Kpz Universality

KPZ Universality Skill

Status: ✅ Production Ready Trit: 0 (ERGODIC - coordinator/neutral) Color: #7B68EE (Medium Slate Blue) Principle: Universal scaling limit of random interface growth Frame: 1:2:3 scaling (space:time:fluctuation), Tracy-Widom fluctuations

Overview

The Kardar-Parisi-Zhang (KPZ) universality class describes the universal scaling behavior of random interface growth models. This skill provides:

1. KPZ Fixed Point: The universal Markov process governing height fluctuations
2. Directed Landscape: The common noise coupling all initial conditions
3. Scaling Theory: The 1:2:3 exponents and Tracy-Widom limits
4. Model Zoo: TASEP, PNG, LPP, polymers, random matrices

The KPZ Equation (SPDE)

∂h/∂t = ν∇²h + (λ/2)(∇h)² + √D ξ(x,t)

Where:
  h(x,t) = height function
  ν = diffusion coefficient (smoothing)
  λ = nonlinearity (growth rate)
  D = noise strength
  ξ = space-time white noise

Mathematical Challenge

The equation is ill-posed due to roughness:

• ∇h is not a function but a distribution
• (∇h)² is not well-defined
• Requires renormalization or regularity structures (Hairer 2013)

The KPZ Fixed Point

The KPZ fixed point is the universal Markov process that emerges under scaling:

h^ε(x,t) = ε^(1/2) h(ε^(-1)x, ε^(-3/2)t) →  KPZ fixed point

Scaling exponents:
  α = 1/2  (roughness)
  β = 1/3  (growth)
  z = 3/2  (dynamic)
  
Relation: α + z = 2, β = α/z

Tracy-Widom Fluctuations

For narrow-wedge initial condition:

P(h(0,t) ≤ s) → F_GUE(s) as t → ∞

Where F_GUE = Tracy-Widom GUE distribution

The Directed Landscape

The directed landscape L(x,s; y,t) is the scaling limit of last passage percolation:

L: {(x,s; y,t) : s < t} → ℝ ∪ {-∞}

Key properties:
1. Metric composition: L(x,s; y,t) = max_z [L(x,s; z,r) + L(z,r; y,t)]
2. Airy sheet initial data: L(x,0; y,1) = Airy sheet
3. Stationarity: L(x+a, s+c; y+a, t+c) =^d L(x,s; y,t)

Coupling via Directed Landscape

All KPZ models with same noise can be coupled:

# Different initial conditions, same noise
h_flat(x, t) = max_y [h₀_flat(y) + L(y, 0; x, t)]
h_wedge(x, t) = max_y [h₀_wedge(y) + L(y, 0; x, t)]

Interface with Colored Vertex Models

The colored stochastic six-vertex model connects KPZ to quantum integrable systems:

Yang-Baxter ← Fusion → Cuboson Weights
     ↓                      ↓
Six-Vertex  ← Gibbs →  Height Function
     ↓                      ↓
KPZ Fixed Point ← Limit → Directed Landscape

See: yang-baxter-integrability, colored-vertex-model skills.

Core Capabilities

1. KPZ Height Simulation

using KPZUniversality

# Simulate KPZ equation via directed polymers
sim = simulate_kpz(
    initial_condition = :wedge,  # or :flat, :stationary
    L = 1000,                    # system size
    T = 100.0,                   # final time
    discretization = :polymer    # or :weakly_asymmetric
)

# Extract height at time T
h = height_at(sim, T)

# Check Tracy-Widom convergence
@test kolmogorov_smirnov(rescaled_fluctuations(h), tracy_widom_gue()) < 0.1

2. Directed Landscape Sampling

# Sample directed landscape on grid
landscape = sample_directed_landscape(
    x_range = -10:0.1:10,
    t_range = 0:0.1:10,
    resolution = 0.01
)

# Evaluate path weight
weight = landscape(x_start, t_start, x_end, t_end)

# Compute geodesic
geo = geodesic(landscape, (x1, t1), (x2, t2))

3. Last Passage Percolation

# Geometric LPP (exponential weights)
lpp = geometric_lpp(
    n = 100,              # grid size
    weights = :exponential
)

# Compute passage time
G = passage_time(lpp, (1, 1), (n, n))

# Tracy-Widom rescaling
chi = (G - 4n) / (2^(4/3) * n^(1/3))

4. TASEP Dynamics

# Totally Asymmetric Simple Exclusion Process
tasep = TASEP(
    L = 100,              # system size
    initial = :step,      # step initial condition
    rate = 1.0
)

