KPZ and SHE Main Results Critical Regime Directed Polymer and SHE On the 2d KPZ and Stochastic Heat Equation via directed polymers Francesco Caravenna Universit` a degli Studi di Milano-Bicocca Etats de la Recherche: M´ ecanique Statistique Paris, IHP ∼ 10-14 December 2018 Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 1 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Collaborators Nikos Zygouras (Warwick) and Rongfeng Sun (NUS) Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 2 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Overview I will talk about two stochastic PDEs on R d (mainly d = 2) ◮ Kardar-Parisi-Zhang Equation (KPZ) ◮ Multiplicative Stochastic Heat Equation (SHE) In a nutshell ◮ KPZ and SHE ill-defined due to singular terms ◮ Regularized versions (mollified, or discretized) ◮ Do regularized solutions converge? (as regularization is removed) Not a minicourse in stochastic analysis! � Statistical Mechanics Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 3 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Overview Main focus on dimension d = 2. Recent progress on “subcritical” regime . . . and some results in the “critical” regime (many questions still open!) Edwards-Wilkinson fluctuations Regularized solutions converge to explicit Gaussian random field Plan ◮ Main results + general picture in dim. d = 1, d = 2, d ≥ 3 ◮ Connection and intuition with Directed Polymer ◮ Sketch of the proof + main tools Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 4 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE References ◮ [CSZ 17] Universality in marginally relevant disordered systems AAP 2017 ◮ [CSZ 18a] On the moments of the (2+1)-dimensional directed polymer and Stochastic Heat Equation in the critical window arXiv, Aug 2018 ◮ [CSZ 18b] The two-dimensional KPZ equation in the entire subcritical regime arXiv, Dec 2018 ( d = 2) [Bertini Cancrini 98] [Chatterjee Dunlap 18] ( d ≥ 3) [Magnen Unterberger 18] [Gu Ryzhik Zeitouni 18] [Mukherjee Shamov Zeitouni 16] [Comets Cosco Mukherjee 18] Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 5 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE KPZ Equation Random interface growth [Kardar-Parisi-Zhang PRL’86] ∂ t h ( t , x ) = 1 2∆ h ( t , x ) + 1 2 |∇ h ( t , x ) | 2 + β ξ ( t , x ) (KPZ) h ( t , x ) = interface height at time t ≥ 0, space x ∈ R d ξ ( t , x ) = space-time white noise ( δ -correlated Gaussian field � Continuum analogue of i.i.d. random field) β > 0 noise strength Singular term |∇ h ( t , x ) | 2 undefined ( ∇ h is a distribution) Take ξ ( t , x ) smooth. KPZ is linearized by Cole-Hopf transformation u ( t , x ) := e h ( t , x ) Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 7 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Stochastic Heat Equation (SHE) t ≥ 0, x ∈ R d Multiplicative Stochastic Heat Equation (SHE) ∂ t u ( t , x ) = 1 2∆ u ( t , x ) + β u ( t , x ) ξ ( t , x ) (SHE) Linear equation, in principle easier SHE well-posed in d = 1 by Ito theory (stochastic integration) Initial datum u (0 , x ) ≡ 1 (for simplicity) � t � Mild formulation: u ( t , x ) = 1 + β g t − s ( x − y ) u ( s , y ) ξ (d s , d y ) 0 R 1 e − x 2 where g t ( x ) = √ heat kernel on R 2 t 2 π t Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 8 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE One space dimension d = 1 ◮ SHE solution u ( t , x ) well-defined, random continuous function ◮ Continuous and strictly positive [Mueller 91 ] Explicit Wiener chaos representation �� ∞ k � � β k u ( t , x ) = 1 + g t 1 ( x 1 ) g t 2 − t 1 ( x 2 − x 1 ) . . . ξ (d t i , d x i ) k =1 i =1 0 < t 1 <...