E_\omega[\operatorname{Var}(H\mid\omega)], ]

and

[ \operatorname{Cov}(H_1,H_2) =\operatorname{Var}_\omega(\mathbb E[H\mid\omega]). ]

Every bootstrap therefore resamples whole environments. Paired replicas and all observables from one environment remain in the same resampling block.

Repository map

.
├── aztec/                 active Julia package, data, results, and CLIs
│   ├── src/               samplers, graph constructions, and observables
│   ├── scripts/           campaign, merge, analysis, and plotting commands
│   ├── configs/           deterministic smoke, pilot, and production schedules
│   ├── test/              exact, statistical, and end-to-end checks
│   ├── data/              compact retained input datasets
│   ├── results/           reviewed tables, reports, and vector figures
│   └── reference/         historical prototype retained for provenance
├── docs/                  research overview, results, roadmap, reproducibility
├── hpc/                   Slurm wrappers and cluster workflow documentation
├── archive/               self-contained earlier LERW experiments
├── .github/workflows/     continuous integration
└── CONTRIBUTING.md        scientific and engineering contribution rules

Start with the documentation index, then use the package guide for model-specific commands.

Reproducibility guarantees

• deterministic seed derivation from campaign identifiers;
• independent random streams where required by the estimand;
• environment-level pairing preserved end to end;
• atomic, restart-safe output batches;
• explicit campaign metadata and execution provenance;
• exact enumeration and detailed-balance checks at small sizes;
• exact finite-volume dimer moments from selected inverse Kasteleyn entries;
• reference-vs-optimized observable tests;
• uncertainty and model comparisons repeated across fit windows;
• controls analyzed with the same pipeline as disordered models.

See Reproducibility for data flow, validation levels, and the commands used to regenerate retained analyses.

Documentation

• Research overview — models, observables, and statistical definitions.
• Results ledger — chronological numerical evidence and caveats.
• Roadmap — current validation work and decision gates.
• Implementation guide — sampler and data-flow details.
• Square-grid model contract — graph, matching, height, and environment conventions.
• HPC workflow — generic Hamilton/Slurm setup and submission.

Scope

The repository supports numerical research and reproducible analysis. Large raw HPC traces are intentionally kept outside Git; compact retained datasets, derived tables, figures, checksums, and exact campaign configurations are kept here when practical.

Citation metadata for the software is available in CITATION.cff.