KPZ Equation, the Hopf–Cole, and the Universality¶
Job: develop the KPZ equation / the Hopf–Cole / the SHE, the KPZ universality class (\(1/3\) exponent / Tracy–Widom / TASEP), the scaling / 1:2:1 / RG / Airy, and the directed polymers / strong-weak disorder — with the KPZ narrow wedge \(h(t,0)\sim ct^{1/3}F_2\) / Tracy–Widom / Airy as the worked example.
Core¶
The KPZ equation and the Hopf–Cole¶
The KPZ equation \(\partial_t h=\partial_x^2 h+(\partial_x h)^2+\xi\) (the Kardar–Parisi–Zhang, the interface growth, the non-linear + the noise). The Hopf–Cole \(h=\log Z\) with the SHE \(\partial_t Z=\partial_x^2 Z+Z\xi\) (the Cole–Hopf solution, the \(Z\) the stochastic heat equation, the \(h=\log Z\) the KPZ via the Hopf–Cole).
| KPZ / Hopf–Cole | Description |
|---|---|
| KPZ equation | \(\partial_t h=\partial_x^2 h+(\partial_x h)^2+\xi\) (the interface growth, the non-linear + the noise) |
| Hopf–Cole / SHE | \(h=\log Z\), \(\partial_t Z=\partial_x^2 Z+Z\xi\) (the Cole–Hopf solution, the \(Z\) the SHE) |
| Regularity structures / direct | the direct solution of the KPZ via the regularity structures (the renormalization, the match to the Cole–Hopf after) |
The KPZ universality and the Tracy–Widom¶
| KPZ universality / Tracy–Widom | Description |
|---|---|
| KPZ universality class | the fluctuations of the interface growth the KPZ (the height \(h(t,x)\sim c t^{1/3}\) the \(1/3\) exponent, the Tracy–Widom the fluctuations) |
| Narrow wedge / flat | the narrow wedge the \(h(0,x)=\delta_0\) the droplet, the flat the \(h(0,x)=0\) (the fluctuations the Tracy–Widom \(F_2\) the narrow wedge, the Baik–Rains the flat) |
| TASEP / corner growth | the TASEP / the PNG / the corner growth the KPZ universality (the convergence to the KPZ / the Tracy–Widom) |
The scaling and the renormalization group¶
| Scaling / 1:2:1 / RG / Airy | Description |
|---|---|
| Scaling | the KPZ the scaling \(x\to\lambda x\), \(t\to\lambda^2 t\), \(h\to\lambda h\) (the 1:2:1 scaling, the KPZ the fixed point) |
| RG / fixed point | the KPZ the RG (the fixed point, the universality, the random matrices / Tracy–Widom) |
| 1:2:1 and Airy | the KPZ the 1:2:1 (the Airy process / the Tracy–Widom, the narrow wedge) |
The directed polymers and the KPZ¶
| Directed polymers / strong-weak | Description |
|---|---|
| Directed polymers | \(Z_n(\beta,x)=\sum_\gamma e^{-\beta\sum_{(t,y)\in\gamma}\xi(t,y)}\) (the polymer / the random environment, the \(h=\log Z\) the KPZ) |
| Continuous / Brownian | the continuous limit the KPZ / the Hopf–Cole (the Alberts–Khan–Quastel) |
| Strong / weak disorder | the \(\beta\) the disorder (the strong / the weak, the phase transition, the Comets–Yoshida) |
See it¶
KPZ narrow wedge \(h(t,0)\sim ct^{1/3}F_2\) and the Tracy–Widom / the Airy. The KPZ equation with the narrow wedge initial \(h(0,x)=\delta_0\) (the droplet, the narrow wedge). The solution \(h(t,0)=\log Z(t,0)\) where \(Z\) the SHE with the narrow wedge \(Z(0,x)=\delta_0\). The fluctuations: \(h(t,0)\sim c t^{1/3}\chi_2\) (the \(1/3\) exponent, the Tracy–Widom \(F_2\) / the \(\chi_2\) the Tracy–Widom GUE, the Sasamoto–Spohn / the Amir–Corwin–Quastel). The Airy: the spatial fluctuations \(h(t,x)\sim c t^{1/3}\mathcal{A}(x/(ct^{2/3}))\) (the Airy process \(\mathcal{A}\), the KPZ fixed point / the Airy, the 1:2:1 scaling). The key point: the KPZ narrow wedge the Tracy–Widom / the \(1/3\) exponent, the Airy process the spatial — the KPZ universality the random matrices / Tracy–Widom the fluctuations, the interface growth the KPZ the universal.
Check¶
Show that the Hopf–Cole \(h=\log Z\) transforms the KPZ into the SHE, and explain the Cole–Hopf solution.
