Ubiquitous KPZ

The KPZ Equation

\displaystyle \frac{\partial h}{\partial t} = \nu\frac{\partial^2 h}{\partial x^2} + \frac{\lambda}{2}\left(\frac{\partial h}{\partial x}\right)^2 + \xi(x,t)

Happy Birthday KPZ

Growing Interfaces

\displaystyle \frac{\partial h}{\partial t} = \nu\frac{\partial^2 h}{\partial x^2} + \frac{\lambda}{2}\left(\frac{\partial h}{\partial x}\right)^2 + \xi(x,t)

  • \nu\partial_x^2 h describes diffusion.
  • \xi(x,t) is noise.
  • What about \frac{\lambda}{2}(\partial_x h)^2? Who ordered that?

The Nonlinear Term

Adding material at rate v to the surface causes it to rise by

\displaystyle v\delta t\sqrt{1+(\partial_x h)^2}

in time \delta t


Turbulent liquid crystals. Takeuchi & Sano, 2010

Other Systems

  • KPZ used to describe bacterial colony growth, flame propagation, etc.

  • The physics of each situation seems different.

  • What do we mean to say that each is described by KPZ?

Statistical Physics

 The Big Idea

  • How do we understand macroscopic matter in terms of microscopic constituents?

  • Talking about a uniform state of matter requires that widely separated regions are nearly independent.

  • Living matter is quite different in this respect.

A Phase Diagram

Critical Point

 Lattice Gas

Ising Model

\displaystyle H = -J\sum_{\langle j\,k\rangle} \sigma_j \sigma_k + h \sum_j \sigma_j,\qquad \sigma_j = \pm 1

 At Critical Point

  • System is scale invariant (a fractal).
  • Correlation function is power law \displaystyle \langle \sigma(\mathbf{x})\sigma(\mathbf{y})\rangle = \frac{1}{|\mathbf{x}-\mathbf{y}|^\eta}, \qquad |\mathbf{x}-\mathbf{y}| \to \infty
  • \eta is a critical exponent
  • Exact solution in 2D: \eta=1/4.
  • No exact solution in 3D.


  • Ising model is clearly ridiculously oversimplified.
  • Why don’t we study more complicated models?
  • Near the critical point, the physics is the same!
  • More precisely, critical exponents coincide.

Random Walk

Heat Equation

\displaystyle \frac{\partial \phi}{\partial t} = D \frac{\partial^2 \phi}{\partial x^2}

Equation ‘smoothes’ \phi(x,t)

At Long Times…

\displaystyle \phi(x,t)\longrightarrow \frac{\phi_0}{\sqrt{t}} \exp\left(-\frac{x^2}{4Dt}\right)

…no matter what you start with!

  • Shows key property of diffusion that x\sim \sqrt{t}
  • T \sim L^z with dynamical critical exponent z=2.
  • This wouldn’t be changed if we added terms with more derivatives e.g. \frac{\partial^4 \phi}{\partial x^4}.

Scaling for KPZ

\displaystyle h(x,t) - vt \sim t^{1/3}

  • Spatial scale of fluctuations is L\sim t^{2/3}, so z=3/2
  • Different from diffusion, but universal.

More Connections


  • From a theorist’s viewpoint!
  • Particles on sites, one per site.
  • Particles hop to right with some rate, if they can.

From hops to heights…

\displaystyle \frac{\partial h}{\partial x} = \frac{1}{2}-\rho


Think in terms of local density \rho(x,t) and current j(x,t).

  • They obey the continuity equation

    \displaystyle \frac{\partial \rho}{\partial t} = - \frac{\partial j}{\partial x}

  • Velocity v\propto (1-\rho), so

    \displaystyle j = \rho v = \rho(1-\rho)

  • Putting it together gives Burgers’ equation

    \displaystyle \frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial x}\rho(1-\rho)

\displaystyle \frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial x}\rho(1-\rho)

  • As we saw, \rho-\frac{1}{2}=\frac{\partial h}{\partial x}, so this is

    \displaystyle \frac{\partial h}{\partial t} = \frac{\partial h}{\partial x} + \left(\frac{\partial h}{\partial x}\right)^2

  • Compare KPZ

    \displaystyle \frac{\partial h}{\partial t} = \frac{\lambda}{2}\left(\frac{\partial h}{\partial x}\right)^2 + \nu\frac{\partial^2 h}{\partial x^2} + \xi(x,t)

  • Noise and diffusion are higher order effects

Polynuclear Growth

Spacetime View

  • Polymer is stretched out in (1,1) direction, but attracted to points.

  • Wants to maximize number visited, subject to moving up and to the right.

This is the directed polymer in a random potential. It’s equivalent to KPZ!


Remember T \sim L^z. Here T = x+y, L=x-y, so z=3/2


  • Model for propagation of light in random refractive index, or fluctuating spacetime geometry.

Solving the KPZ Equation

Can We Solve KPZ?

  • This means: calculate the statistical properties exactly

  • Nope

  • Actually, it’s a struggle to define it!

Martin Hairer won the Fields medal in 2014 for giving precise meaning to the KPZ equation.

  • Thus we resort to models. If we believe that scaling behaviour is insensitive to details of system, we are free to choose!

  • Such models have played a vital role in our understanding of critical phenomena.

Back to directed polymer

Permutation (23154)

Longest increasting subsequences are (2,3,5) and (2,3,4).

The fluctuations of height of a KPZ surface, or of the energy of a random polymer, are equivalent to the statistics of the longest increasing subsequence of a random permutation!

  • Statistical physics reduced to combinatorics.

Effect of Geometry

  • Exact solution describe different probability distributions.
  • Same h(x,t) - vt \sim t^{1/3} scaling.

What Don’t We Know?

  • Almost anything about higher dimensions (e.g. growth on a 2D interface). Critical exponents and scaling functions.

  • A rough-to-smooth phase transition?

  • Any generalization of the ‘tricks’ (e.g. random permutation) used in 1D.