# Run simulation
simulate!(tasep, T = 100.0)

# Current fluctuations
J = current(tasep)

Integration with Gay-MCP (Colored Projections)

KPZ models with colored particles connect directly to Gay-MCP:

using GayMCP, KPZUniversality

# Colored TASEP with GF(3) conservation
ctasep = ColoredTASEP(
    n_colors = 3,
    seed = gay_seed(1069)
)

# Project to single color
single_color = project(ctasep, color = 1)
# Result satisfies ordinary TASEP statistics

# Verify GF(3) conservation
@test sum(color_trit.(ctasep.particles)) % 3 == 0

Integration with Langevin-Dynamics

KPZ equation is a Langevin equation for interface growth:

using LangevinDynamics, KPZUniversality

# Convert KPZ to standard Langevin form
langevin = kpz_to_langevin(
    kpz_params = (ν = 1.0, λ = 2.0, D = 1.0),
    discretization = :imhof_mannella
)

# Solve with instrumented noise
sol, audit = solve_langevin(langevin, seed = gay_seed(42))

# Verify Gibbs property
@test check_gibbs_property(sol, :inter_color)

Integration with Fokker-Planck

The Fokker-Planck equation for KPZ describes probability flow:

using FokkerPlanck, KPZUniversality

# Fokker-Planck for height distribution
fp = kpz_fokker_planck(
    initial_distribution = :delta_wedge,
    temperature = T
)

# Stationary measure (Louisville quantum gravity)
stationary = louisville_measure(fp)

# Verify convergence to stationary
@test kl_divergence(evolve(fp, t), stationary) < 0.01

GF(3) Triad Assignment

Conservation: (-1) + (0) + (+1) = 0 ✓

Configuration

# kpz-universality.yaml
simulation:
  discretization: polymer  # polymer, weakly_asymmetric, hopf_cole
  grid_size: 1000
  time_steps: 10000
  seed: 0xDEADBEEF

scaling:
  alpha: 0.5      # roughness
  beta: 0.333     # growth
  z: 1.5          # dynamic

verification:
  tracy_widom_test: true
  airy_process_test: true
  directed_landscape_test: true

Commands

# Simulate KPZ
just kpz-simulate initial=wedge T=100

# Sample directed landscape
just kpz-landscape n=1000

# Run LPP
just kpz-lpp weights=exponential n=100

# TASEP simulation
just kpz-tasep L=100 T=100

# Verify Tracy-Widom
just kpz-verify-tw

Related Skills

• yang-baxter-integrability (-1): Quantum integrability structure
• colored-vertex-model (+1): Colored stochastic models
• louisville-quantum-gravity (+1): Stationary measures
• last-passage-percolation (0): Discrete KPZ model
• langevin-dynamics (0): SDE perspective
• fokker-planck-analyzer (-1): Probability evolution
• modelica (0): Acausal dynamics modeling

Research References

1. Corwin (2012): "The Kardar-Parisi-Zhang equation and universality class"
2. Quastel-Remenik (2023): "KPZ fixed point and directed landscape"
3. Matetski-Quastel-Remenik (2021): "The KPZ fixed point"
4. Dauvergne-Ortmann-Virág (2021): "The directed landscape"
5. Borodin-Corwin (2014): "Macdonald processes"

Skill Name: kpz-universality Type: Stochastic Interface Growth Trit: 0 (ERGODIC) Key Property: 1:2:3 scaling, Tracy-Widom fluctuations Status: ✅ Production Ready

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule:

Trit: 0 (ERGODIC)
Home: Stochastic
Poly Op: ⊗
Kan Role: Adj
Color: #7B68EE

Height Function as Lens

The height function h(x,t) forms a lens in the polynomial category:

Height = (SpaceTime, Fluctuation)
get: SpaceTime → Height
put: SpaceTime × Noise → Height

Directed Landscape as Profunctor

The directed landscape L(x,s; y,t) is a profunctor:

L: Space × Time ↛ Space × Time
L(x,s; y,t) = max passage weight from (x,s) to (y,t)

This profunctor satisfies metric composition (Yoneda-like):

L(x,s; y,t) = max_z [L(x,s; z,r) + L(z,r; y,t)]

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

Autopoietic Marginalia

The interface grows. The fluctuations universalize. The directed landscape couples all.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record Tracy-Widom deviations
• REMEMBERING (0): Connect to other integrable systems
• WORLDING (+1): Evolve scaling predictions

Add Interaction Exemplars here as the skill is used.