Li-Cheng Tsai Rutgers University Stochastic Analysis, Random Fields - PowerPoint PPT Presentation

Lower-tail large deviations of the KPZ equation Li-Cheng Tsai Rutgers University Stochastic Analysis, Random Fields and Integrable Probability The 12th Mathematical Society of Japan, Seasonal Institute Li-Cheng Tsai Lower-tail LDs of KPZ

The Kardar–Parisi–Zhang (KPZ) equation Random growth with smoothing effect and slope dependence 2 ( ∂ x h ) 2 + ξ ∂ t h = 1 2 ∂ xx h + 1 ξ = ξ ( t, x ) = spacetime white noise Li-Cheng Tsai Lower-tail LDs of KPZ

The Kardar–Parisi–Zhang (KPZ) equation • Define h ( t, x ) := log Z ( t, x ) . Li-Cheng Tsai Lower-tail LDs of KPZ

The Kardar–Parisi–Zhang (KPZ) equation • Define h ( t, x ) := log Z ( t, x ) . • This talk: Z (0 , x ) = δ ( x ) . For small t ≪ 1 , 2 πt e − x 2 1 2 t . Z ( t, x ) ≈ √ Li-Cheng Tsai Lower-tail LDs of KPZ

t → ∞ behaviors: centering, fluctuations, and tails Li-Cheng Tsai Lower-tail LDs of KPZ

t → ∞ behaviors: centering, fluctuations, and tails [Amir Corwin Quastel 10], [Calabrese Le Doussal Rosso 10], [Dotsenko 10], [Sasamoto Spohn 10] For Z (0 , x ) = δ ( x ) , as t → ∞ , t − 1 3 ( h (2 t, 0) + t 12 ) = ⇒ GUE Tracy Widom Li-Cheng Tsai Lower-tail LDs of KPZ

t → ∞ behaviors: centering, fluctuations, and tails 2 Φ ± ( z ) = rate functions Speed t v.s. t 2 e h (2 t, 0) = Z (2 t, 0) � 2 t � ξ ( s,b (2 t − s ))d s � = E BB e 0 Li-Cheng Tsai Lower-tail LDs of KPZ

Perturbative versus non-perturbative [Amir Corwin Quastel 10], [Calabrese Le Doussal Rosso 10], [Dotsenko 10], [Sasamoto Spohn 10] � � t �� � � 12 + tz Z (2 t, 0) exp − e = det I − K t,z E L 2 ( R + ) ( − 1) n � det( I − K t,z ) := 1 + � ∞ + det( K t,z ( x i , x j )) n i,j =1 d n x R n n =1 n ! � R + (1 + exp( − t 1 / 3 λ − tz )) − 1 Ai( x + r )Ai( x ′ + r )d r K t,z ( x, x ′ ) := Li-Cheng Tsai Lower-tail LDs of KPZ

Perturbative versus non-perturbative [Amir Corwin Quastel 10], [Calabrese Le Doussal Rosso 10], [Dotsenko 10], [Sasamoto Spohn 10] � � 12 + tz + h (2 t, 0) �� t � � exp − e = det I − K t,z E L 2 ( R + ) ( − 1) n � det( I − K t,z ) := 1 + � ∞ + det( K t,z ( x i , x j )) n i,j =1 d n x R n n =1 n ! Li-Cheng Tsai Lower-tail LDs of KPZ

Perturbative versus non-perturbative [Amir Corwin Quastel 10], [Calabrese Le Doussal Rosso 10], [Dotsenko 10], [Sasamoto Spohn 10] � � � � h (2 t, 0) + t 12 < tz ≈ det I − K t,z P L 2 ( R + ) ( − 1) n det( I − K t,z ) := 1 + � ∞ � + det( K t,z ( x i , x j )) n i,j =1 d n x n =1 R n n ! Li-Cheng Tsai Lower-tail LDs of KPZ

Perturbative versus non-perturbative [Amir Corwin Quastel 10], [Calabrese Le Doussal Rosso 10], [Dotsenko 10], [Sasamoto Spohn 10] � � � � h (2 t, 0) + t 12 < tz ≈ det I − K t,z P L 2 ( R + ) det( I − K t,z ) := 1 + � ∞ ( − 1) n � + det( K t,z ( x i , x j )) n i,j =1 d n x R n n =1 n ! • Upper tail z > 0 as t → ∞ , we have K t,z → 0 - Perturbative: det( I − K t,z ) = 1 − Tr( K t,z ) + . . . 3 - [Le Doussal Majumdar Schehr 16] predicted Φ + ( z ) = 4 3 z 2 - Proof in progress Li-Cheng Tsai Lower-tail LDs of KPZ

