The 1D KPZ equation and its universality T. Sasamoto 17 Aug 2015 @ Kyoto 1
Plan • The KPZ equation • Exact solutions Height distribution Stationary space-time two point correlation function • A few recent developments Dualities Free-fermionic structures • Universality Brownian motions with oblique reflection KPZ in Hamiltonian dynamics 2
1. Basics of the KPZ equation: Surface growth • Paper combustion, bacteria colony, crystal growth, etc • Non-equilibrium statistical mechanics • Stochastic interacting particle systems • Connections to integrable systems, representation theory, etc 3
Simulation models Ex: ballistic deposition Height fluctuation O ( t β ) , β = 1 / 3 A B ↓ ↓ ↓ 100 "ht10.dat" "ht50.dat" "ht100.dat" ↕ 80 60 A ′ 40 B ′ 20 0 0 10 20 30 40 50 60 70 80 90 100 Flat 4
KPZ equation h ( x, t ) : height at position x ∈ R and at time t ≥ 0 1986 Kardar Parisi Zhang (not Knizhnik-Polyakov-Zamolodchikov) √ 2 λ ( ∂ x h ( x, t )) 2 + ν∂ 2 ∂ t h ( x, t ) = 1 x h ( x, t ) + Dη ( x, t ) where η is the Gaussian noise with mean 0 and covariance ⟨ η ( x, t ) η ( x ′ , t ′ ) ⟩ = δ ( x − x ′ ) δ ( t − t ′ ) • Dynamical RG analysis: → β = 1 / 3 (KPZ class) • A simplest nonequilibrium model with nonlinearity, noise and ∞ -degrees of freedom • By a simple scaling we can and will do set ν = 1 2 , λ = D = 1 . 5
Most Famous(?) KPZ • MBT-70 ／ KPz 70 Tank developed in 1960s by US and West Germany. MBT(MAIN BATTLE TANK)-70 is the US name and KPz(KampfPanzer)-70 is the German name. 6
New most famous KPZ in Japan(?) A sushi restaurant franchise with character ”kappa” (an imaginary creature) [address: kpz.jp] 7
A discrete model: ASEP ASEP = asymmetric simple exclusion process q p q p q ⇒ ⇒ · · · ⇐ ⇐ ⇐ · · · -3 -2 -1 0 1 2 3 • TASEP(Totally ASEP, p = 0 or q = 0 ) • N ( x, t ) : Integrated current at ( x, x + 1) upto time t ⇔ height for surface growth • In a certain weakly asymmetric limit ASEP ⇒ KPZ equation 8
2. Exact solutions: Cole-Hopf transformation If we set Z ( x, t ) = exp ( h ( x, t )) this quantity (formally) satisfies ∂ 2 Z ( x, t ) ∂tZ ( x, t ) = 1 ∂ + η ( x, t ) Z ( x, t ) ∂x 2 2 This can be interpreted as a (random) partition function for a directed polymer in random environment η . h(x,t) 2 λ t/ δ x δ → 0 c δ e −| x | /δ The polymer from the origin: Z ( x, 0) = δ ( x ) = lim corresponds to narrow wedge for KPZ. 9
Replica approach Dotsenko, Le Doussal, Calabrese Feynmann-Kac expression for the partition function, ∫ t ( ) 0 η ( b ( s ) ,t − s ) ds Z ( b ( t ) , 0) Z ( x, t ) = E x e Because η is a Gaussian variable, one can take the average over the noise η to see that the replica partition function can be written as (for narrow wedge case) ⟨ Z N ( x, t ) ⟩ = ⟨ x | e − H N t | 0 ⟩ where H N is the Hamiltonian of the (attractive) δ -Bose gas, N N ∂ 2 H N = − 1 − 1 ∑ ∑ δ ( x j − x k ) . ∂x 2 2 2 j j =1 j ̸ = k 10
We are interested not only in the average ⟨ h ⟩ but the full distribution of h . We expand the quantity of our interest as − e − γ t s ) N ∞ ( γ 3 ⟨ e − e h (0 ,t )+ t 24 − γts ⟩ = t ∑ Z N (0 , t ) e N ⟨ ⟩ 12 N ! N =0 Using the integrability (Bethe ansatz) of the δ -Bose gas, one gets explicit expressions for the moment ⟨ Z N ⟩ and see that the generating function can be written as a Fredholm determinant. But for the KPZ, ⟨ Z N ⟩ ∼ e N 3 ! Note that the δ -Bose gas is exactly solvable but is in general not a free fermion model. 11
Explicit determinantal formula Thm (2010 TS Spohn, Amir Corwin Quastel ) For the initial condition Z ( x, 0) = δ ( x ) (narrow wedge for KPZ) ⟨ e − e h (0 ,t )+ t 24 − γts ⟩ = det(1 − K s,t ) L 2 ( R + ) where γ t = ( t/ 2) 1 / 3 and K s,t is ∫ ∞ d λ Ai( x + λ )Ai( y + λ ) K s,t ( x, y ) = e γ t ( s − λ ) + 1 −∞ A determinant for non-free-fermion model? Why Fermi distribution? 12
Explicit formula for the height distribution Thm h ( x, t ) = − x 2 / 2 t − 12 γ 3 1 t + γ t ξ t where γ t = ( t/ 2) 1 / 3 . The distribution function of ξ t is ∫ ∞ − e γ t ( s − u ) ] [ F t ( s ) = P [ ξ t ≤ s ] = 1 − exp −∞ ( ) × det(1 − P u ( B t − P Ai ) P u ) − det(1 − P u B t P u ) d u where P Ai ( x, y ) = Ai( x )Ai( y ) , P u is the projection onto [ u, ∞ ) and the kernel B t is ∫ ∞ d λ Ai( x + λ )Ai( y + λ ) B t ( x, y ) = e γ t λ − 1 −∞ 13
