The KPZ fixed point Jeremy Quastel University of Toronto joint - PowerPoint PPT Presentation

The KPZ fixed point Jeremy Quastel University of Toronto joint work with Konstantin Matetski (University of Toronto) and Daniel Remenik (Universidad de Chile) ICMP, Montreal 23 July 2018

1d randomly stirred fluid: Stochastic Burgers equation ∂ t u = ∂ x u 2 + ∂ 2 x u + ∂ x ξ t > 0 , x ∈ R ξ ( t , x ) space-time white noise Forster-Nelson-Stephen 77: Dynamic renormalization group ǫ − 1 / 2 u ( ǫ − 3 / 2 t , ǫ − 1 x ) → Non-linear long time fixed pt Anomalous fluctuations → A new universality class ? 1/16

t ( some shape ) + t χ Fluctuations 1 + 1 d: Fluctuation exponent χ = 1 / 3 2/16

Kardar-Parisi-Zhang (KPZ) Equation 86 ∂ t h = ( ∂ x h ) 2 + ∂ 2 + ξ h ( t , x ) x h ���� � �� � ���� � �� � space − time height at time lateral growth relaxation white noise t position x ∂ x h = u � stoch Burgers eqn ∂ t u =2 u ∂ x u + ∂ 2 x u + ∂ x ξ Eden growth Ballistic aggregration ASEP 3/16

Burning paper Coffee Stains Tumour growth (?) Bacterial growth 4/16

A special discretization of KPZ equation (TASEP) h ( x + 1) = h ( x ) ± 1, x ∈ Z local max �→ local min at rate 1 � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � � × � � ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐✐ ✐ ✐ ✐ ✐✐ ✐ ✐ X(3) X(2) X(1) X(0) X(-1) X(-2) � � − 2 1 ∧ = 1 ( ∇ − h )( ∇ + h ) − 1 + ∆ h 2 symmetric random walk invariant (except for height shift) Lab mouse of non-equilibrium stat mech since the late 60’s 5/16

A special discretization of KPZ equation (TASEP) h ( x + 1) = h ( x ) ± 1, x ∈ Z local max �→ local min at rate 1 � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � � × � � ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐✐ ✐ ✐ ✐ ✐✐ ✐ ✐ X(3) X(2) X(1) X(0) X(-1) X(-2) � � − 2 1 ∧ = 1 ( ∇ − h )( ∇ + h ) − 1 + ∆ h 2 symmetric random walk invariant (except for height shift) Lab mouse of non-equilibrium stat mech since the late 60’s 6/16

Asymptotic fluctuations depend on initial data In TASEP special initial data could be computed 00’s Johansson, Spohn, Borodin, Sasamoto,... Corner h ( t , x ) ∼ c 1 t − c 2 x 2 t + c 3 t 1 / 3 F GUE Flat h ( t , x ) ∼ c 3 t + c 4 t 1 / 3 F GOE F GUE / F GOE are the rescaled top eigenvalues of a matrix from the Gaussian Unitary/Orthogonal Ensembles 7/16

Asymptotic fluctuations depend on initial data In TASEP special initial data could be computed 00’s Johansson, Spohn, Borodin, Sasamoto,... Corner h ( t , x ) ∼ c 1 t − c 2 x 2 t + c 3 t 1 / 3 A 2 ( t − 2 / 3 x ) Flat h ( t , x ) ∼ c 3 t + c 4 t 1 / 3 A 1 ( t − 2 / 3 x ) Airy 2 / Airy 1 are special stochastic processes Equivalent to h ǫ ( t , x ) = ǫ 1 / 2 h ( ǫ − 3 / 2 t , ǫ − 1 x ) − c ǫ − 3 / 2 t 8/16

Conjectural KPZ universality class KPZ fixed point is space-time random field at the centre, the non-linear fixed point invariant under h ǫ ( t , x ) = ǫ 1 / 2 h ( ǫ − 3 / 2 t , ǫ − 1 x ) KPZ eq limit of ǫ 1 / 2 h ( ǫ − 2 t , ǫ − 1 x ) with nonlinearity/noise of order ǫ 1 / 2 9/16

What is the fixed point? Formally, solution of Hopf equation ∂ t h = ( ∂ x h ) 2 , or ∂ x h = u solves Burgers equation ∂ t u = ∂ x u 2 , but they are not well posed Instead, we describe it as a Markov process whose transition probabilities can be linearized, an integrable Markov process This is done by first solving TASEP, which also turns out to be an integrable Markov process , then taking limit At TASEP level there are well-posed equations (the Kolmogorov or backward equation, or the Fokker-Planck or forward equation) so the solution can be checked Method works for some variants of TASEP, but the fact that there is only one such fixed point is still conjectural 10/16

