1.2.2. Kardar-Parisi-Zhang (KPZ) equation
(7)
- v0 can be absorbed.
- Scaling exponents:
1d: α = 1/2, β = 1/3, z = 3/2 (shown below)
2d: α ≈ 0.39, β ≈ 0.24, z ≈ 1.6
Note: naive calculation done below Eq.(4) gives wrong results here.
Coefficients can also be scale-dependent.
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1.2.3. Quenched KPZ equation
If noise comes from spatial heterogeneity, one should use the quenched KPZ eq.
(8)
- F cannot be absorbed (because noise depends on h)
- Pinning-depinning transition
F > Fc: interface grows, KPZ scaling (η(x,h) is equivalent to η(x,t))
F < Fc: interface pinned
F ≈ Fc: critical scaling α = β ≈ 0.63 “quenched KPZ class”
[homework (advanced)]
with exponents for directed percolation class.
In fact,
Read [4] and understand the relation to directed percolation.
1.2.4. Conserved growth
Consider particle deposition. If
(time scale of particle motion) << (time scale of deposition)
then the particle number is effectively conserved.
 Mullins-Herring (MH) equation
(9)
(MH class)
[homework] Derive these exponents.
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In practice, nonlinearity often plays role, then we obtain:
(10)
Confirmed in numerics and some experiments. Molecular beam epitaxy (MBE) class.
2. Basics on KPZ equation
In the following, we set v0=0.
2.1. Relation to noisy Burgers equation
(11)
This is the noisy Burgers equation (if λ=1).
Galilei symmetry: invariant under
 For KPZ, invariant under
(statistical tilt symmetry)
Galilei symmetry remains valid under scale transformation (4)
 λ is scale-invariant,
(12)
[homework] Show Eq.(12) from Eqs.(4), (7), and scale invariance of λ.
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2.2. Stationary KPZ interface (1d)
Fokker-Planck equation for KPZ eq.
(13)
For 1d (only), stationary solution (5) for the EW eq.,
(Brownian motion) is also the stationary solution for the KPZ eq.
[homework] Show this claim.
∴ α = 1/2.
Using Eq.(12), z = 3/2, β = 1/3.
2.3. EW  KPZ crossover
Starting from flat initial condition h(x,0)=0,
is small at short times,
so KPZ eq. behaves like EW eq.
2.4. Well-definedness of KPZ equation
In fact, KPZ equation is ill-defined. (
is incompatible with white noise)
Using the Cole-Hopf transformation
(14)
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KPZ eq. (7) can be formally rewritten as: (stochastic heat equation, SHE)
(15)
with
(16)
Eq.(15) is a linear equation with multiplicative noise.  Ito or Stratonovich product?
Transformation from Eqs.(14) to (15) is valid for Stratonovich product, but then
with
By contrast, SHE with Ito product has a well-defined solution Z(x,t).
Then, h(x,t) obtained by Eq.(14) is called the “solution” of KPZ eq.
[homework (advanced)]
Consider colored noise
with
Then argue how fast h(x,t) grows in KPZ eq.
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3. Exact results on 1d KPZ class [5,6]
3.1. Polynuclear growth (PNG) model
Rules:
(1) Random nucleation (at constant nucleation rate ρ)
(2) Lateral expansion (at constant speed v)
We set ρ=1 and v=1.
3.2. PNG circular interface [6]
1st nucleation at (x,t)=(0,0), subsequent nucleations occur only on top of the 1st plateau.
Space-time plot
h(0,t)
= # of lines to pass from (0,0) to (0,t)
= Max # of points passed by directed polymer
with fixed end points (0,0) & (0,t)
(= its ground-state energy under random potential
U(x,t) = -Σδ(x-xn)δ(t-tn) )
= Length of longest increasing subsequences
of random permutations
with Poisson-distributed length (Ulam’s problem)
= … (combinatorics, Young tableau, etc.)
(17)
largest-eigenvalue distribution of GUE random matrices
(GUE Tracy-Widom distribution)
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[homework] Draw a new space-time plot and convince yourself of each step of Eq.(17).
h(x,t) behaves analogously, but
’s area is
Semicircle mean profile.
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