Finite time KPZ distribution and TW 0.5 0.4 0.3 0.2 0.1 0.0 � 6 � 4 � 2 0 2 s : exact KPZ density F ′ t ( s ) at γ t = 0 . 94 −− : Tracy-Widom density • In the large t limit, F t tends to the GUE Tracy-Widom distribution F 2 from random matrix theory. 14
Tracy-Widom distributions For GUE (Gaussian unitary ensemble) with density P ( H ) dH ∝ e − Tr H 2 dH for H : N × N hermitian matrix, the joint eigenvalue density is (with ∆( x ) Vandelmonde) 1 e − x 2 Z ∆( x ) 2 ∏ i i GUE Tracy-Widom distribution √ [ ] x max − 2 N = F 2 ( s ) = det(1 − P s K 2 P s ) lim 2 − 1 / 2 N − 1 / 6 < s N →∞ P where P s : projection onto [ s, ∞ ) and K 2 is the Airy kernel ∫ ∞ K 2 ( x, y ) = d λ Ai( x + λ )Ai( y + λ ) 0 There is also GOE TW ( F 1 ) for GOE (Gaussian orthogonal ensemble, real symmetric matrices, for flat surface) 15
Probability densities of Tracy-Widom distributions F ′ 2 (GUE), F ′ 1 (GOE) 16
Stationary 2pt correlation Not only the height/current distributions but correlation functions show universal behaviors. • For the KPZ equation, the Brownian motion is stationary. h ( x, 0) = B ( x ) where B ( x ) , x ∈ R is the two sided BM. • Two point correlation h t 2/3 t 1/3 ∂ x h ( x , t ) ∂ x h (0,0) x o 17
Figure from the formula Imamura TS (2012) ⟨ ∂ x h ( x, t ) ∂ x h (0 , 0) ⟩ = 1 2(2 t ) − 2 / 3 g ′′ t ( x/ (2 t ) 2 / 3 ) The figure can be drawn from the exact formula (which is a bit involved though). 2.0 γ t = 1 γ t = ∞ 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 y 1 t ( y ) for γ t := ( t Stationary 2pt correlation function g ′′ 3 = 1 . 2 ) The solid curve is the scaling limit g ′′ ( y ) . 18
3.1 Dualities for asymmetric processes 2012-2015 Borodin-Corwin-TS Rigorous replica approach i q N ( x i ,t ) ⟩ satisfies • For ASEP the n -point function like ⟨ ∏ the n particle dynamics of the same process (Duality). This is a discrete generalization of δ -Bose gas for KPZ. One can apply the replica approach to get a Fredholm det expression for generating function for N ( x, t ) . • Rigorous replica: the one for KPZ (which is not rigorous) can be thought of as a shadow of the rigorous replica for ASEP. • Stationary case(Borodin Corwin Ferrari Veto (2014)), Flat case (Quastel et al (2014), Generalized models ( q -Hahn, six-vertex, ...), Plancherel theorem,... • For ASEP, the duality is related to U q ( sl 2 ) symmetry. 19
More general formulation • Dualities have been an important tool in statistical mechanics (e.g. Kramers-Wannier duality for Ising model). • For symmetric processes, the duality has been used to study its various properties. For symmetric simple exclusion process (SSEP), the n -point function satisfies the n -body problem. This is related to the SU (2) symmetry. Another well-known example with duality is the Kipnis- Marchioro-Pressutti (KMP) model of stochastic energy transfer. Its duality is related to the SU (1 , 1) symmetry. 20
• As explained, the duality for ASEP is useful to study its current distribution. Its duality is related to U q ( sl 2 ) . • Carinci Giardina Redig TS (2014,2015) presented a general scheme to construct a duality from a (deformed) symmetry of the process. As an application they have constructed a new process with U q ( su (1 , 1)) symmetry and an asymmetric version of the KMP process. 21
3.2 A determinantal structure for a finite temperature polymer 2001 O’Connell Yor Semi-discrete directed polymer in random media B i , 1 ≤ i ≤ N : independent Brownian motions Energy of the polymer π E [ π ] = B 1 ( s 1 ) + B 2 ( s 1 , s 2 ) + · · · + B N ( s N − 1 , t ) with B j ( s, t ) = B j ( t ) − B j ( s ) , j = 2 , · · · , N for s < t Partition function ( β = 1 /k B T : inverse temperature ) ∫ e βE [ π ] ds 1 · · · ds N − 1 Z N ( t ) = 0 <s 1 < ··· <s N − 1 <t In continuous limit, this becomes the polymer for KPZ equation. 22
Zero-temperature limit In the T → 0 (or β → ∞ ) limit f N ( t ) := lim β →∞ F N ( t ) = 0 <s 1 < ··· <s N − 1 <t E [ π ] max 2001 Baryshnikov Connection to random matrix theory N ∫ ∏ Prob ( f N (1) ≤ s ) = dx j · P GUE ( x 1 , · · · , x N ) , ( −∞ ,s ] N j =1 e − x 2 N j / 2 ∏ ∏ ( x k − x j ) 2 P GUE ( x 1 , · · · , x N ) = √ · j ! 2 π j =1 1 ≤ j<k ≤ N where P GUE ( x 1 , · · · , x N ) is the probability density function of the eigenvalues in the Gaussian Unitary Ensemble (GUE) 23
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