Replica analysis of the 1D KPZ equation T. Sasamoto (Based on - PowerPoint PPT Presentation

Replica analysis of the 1D KPZ equation T. Sasamoto (Based on collaborations with T. Imamura) 5 Dec 2011 @ Kochi References: arxiv:1105.4659, 1111.4634 1

1. Introduction: 1D surface growth • Paper combustion, bacteria colony, crystal growth, liquid crystal turbulence • Non-equilibrium statistical mechanics • Stochastic interacting particle systems • Integrable systems 2

Kardar-Parisi-Zhang(KPZ) equation 1986 Kardar Parisi Zhang √ 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 covariance ⟨ η ( x, t ) η ( x ′ , t ′ ) ⟩ = δ ( x − x ′ ) δ ( t − t ′ ) • The Brownian motion is stationary. • Dynamical RG analysis: h ( x = 0 , t ) ≃ vt + cξt 1 / 3 KPZ universality class • Now revival: New analytic and experimental developments 3

A discrete model: ASEP as a surface growth model ASEP(asymmetric simple exclusion process) q p q p q Mapping to surface growth 4

Stationary measure ASEP · · · Bernoulli measure: each site is independent and occupied with prob. ρ ( 0 < ρ < 1 ). Current is ρ (1 − ρ ) . · · · · · · ρ ρ ρ ρ ρ ρ ρ -3 -2 -1 0 1 2 3 Surface growth · · · Random walk height profile 5

Surface growth and 2 initial conditions besides stationary Flat Droplet ↕ ↕ ↕ ↕ Wedge Step Alternating Integrated current N ( x, t ) in ASEP ⇔ Height h ( x, t ) in surface growth 6

Current distributions for ASEP with wedge initial conditions 2000 Johansson (TASEP) 2008 Tracy-Widom (ASEP) N (0 , t/ ( q − p )) ≃ 1 4 t − 2 − 4 / 3 t 1 / 3 ξ TW Here N ( x = 0 , t ) is the integrated current of ASEP at the origin and ξ TW obeys the GUE Tracy-Widom distributions; F TW ( s ) = P [ ξ TW ≤ s ] = det(1 − P s K Ai P s ) 0.5 0.4 0.3 where K Ai is the Airy kernel 0.2 0.1 ∫ ∞ 0.0 K Ai ( x, y ) = d λ Ai( x + λ )Ai( y + λ ) � 6 � 4 � 2 0 2 s 0 7

Current Fluctuations of ASEP with flat initial conditions: GOE TW distribution More generalizations: stationary case: F 0 distribution, multi-point fluctuations, etc They can be measured experimentally! The KPZ equation itself can be treated analytically! 8

Random matrix theory GUE (Gaussian Unitary Ensemble) hermitian matrices   · · · u 11 u 12 + iv 12 u 1 N + iv 1 N   u 12 − iv 12 · · · u 22 u 2 N + iv 2 N     A = . . . ...   . . .  . . .      u 1 N − iv 1 N u 2 N − iv 2 N · · · u NN u jj ∼ N (0 , 1 / 2) u jk , v jk ∼ N (0 , 1 / 4) The largest eigenvalue x max · · · GUE TW distribution GOE (Gaussian Orthogonal Ensemble) real symmetric matrices · · · GOE TW distribution 9

Experiments by liquid crystal turbulence 2010-2011 Takeuchi Sano 10

See Takeuchi Sano Sasamoto Spohn, Sci. Rep. 1,34(2011) 11

The narrow wedge KPZ equation 2010 Sasamoto Spohn, Amir Corwin Quastel • Narrow wedge initial condition • Based on (i) the fact that the weakly ASEP is KPZ equation (1997 Bertini Giacomin) and (ii) a formula for step ASEP by 2009 Tracy Widom • The explicit distribution function for finite t • The KPZ equation is in the KPZ universality class Before this 2009 Bala´ zs, Quastel, and Sepp¨ al¨ ainen The 1/3 exponent for the stationary case 12