The KPZ fixed point We give formulas for fixed point; formulas for TASEP similar with Brownian motion replaced by random walk Markov transition probabilities are given by determinants � � Prob h ( t , x 1 ) ≤ a 1 , . . . , h ( t , x M ) ≤ a M | h (0 , · ) = h 0 ( · ) = det( I − K h 0 , t , x , a ) Fredholm determinant of a compact (trace class) operator ∞ � 1 � � � n det( I + K ) L 2 ( S ,µ ) = 1 + S n det K ( u i , u j ) i , j =1 d µ ( u 1 ) · · · d µ ( u n ) n ! n =1 Stochastic integrable system: dynamics linearized by a novel Brownian scattering transform h �→ K h 11/16

Brownian scattering transform For h upper semi-continuous define for B Brownian motion � � P hit h − L , L ( u 1 , u 2 ) = P B ( − L )= u 1 , B ( L )= u 2 B hits hypo ( h ) on [ − L , L ] The Brownian scattering transform of h is L →∞ e − L ∂ 2 P hit h − L , L e − L ∂ 2 K h = lim Looks terrible because of backwards heat equation, but we only ever use t 3 ∂ 3 K h , t = U t K h 0 U − 1 U t = e t 2 x 3 Ok! e x ∂ 2 + t 3 t 2 − ( z 1 − z 2) x 3 ∂ 3 ( z 1 , z 2 ) = t − 1 / 3 e Ai( − t − 1 / 3 ( z 1 − z 2 ) + t − 4 / 3 x 2 ) t even for x < 0 as long as t � = 0 ! 12/16

KPZ fixed pt formula � � � � I − χ a U t K ext h 0 U − 1 h ( t , x 1 ) ≤ a 1 , . . . , h ( t , x M ) ≤ a M = det χ a P h 0 t h ( x i , · ; x j , · ) = − e ( x j − x i ) ∂ 2 ✶ x i < x j + e − x i ∂ 2 K h 0 e x j ∂ 2 K ext χ a ( x i , u ) = ✶ u > a i Conjecturally unique local dynamics satisfying dist (1:2:3 scaling invariant) α h ( α − 3 t , α − 2 x ; α h 0 ( α − 2 x )) = h ( t , x ; h 0 ) 1 � � � � (Skew time reversible) P h ( t , x ; g ) ≤ − f ( x ) = P h ( t , x ; f ) ≤ − g ( x ) 2 dist (Stationarity in space) h ( t , x + u ; h 0 ( x − u )) = h ( t , x ; h 0 ); 3 For TASEP (and a few related models) if the rescaled height function converges to h 0 as upper semicont fns ǫ 1 / 2 � � h (2 ǫ − 3 / 2 t , 2 ǫ − 1 x ) + ǫ − 3 / 2 t − − → h ( t , x ; h 0 ) in distr ǫ → 0 older 1 2 − in x , 1 For t > 0, h ( t , x ) locally Brownian, H¨ 3 − in t For eg, flat, narrow wedge, Brownian scattering transform is easy to compute and recovers the known formulas for the Airy processes. 13/16

Biorthogonalization of TASEP One-sided i.d. X (0) = X ( − 1) = · · · = + ∞ , 1 ≤ n 1 < n 2 < · · · < n M Borodin, Ferrari, Pr¨ ahofer and Sasamoto’07: � � � � P X 0 X t ( n j ) > a j , j = 1 , . . . , M = det I − ✶ x i ≤ a i K t ✶ x i ≤ a i ℓ 2 ( { n 1 ,..., n M }× Z ) n j � n i − k ( x i ) Φ n j K t ( n i , x i ; n j , x j ) = − Q n j − n i ( x i , x j ) ✶ n i < n j + Ψ n i n j − k ( x j ) k =1 � � 1 − w � k 1 dw Ψ n e t ( w − 1 / 2) Q ( x , y ) = ✶ x > y k ( x ) = 2 π i w x − X 0 ( n − k )+1 w Γ 0 Ψ n k Charlier polynomials. Φ n k are defined implicitly by 1 Biorthogonality: � x ∈ Z Ψ n ℓ ( x )Φ n k ( x ) = ✶ ℓ = k 2 Φ n k is a polynomial of degree k Φ n k were only known for particles equally spaced in a block. We find them in general 14/16

Exact fluctuations starting from h 0 ( x ) = x 2 explains recent experimental results of Takeuchi on fluctuations of inward growing fronts (incorrect claims that it has flat fluctuations, F GOE ) Here we have finite time ( t = 50 s ) blowup. If initial data has (say) linear growth at ∞ then solutions for all time 15/16

Conclusions At least now we can state the KPZ Universality conjecture: All models in the class should converge to this fixed point under the 1:2:3 scaling We have proved it now for TASEP and several generalizations The solution of TASEP (after 50 yrs!) has ramifications well beyond the KPZ fixed point. TASEP is a microscopic (as opposed to continuum) model for (single lane) traffic (Burgers equation). With Mustazee Rahman (MIT) we are have results for eg. position of shock (typically the difference of two GOE’s. Earlier special cases by Ferrari-Nejjar.) 16/16