Two-dimensional Stochastic Interface Growth Fabio Toninelli CNRS - PowerPoint PPT Presentation

Two-dimensional Stochastic Interface Growth Fabio Toninelli CNRS and Universit´ e Lyon 1 XIX ICMP, Montr´ eal F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 1 / 30

Random discrete interfaces and growth 2d discrete interfaces = ⇒ random tilings, dimer model Stochastic growth (random deposition). Large scales = ⇒ non-linear PDEs, stochastic PDEs, ... An interesting story: Wolf’s conjecture on universality classes of 2d interface growth F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 2 / 30

Random discrete interfaces and growth Links with: macroscopic shapes facet singularities massless Gaussian field (GFF) F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 3 / 30

Interfaces, tilings & dimers z 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 0 0 0 0 x y 0 0 0 0 0 0 0 Discrete monotone interface Lozenge tiling of the plane Dimer model (perfect matching of planar bipartite graph) Link with spin systems: ground state of 3d Ising model F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 4 / 30

Tilings & interlaced particles Lozenge tiling ⇔ Interlaced particle system X X n +1 ℓ − 1 X n ℓ +1 X n ℓ X n − 1 ℓ +1 X n ℓ − 1 ℓ X n − 1 ℓ +1 < X n ℓ < X n ℓ +1 The whole interface/dimer/lozenge picture is still there F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 5 / 30

A stochastic deposition model Continuous-time Markov process. Updates: rate = p h rate = p h − 1 rate = 1 − p rate = 1 − p Jumps respect interlacing conditions F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 6 / 30

A stochastic deposition model Continuous-time Markov process. Updates: rate = p h rate = p h − 1 rate = 1 − p rate = 1 − p Jumps respect interlacing conditions symmetric case p = 1 / 2: uniform measure is stationary & reversible p � = 1 / 2: growth model, irreversibility. Interesting in infinite volume (or with periodic boundary conditions) equivalent to zero temperature Glauber dynamics of 3d Ising p ↔ magnetic field F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 6 / 30

Interface growth: phenomenological picture Speed of growth v = v ( ρ ): asymptotic growth rate for interface of slope ρ ∈ R d (for us, d = 2) h ( · , 0) y = ρx x F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 7 / 30

Interface growth: phenomenological picture Speed of growth v = v ( ρ ): asymptotic growth rate for interface of slope ρ ∈ R d h ( · , t ) t > 0 h ( · , 0) y = ρx x v ( ρ ) = lim t →∞ h ( x,t ) − h ( x, 0) t F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 8 / 30

Interface growth: phenomenological picture As t → ∞ , law of gradients x ∈ Z d , i = 1 , . . . , d ∇ h ≡ ( h ( x + ˆ e i ) − h ( x )) , should tend to limit stationary, non-reversible measure π ρ E. g. v ( ρ ) = p × π ρ ( ) − (1 − p ) × π ρ ( ) Roughness exponent α : at large distances � Var π ρ ( h ( x ) − h ( y )) ∼ c 1 + c 2 | x − y | α Growth exponent β : at large times, � Var( h ( x, t ) − h ( x, 0)) ∼ c 3 + c 4 t β F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 9 / 30

Fluctuation field and link with the KPZ equation Heuristics: large-scales behavior of fluctuations � Kardar-Parisi-Zhang equation relaxes large fluctuations tunes strength of non-linearity. Useful in perturbation theory ∂ t h ( x, t ) = ∆ h ( x, t ) + λ ( ∇ h ( x, t ) , H ∇ h ( x, t )) + ξ smooth ( x, t ) d × d symmetric matrix smoothed space-time white noise Quadratic non-linearity from second-order Taylor expansion of hydrodynamic PDE. H = D 2 v ( ρ ) (Hessian of speed of growth) F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 10 / 30

Fluctuation field and link with the KPZ equation ∂ t h ( x, t ) = ∆ h ( x, t ) + λ ( ∇ h ( x, t ) , H ∇ h ( x, t )) + ξ smooth ( x, t ) Linear case ( λ = 0): Edwards-Wilkinson (EW) equation. Stationary state: massless Gaussian field. α EW = (2 − d ) / 2 , β EW = (2 − d ) / 4 . d = 1: KPZ ’86 predicted relevance of non-linearity. β = 1 3 � = β EW Confirmed by exact solutions (1-d KPZ universality class: universal non-Gaussian limit laws, ...) d ≥ 3: predicted irrelevance of small non-linearity, transition at λ c . ⇒ see Magnen-Unterberger ’17, Gu-Ryzhik-Zeitouni ’17 for λ ≪ 1 F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 11 / 30