Narrow wedge initial condition Scalings h → λ x → α 2 x, t → 2 να 4 t, 2 ν h where α = (2 ν ) − 3 / 2 λD 1 / 2 . We can and will do set ν = 1 2 , λ = D = 1 . We consider the droplet growth with macroscopic shape  − x 2 / 2 t for | x | ≤ t/δ ,  h ( x, t ) = (1 / 2 δ 2 ) t − | x | /δ for | x | > t/δ  which corresponds to taking the following narrow wedge initial conditions: h ( x, 0) = −| x | /δ , δ ≪ 1 13

h(x,t) 2 λ t/ δ x 14

Distribution h ( x, t ) = − x 2 / 2 t − 12 γ 3 1 t + γ t ξ t where γ t = (2 t ) − 1 / 3 . The distribution function of ξ t ∫ ∞ − 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 ) . 15

P u is the projection onto [ u, ∞ ) and the kernel B t is ∫ ∞ d λ (e γ t λ − 1) − 1 B t ( x, y ) = K Ai ( x, y ) + 0 ( ) × Ai( x + λ )Ai( y + λ ) − Ai( x − λ )Ai( y − λ ) . 16

Developments (not all!) • 2010 Calabrese Le Doussal Rosso, Dotsenko Replica • 2010 Corwin Quastel Half-BM by step Bernoulli ASEP • 2010 O’Connell A directed polymer model related to quantum Toda lattice • 2010 Prolhac Spohn Multi-point distributions by replica • 2011 Calabrese Le Dossal Flat case by replica • 2011 Corwin et al Tropical RSK for inverse gamma polymer • 2011 Borodin Corwin Macdonald process • 2011 Imamura Sasamoto Half-BM and stationary case by replica 17

Replica analysis of KPZ equation • Rederivation of the narrow wedge distribution by 2010 Calabrese Le Doussal Rosso, Dotsenko. Arrives at the correct formula by way of a divergent sum. Now there is a rigorous version for a discrete model. • In a sense simpler than through ASEP • Suited for generaliations Multipoint distributions (2010 Prolhac Spohn), Flat case (2011 Calabrese Le Dossal ), Half-BM (2011 Imamura Sasamoto), Stationary case (2011 Imamura Sasamoto). 18

2. Stationary case Two sided BM  B − ( − x ) , x < 0 ,  h ( x, 0) = B + ( x ) , x > 0 ,  where B ± ( x ) are two independent standard BMs We consider a generalized initial condition  ˜ B ( − x ) + v − x, x < 0 ,  h ( x, 0) = B ( x ) − v + x, x > 0 ,  where B ( x ) , ˜ B ( x ) are independent standard BMs and v ± are the strength of the drifts. 19

Result For the generalized initial condition with v ± h ( x, t ) + γ 3 [ ] t / 12 ≤ γ t s F v ± ,t ( s ) := Prob ∫ ∞ Γ( v + + v − ) [ ] due − e γt ( s − u ) ν v ± ,t ( u ) 1 − = Γ( v + + v − + γ − 1 d/ds ) −∞ t Here ν v ± ,t ( u ) is expressed as a difference of two Fredholm determinants, 1 − P u ( B Γ t − P Γ 1 − P u B Γ ( ) ( ) ν v ± ,t ( u ) = det Ai ) P u − det t P u , where P s represents the projection onto ( s, ∞ ) , ξ 1 , 1 ξ 2 , 1 ( ) ( ) P Γ Ai ( ξ 1 , ξ 2 ) = Ai Γ Ai Γ , v − , v + , v + , v − Γ Γ γ t γ t 20

