Aubry-André Critical

random-matrix-quantum · 37 views
random-matrix-quantumquasiperiodic

What It Is

Almost-Mathieu operator at critical coupling λ=2 with golden-ratio incommensurate potential --- multifractal eigenstate intensity (deterministic multifractality, canonical exemplar from quasiperiodic origin)

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; few distinct values; low-dimensional; nonstationary / drifting. The atlas additionally detects Higher-Order Statistics:c3_energy, Attractor Reconstruction:lyap_entropy.

What standard analysis sees
tail heaviness1.00
asymmetry1.00
occupancy0.01
short-range corr0.51
long-range memory0.72
spectral colour0.45
periodicity0.42
complexity0.67
time-irreversibility0.44
volatility clustering0.53
multifractality0.84
dimensionality0.04
nonstationarity0.98
What the atlas adds
Higher-Order Statistics:c3_energy+3.4z
third-cumulant burst energy (SOC irreducibility signature)
Attractor Reconstruction:lyap_entropy+2.9z
Kolmogorov-Sinai entropy production rate (sum of positive λ)
Atlas-extreme metrics the standard bank can’t predict for this source
Mostow Rigidity:laplacian_perturbation_stability-6.3zbank-miss 1.8σ
2-adic:mean_distance-4.1zbank-miss 1.2σ
Laplacian:poisson_recovery_error-2.4zbank-miss 3.4σ
Klein Bottle:rank_deficit+2.3zbank-miss 1.6σ
Ulam Spiral (Square):polynomial_concentration-2.1zbank-miss 1.0σ
Klein Bottle:rank_deficit_max+2.1zbank-miss 1.4σ

Composition

dtypefloat64
range[1.677e-14, 0.02408]
unique values16384 / 16384
mean ± std6.1e-05 ± 0.000385

Render Gallery

Native > phase_plot
Native > phase_plotraw
Native > timeseries
Native > timeseriesraw
Native > triple_lattice
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > lag_map
Boltzmann > spin_grid
Boltzmann > spin_grid
Cayley > default
Cayley > default
Chladni > free_plate_nodal
Chladni > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_bottom
Chladni > plate_top
Chladni > plate_top
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > xy_path
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
Hodge–Laplacian > default
Inflation (Substitution) > barcode
Inflation (Substitution) > barcode
Inflation (Substitution) > d_curve
Inflation (Substitution) > d_curve
Information Theory > default
Information Theory > default
Julia Set > escape_histogram
Julia Set > escape_histogram
Klein Bottle > default
Klein Bottle > default
Laplacian > default
Laplacian > default
Logarithmic Spiral > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Möbius-S³ > s2_base_map
Ordinal Partition > default
Ordinal Partition > default
Penrose (Quasicrystal) > phi_spectrum
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > barcode
Persistent Homology > h1_diagram
Persistent Homology > h1_diagram
Predictability > default
Predictability > default
Recurrence Quantification > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Analysis > default
Spectral Graph > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > flux_grid
Symplectic > phase_space
Symplectic > phase_space
Ulam Spiral (Sacks) > default
Ulam Spiral (Sacks) > default
Ulam Spiral (Square) > default
Ulam Spiral (Square) > default
Visibility Graph > default
Visibility Graph > default
Wavelet Cascade > default
Wavelet Cascade > default

Atlas Position

Nearest neighborDistance
Anderson 1D Localized4.40
Fibonacci Tight-Binding4.50
Accel Sit4.75cross-origin

Open in Atlas →

Which Geometries Light Up

2-adic › 2-adic:distance_entropyrank 302/3060.7532
2-adic › 2-adic:mean_distancerank 303/3060.1079
Attractor Reconstruction › Attractor Reconstruction:embedding_divergence_sumrank 3/3063.5017
Attractor Reconstruction › Attractor Reconstruction:lyap_entropyrank 5/3063.5017
Catch24 › Catch24:DN_HistogramMode_10rank 1/3063.6296
Catch24 › Catch24:DN_HistogramMode_5rank 1/3067.3919
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiffrank 302/3060.0142
Catch24 › Catch24:DN_Meanrank 306/3060.0018
Catch24 › Catch24:DN_Spread_Stdrank 306/3060.0133
Chladni › Chladni:temporal_burstinessrank 1/3060.7671
Continued Fraction › Continued Fraction:gauss_kuzmin_distancerank 1/3060.8724
D4 Triality › D4 Triality:triplet_temporalrank 5/3060.4016
Higher-Order Statistics › Higher-Order Statistics:kurt_maxrank 1/3067.7395
Higher-Order Statistics › Higher-Order Statistics:skew_meanrank 1/3063.2447
Hodge–Laplacian › Hodge–Laplacian:source_fractionrank 4/3060.6662
Klein Bottle › Klein Bottle:rank_deficitrank 1/3060.9919
Klein Bottle › Klein Bottle:rank_deficit_maxrank 2/3060.9996
Mostow Rigidity › Mostow Rigidity:distance_rigidityrank 304/3060.0361
Mostow Rigidity › Mostow Rigidity:laplacian_perturbation_stabilityrank 305/3060.5348
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_finerank 302/3060.0011
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_meanrank 303/3060.0011
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_stdrank 303/3060.0002
Predictability › Predictability:cond_entropy_k1rank 302/3060.0135
Projective ℙ² › Projective ℙ²:mean_distancerank 302/3060.0001
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_raterank 1/3061.9883
Spherical S² › Spherical S²:concentrationrank 2/3060.9992
Spherical S² › Spherical S²:hemisphere_balancerank 302/3060.0003
Spherical S² › Spherical S²:angular_spreadrank 304/3060.0453
Spirograph › Spirograph:rational_grid_proximityrank 302/3060.5052
Symplectic › Symplectic:phase_reflection_symmetryrank 1/3060.9999
Symplectic › Symplectic:recurrence_raterank 1/3060.9831
Symplectic › Symplectic:windowed_area_cvrank 2/3069.2084
S² × ℝ (Thurston) › S² × ℝ (Thurston):sphere_concentrationrank 2/3060.9994
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersionrank 303/3060.0071
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentrationrank 302/3060.0204
Visibility Graph › Visibility Graph:degree_r_squaredrank 2/3060.9778
Wasserstein › Wasserstein:dist_from_uniformrank 1/3060.4837
Wasserstein › Wasserstein:concentrationrank 2/30631.7109
Wasserstein › Wasserstein:entropyrank 305/3060.0925
Wavelet Cascade › Wavelet Cascade:cascade_couplingrank 2/3060.9350
Zariski › Zariski:nonsep_fractionrank 2/3060.9767
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):gini_coefficientrank 2/3060.9744
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):gini_coefficientrank 3/3060.9629
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):hapax_ratiorank 5/3060.3175
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Structure Atlas — Open Atlas · GitHub