The critical dimension d = 2 and Wolf’s conjecture ∂ t h ( x, t ) = ∆ h ( x, t ) + λ ( ∇ h ( x, t ) , H ∇ h ( x, t )) + ξ smooth ( x, t ) One-loop perturbative (in λ ) Renormalization-Group analysis (D. Wolf ’91): if det( H ) > 0, non-linearity relevant, α � = α EW , β � = β EW ; if det( H ) ≤ 0, small non-linearity irrelevant. EW Universality class. Conjecture : Two universality classes: Anisotropic KPZ (AKPZ) class: det( D 2 v ( ρ )) ≤ 0. Large-scale fixed point: EW equation. α AKPZ = 0 , β AKPZ = 0. KPZ class: det( D 2 v ( ρ )) > 0. α KPZ � = 0 , β KPZ � = 0. Numerics (Halpin-Healy et al.): in KPZ class, universal exponents α KPZ ≈ 0 . 39 ..., β KPZ ≈ 0 . 24 ... . F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 12 / 30

Back to the deposition process rate = 1 rate = 0 Envelope property: h ( t = 0) = h (1) ∨ h (2) = ⇒ h ( t ) = h (1) ( t ) ∨ h (2) ( t ) h (1) ∨ h (2) h (2) h (1) h (1) h (2) Then, superadditivity argument (T. Sepp¨ al¨ ainen, F. Rezakhanlou) implies that v ( · ) exists and is convex. Natural candidate for KPZ class. No math results on stationary states or critical exponents α KPZ , β KPZ F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 13 / 30

A long-jump variant rate = p Jumps constrained only by interlacement conditions rate = 1 − p A. Borodin & P. Ferrari ’08 F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 14 / 30

A long-jump variant rate = p Jumps constrained only by interlacement conditions rate = 1 − p A. Borodin & P. Ferrari ’08 Should the universality class change? not obvious a priori. In fact, it does change F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 14 / 30

AKPZ signature Theorem (F.T., 15) Stationary states π ρ are “locally uniform” Stationary states free-fermionic (determinantal correlations) Roughness exponent α = 0 : logarithmic fluctuations, scaling to massless Gaussian field Growth exponent β = 0 Var π ρ ( h ( x, t ) − h ( x, 0)) t →∞ = O (log t ) uniform given F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 15 / 30

Speed and Hydrodynamic limit Theorem (M. Legras, F.T. ’17) If ǫ → 0 ǫh ( ǫ − 1 x, t = 0) = φ 0 ( x ) , ∀ x ∈ R 2 lim with φ 0 ( · ) convex, then ǫ → 0 ǫh ( ǫ − 1 x, ǫ − 1 t ) = φ ( x, t ) , lim t > 0 (with high probability as ǫ → 0) where φ solves � ∂ t φ ( x, t ) = v ( ∇ φ ( x, t )) φ ( x, 0) = φ 0 ( x ) . Speed of growth v ( ρ ): explicit and det D 2 v ( ρ ) < 0 F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 16 / 30

Comments on hydrodynamic equation Non-linear Hamilton-Jacobi equation ⇒ singularities in finite time Physically relevant solution: viscosity solution. ǫ → 0 + v ( ∇ φ ) �→ v ( ∇ φ ) + ǫ ∆ φ, v ( · ) non convex ⇒ no variational formula (like “minimal action”) for viscosity solution. For convex profile, variational formula. Technical difficulty: long jumps, possible pathologies (tools: from works of T. Sepp¨ al¨ ainen) F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 17 / 30

Previous results on the model Theorem (A. Borodin, P. Ferrari ’08) For “triangular-array Gibbs-type initial conditions”, hydrodynamic limit and central limit theorem on scale √ log t . 2 uniform given F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 18 / 30

Smooth phases and singularities of v ( · ) For equilibrium 2d discrete interface models, smooth (or “rigid”) (as opposed to: rough) phases at special slopes Exponential decay of correlations, no fluctuation growth: sup x Var( h ( x ) − h (0)) < ∞ , E.g. SOS model at low temperature; dimers (“gas phases”),... F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 19 / 30

Smooth phases and singularities of v ( · ) Questions: AKPZ growth models with smooth stationary states? We implicitly assumed that speed v ( · ) is differentiable ( H = D 2 v in KPZ Eq.) What if it is not? Still Edwards-Wilkinson behavior? Link with smooth stationary states? F. Toninelli (CNRS & Lyon 1) Stochastic Interface Dynamics XIX ICMP, Montr´ eal 20 / 30