∫ ∞ 1 ( ξ 1 + y, 1 ) B Γ 1 − e − γ t y Ai Γ t ( ξ 1 , ξ 2 ) = dy , v − , v + Γ γ t −∞ ξ 2 + y, 1 ( ) × Ai Γ , v + , v − , Γ γ t and Γ ( a, b, c, d ) = 1 Γ ( ibz + d ) ∫ dze iza + i z 3 Ai Γ Γ ( − ibz + c ) , 3 2 π Γ i d b where Γ z p represents the contour from −∞ to ∞ and, along the way, passing below the pole at z = id/b . 21

Height distribution for the stationary KPZ equation ∫ ∞ 1 duγ t e γ t ( s − u )+ e − γt ( s − u ) ν 0 ,t ( u ) F 0 ,t ( s ) = Γ(1 + γ − 1 d/ds ) −∞ t where ν 0 ,t ( u ) is obtained from ν v ± ,t ( u ) by taking v ± → 0 limit. 0.4 γ t = 1 γ t = ∞ 0.3 0.2 0.1 0.0 4 2 0 2 4 s Figure 1: Stationary height distributions for the KPZ equation for γ t = 1 case. The solid curve is F 0 . 22

Stationary 2pt correlation function C ( x, t ) = ⟨ ( h ( x, t ) − ⟨ h ( x, t ) ⟩ ) 2 ⟩ ( ) g t ( y ) = (2 t ) − 2 / 3 C (2 t ) 2 / 3 y, t 2.0 γ t = 1 γ t = ∞ 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 y Figure 2: Stationary 2pt correlation function g ′′ t ( y ) for γ t = 1 . The solid curve is the corresponding quantity in the scaling limit g ′′ ( y ) . 23

Derivation Cole-Hopf transformation 1997 Bertini and Giacomin h ( x, t ) = log ( Z ( x, t )) Z ( x, t ) is the solution of the stochastic heat equation, ∂ 2 Z ( x, t ) ∂Z ( x, t ) = 1 + η ( x, t ) Z ( x, t ) . ∂x 2 ∂t 2 and can be considered as directed polymer in random potential η . cf. Hairer Well-posedness of KPZ equation without Cole-Hopf 24

Feynmann-Kac and Generating function Feynmann-Kac expression for the partition function, [∫ t ( ] ) Z ( x, t ) = E x exp η ( b ( s ) , t − s ) ds Z ( b ( t ) , 0) 0 We consider the N th replica partition function ⟨ Z N ( x, t ) ⟩ and compute their generating function G t ( s ) defined as − e − γ t s ) N ∞ ( γ 3 t ∑ Z N (0 , t ) e N ⟨ ⟩ G t ( s ) = 12 N ! N =0 with γ t = ( t/ 2) 1 / 3 . 25

δ -Bose gas Taking the Gaussian average over the noise η , one finds that the replica partition function can be written as ⟨ Z N ( x, t ) ⟩   ∫ ∞ ∫ x j ( t )= x ∫ t N N ) 2 1 ( dx ∏ ∑ = dy j D [ x j ( τ )] exp  − dτ  2 dτ −∞ x j (0)= y j 0 j =1 j =1 ( N   N ⟨ )⟩ ∑  × ∑ − δ ( x j ( τ ) − x k ( τ )) exp h ( y k , 0)  j ̸ = k =1 k =1 = ⟨ x | e − H N t | Φ ⟩ . 26

H N is the Hamiltonian of the δ -Bose gas, N N ∂ 2 H N = − 1 − 1 ∑ ∑ δ ( x j − x k ) , ∂x 2 2 2 j j =1 j ̸ = k | Φ ⟩ represents the state corresponding to the initial condition. We compute ⟨ Z N ( x, t ) ⟩ by expanding in terms of the eigenstates of H N , ∑ ⟨ x | Ψ z ⟩⟨ Ψ z | Φ ⟩ e − E z t ⟨ Z ( x, t ) N ⟩ = z where E z and | Ψ z ⟩ are the eigenvalue and the eigenfunction of H N : H N | Ψ z ⟩ = E z | Ψ z ⟩ . 27