The KPZ \(\partial_t h=\partial_x^2 h+(\partial_x h)^2+\xi\). The Hopf–Cole: \(Z=e^h\) (the \(h=\log Z\)), the \(\partial_t Z=\partial_t(e^h)=e^h\partial_t h=Z(\partial_x^2 h+(\partial_x h)^2+\xi)\). The \(\partial_x^2 Z=\partial_x(Z\partial_x h)=\partial_x Z\cdot\partial_x h+Z\partial_x^2 h=Z(\partial_x h)^2+Z\partial_x^2 h\) (the \(\partial_x Z=Z\partial_x h\)). So \(\partial_t Z=Z\partial_x^2 h+Z(\partial_x h)^2+Z\xi=\partial_x^2 Z+Z\xi\) (the SHE, the stochastic heat equation with the multiplicative noise \(Z\xi\)). The Cole–Hopf: the \(h=\log Z\) the KPZ via the SHE (the \(Z\) the SHE, the \(h=\log Z\) the Cole–Hopf solution). The regularity structures: the direct solution of the KPZ (the \((\partial_x h)^2\) the ill-defined, the \(h\) the distribution, the regularity structures / the renormalization), the match to the Cole–Hopf after the renormalization (the Hairer, the direct the regularity structures the match to the \(h=\log Z\)).
Explain the 1:2:1 scaling of the KPZ and the Airy process, and the Tracy–Widom.
The KPZ scaling: \(x\to\lambda x\), \(t\to\lambda^2 t\), \(h\to\lambda h\) (the 1:2:1 scaling: the \(x\) the \(\lambda\), the \(t\) the \(\lambda^2\), the \(h\) the \(\lambda\) — the ratio \(1:2:1\)). The scaling: the KPZ the invariant (the equation the invariant under the 1:2:1, the fixed point). The \(1/3\) exponent: the \(h(t,x)\sim t^{1/3}\) (the \(t^{1/3}\) the fluctuations, the \(1/3\) the KPZ exponent, the \(t^{1/3}\cdot\lambda=t^{1/3}\cdot\lambda^{1/3}\)... the \(h\to\lambda h\), \(t^{1/3}\to\lambda^{2/3}t^{1/3}\), the \(\lambda h\sim\lambda^{2/3}t^{1/3}\), the \(h\sim\lambda^{-1/3}t^{1/3}\), the \(1/3\) the self-consistent). The Tracy–Widom: the \(h(t,0)\sim t^{1/3}\chi_2\) (the \(\chi_2\) the Tracy–Widom GUE, the Sasamoto–Spohn / the Amir–Corwin–Quastel). The Airy: the \(h(t,x)\sim t^{1/3}\mathcal{A}(x/t^{2/3})\) (the Airy process \(\mathcal{A}\), the spatial the \(x/t^{2/3}\) the 1:2:1, the Airy the KPZ fixed point). The Tracy–Widom / the Airy the KPZ universality (the random matrices / Tracy–Widom the fluctuations, the Airy the spatial).
Pitfalls¶
- The KPZ the Hopf–Cole (the \(h=\log Z\) the SHE, the Cole–Hopf solution; the direct / the regularity structures the match after the renormalization) — the Cole–Hopf the classical, the regularity structures the direct the match after the renormalization (the Hairer).
- The KPZ universality the \(1/3\) / the Tracy–Widom (the narrow wedge, the flat the Baik–Rains) — the narrow wedge the \(F_2\) (the GUE Tracy–Widom), the flat the Baik–Rains (the different).
- The 1:2:1 scaling the KPZ the invariant (the \(x:t:h=1:2:1\)), the Airy the spatial (\(x/t^{2/3}\)), the Tracy–Widom the point (\(h/t^{1/3}\)).
- The directed polymers / the strong-weak disorder (the \(\beta\) the disorder, the phase transition, the Comets–Yoshida) — the KPZ the polymer the \(h=\log Z\), the strong / weak the \(\beta\).
- The TASEP / the corner growth the KPZ universality (the convergence to the KPZ / the Tracy–Widom) — the discrete the KPZ, the Tracy–Widom the universal.
Refs¶
- Kardar–Parisi–Zhang, "Dynamic scaling for growing interfaces" (PRL 1986); the KPZ equation
- Sasamoto–Spohn, "The one-dimensional KPZ equation and the Airy process" (2010); Amir–Corwin–Quastel, "On the distribution of the solution to the KPZ equation" (2011)
- Baik–Rains, "Limiting distribution for the KPZ equation with flat initial conditions" (2000); Alberts–Khan–Quastel, "The continuum directed random polymer" (2014)
- Hairer, "Solving the KPZ equation" (Annals 2013); the regularity structures / the direct
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SPDE index · Foundations · Regularity Structures · Paracontrolled & Renormalization · Stochastic NS & Quantization · RMT / Tracy–Widom · PDE