< t k < t ( x 1 ,..., x k ) ∈ R k Forget the definition of KPZ equation, focus on its solution Cole-Hopf “solution” of KPZ h ( t , x ) := log u ( t , x ) This is indeed the “right” solution Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 9 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE One space dimension d = 1 In support of KPZ Cole-Hopf solution ◮ Arises as a limit of interacting particle systems [Bertini Giacomin ’97] ◮ Fluctuations of 1d exactly solvable models of interface growth � KPZ universality class Surveys: [Corwin ’12] [Quastel Spohn ’15] Robust justification by solution theories for singular stochastic PDEs ◮ Regularity Structures [Hairer ’13] [Hairer ’14] ◮ Paracontrolled Distributions [Gubinelli Imkeller Perkowski ’15] ◮ Energy Solutions [Goncalves Jara ’14] ◮ Renormalization Approach [Kupiainen ’16] All these approaches only work for KPZ in d = 1 (sub-critical) Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 10 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE The general setting General dimensions d : how to find a “solution” of KPZ and SHE ? Mollify (regularize) the white noise ξ ( t , x ) in space on scale ε > 0 � � ξ ε ( t , x ) := ξ ( t , · ) ∗ j ε ( x ) ◮ j ε ( x ) := ε − d j ( ε − 1 x ) j ∈ C ∞ c ( R d ) probability density � t ◮ t �→ W ε ( t , x ) := ξ ε (d s , x ) Brownian motions 0 ε := ε − d � j � 2 (correlated in x , variance σ 2 L 2 ) Replace ξ by ξ ε (KPZ) and (SHE) well-posed by Ito theory � Do mollified solutions h ε ( t , x ) and u ε ( t , x ) have a limit as ε ↓ 0 ? Disorder strength β = β ε needs to be renormalized! Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 11 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Mollified equations Mollified SHE ∂ t u ε = 1 2∆ u ε + β ε u ε ξ ε ( ε -SHE) u ε (0 , · ) ≡ 1 By Ito’s formula h ε ( t , x ) := log u ε ( t , x ) satisfies Mollified KPZ ∂ t h ε = 1 2∆ h ε + 1 2 |∇ h ε | 2 + β ε ξ ε − C ε ( ε -KPZ) h ε (0 , · ) ≡ 0 ε ε − d � j � 2 C ε := β 2 ε σ 2 ε = β 2 L 2 Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 12 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Key problem Can we choose β ε ∈ (0 , ∞ ) so that u ε ( t , x ) and h ε ( t , x ) admit non-trivial limits as ε ↓ 0 ? YES! (. . . ) ˆ β (fixed) d = 1 √ 2 π ˆ β ˆ � d = 2 β ε = β ∈ (0 , ∞ ) log ε − 1 d − 2 ˆ β ε d ≥ 3 2 Note that β ε → 0 for d = 2 and d ≥ 3 Choice of β ε will be clear later ( � directed polymers) Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 13 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Main result I. Phase transition for SHE √ 2 π ˆ β ˆ Space dimension d = 2 β ε = � β ∈ (0 ∞ ) log ε − 1 Theorem (SHE one-point distribution) [CSZ 17] Phase transition (“weak to strong disorder”) with critical value ˆ β c = 1 � � if ˆ β Z − 1 2 σ 2 exp σ ˆ β < 1 ˆ Fix t > 0 , x ∈ R 2 : d β u ε ( t , x ) − − → if ˆ ε ↓ 0 0 β ≥ 1 σ 2 1 Z ∼ N (0 , 1) β := log ˆ 1 − ˆ β 2 Subcritical regime ˆ β < 1. For distinct x 1 , . . . , x n ∈ R 2 u ε ( t , x i ) become asymptotically independent (!) as ε ↓ 0 Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 15 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Main result I. Phase transition for KPZ √ 2 π ˆ β ˆ Space dimension d = 2 β ε = � β ∈ (0 ∞ ) log ε − 1 Theorem (KPZ one point distribution) [CSZ 17] Phase transition (“weak to strong disorder”) with critical value ˆ β c = 1 if ˆ β Z − 1 2 σ 2 σ ˆ β < 1 ˆ Fix t > 0 , x ∈ R 2 : d β h ε ( t , x ) − − → if ˆ ε ↓ 0 − ∞ β ≥ 1 σ 2 1 Z ∼ N (0 , 1) β := log ˆ 1 − ˆ β 2 Subcritical regime ˆ β < 1. For distinct x 1 , . . . , x n ∈ R 2 h ε ( t , x i ) become asymptotically independent (!) as ε ↓ 0 Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 16 / 37
KPZ and SHE Main Results Critical Regime Directed Polymer and SHE Sub-critical regime ˆ β < 1 For ˆ β < 1 u ε ( t , x ) and h ε ( t , x ) are very irregular functions of x Look at u ε ( t , · ) and h ε ( t , · ) as random distributions on R 2 E [ h ε ( t , x )] = − 1 E [ u ε ( t , x )] ≡ 1 2 σ 2 β + o (1) as ε ↓ 0 ˆ Law of large numbers d d u ε ( t , · ) h ε ( t , · ) − 1 2 σ 2 − − → 1 − − → as distributions ˆ β ε ↓ 0 ε ↓ 0 � � d ∀ φ ∈ C c ( R 2 ) : R 2 u ε ( t , x ) φ ( x ) d x − − → R 2 φ ( x ) d x ε ↓ 0 � � � � d R 2 h ε ( t , x ) φ ( x ) d x − 1 2 σ 2 − − → R 2 φ ( x ) d x ˆ β ε ↓ 0 Fluctuations? Francesco Caravenna 2d KPZ and SHE via directed polymers 10-14 December 2018 17 / 37
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