Perturbative versus non-perturbative [Amir Corwin Quastel 10], [Calabrese Le Doussal Rosso 10], [Dotsenko 10], [Sasamoto Spohn 10] � � � � h (2 t, 0) + t 12 < tz ≈ det I − K t,z P L 2 ( R + ) det( I − K t,z ) := 1 + � ∞ ( − 1) n � + det( K t,z ( x i , x j )) n i,j =1 d n x R n n =1 n ! • Upper tail z > 0 as t → ∞ , we have K t,z → 0 - Perturbative: det( I − K t,z ) = 1 − Tr( K t,z ) + . . . 3 - [Le Doussal Majumdar Schehr 16] predicted Φ + ( z ) = 4 3 z 2 - Proof in progress • Lower tail z < 0 , K t,z �→ I as t → ∞ - Non -perturbative Li-Cheng Tsai Lower-tail LDs of KPZ

The lower-tail of h (2 t, 0) Physics results • [Kolokolov Korshunov 07] and [Meerson Katzav Vilenkin 16] predicted small/large | z | behaviors Math results • [Corwin Ghosal 18] obtained bounds ( ∀ t ≥ t 0 ) capturing small/large | z | behaviors Li-Cheng Tsai Lower-tail LDs of KPZ

The lower-tail of h (2 t, 0) Physics results • [Kolokolov Korshunov 07] and [Meerson Katzav Vilenkin 16] predicted small/large | z | behaviors • [Sasorov Meerson Prolhac 17] predicted 5 4 4 2 1 15 π 6 (1 − π 2 z ) 2 − 2 π 2 z 2 Φ − ( z ) = 15 π 6 + 3 π 4 z − by WKB approx of an integral-diff eqn Math results • [Corwin Ghosal 18] obtained bounds ( ∀ t ≥ t 0 ) capturing small/large | z | behaviors Li-Cheng Tsai Lower-tail LDs of KPZ

The lower-tail of h (2 t, 0) Physics results • [Kolokolov Korshunov 07] and [Meerson Katzav Vilenkin 16] predicted small/large | z | behaviors • [Sasorov Meerson Prolhac 17] predicted 5 4 4 2 1 15 π 6 (1 − π 2 z ) 2 − 2 π 2 z 2 Φ − ( z ) = 15 π 6 + 3 π 4 z − by WKB approx of an integral-diff eqn • [Corwin Ghosal Krajenbrink Le Doussal Tsai 18] same Φ − by log/Coulomb gas • [Krajenbrink Le Doussal Prolhac 18] same Φ − by cumulant expansion Math results • [Corwin Ghosal 18] obtained bounds ( ∀ t ≥ t 0 ) capturing small/large | z | behaviors Li-Cheng Tsai Lower-tail LDs of KPZ

The lower-tail of h (2 t, 0) Physics results • [Kolokolov Korshunov 07] and [Meerson Katzav Vilenkin 16] predicted small/large | z | behaviors • [Sasorov Meerson Prolhac 17] predicted 5 4 4 2 1 15 π 6 (1 − π 2 z ) 2 − 2 π 2 z 2 Φ − ( z ) = 15 π 6 + 3 π 4 z − by WKB approx of an integral-diff eqn • [Corwin Ghosal Krajenbrink Le Doussal Tsai 18] same Φ − by log/Coulomb gas • [Krajenbrink Le Doussal Prolhac 18] same Φ − by cumulant expansion Math results • [Corwin Ghosal 18] obtained bounds ( ∀ t ≥ t 0 ) capturing small/large | z | behaviors • [Tsai 18] proof of Φ − by stochastic Airy operator Li-Cheng Tsai Lower-tail LDs of KPZ

Result Theorem (Tsai 18) Consider the IC Z (0 , x ) = δ ( x ) . For z < 0 , as t → ∞ , � � 1 P [ h (2 t, 0) + t lim t 2 log 12 < tz ] = − Φ − ( z ) t →∞ 5 4 4 2 1 15 π 6 (1 − π 2 z ) 2 − 2 π 2 z 2 . where Φ − ( z ) := 15 π 6 + 3 π 4 z − Li-Cheng Tsai Lower-tail LDs of KPZ

Exponential functional of Airy Point Process [Borodin Gorin 16] � ∞ 1 � t � � 12 + tz � e − Z (2 t, 0) e E = E Airy 1 + e − t 1 / 3 ( λ i + t 2 / 3 z ) i =1 λ 1 < λ 2 < . . . ∈ R (space-reversed) Airy Point Process Li-Cheng Tsai Lower-tail LDs of KPZ

Exponential functional of Airy Point Process [Borodin Gorin 16] t � 12 + tz � � i =1 ψ t,z ( λ i ) � e − Z (2 t, 0) e e − � ∞ E = E Li-Cheng Tsai Lower-tail LDs of KPZ

Exponential functional of Airy Point Process [Borodin Gorin 16] � � � i =1 ψ t,z ( λ i ) � e − � ∞ h (2 t, 0) + t P 12 < tz ≈ E Li-Cheng Tsai Lower-tail LDs of KPZ

Laplace’s method / Varadhan’s lemma — a general picture � � i =1 ψ t,z ( λ i ) � e − ψ t,z ( ρ ) e − penalty ( ρ ) d ρ e − � ∞ = E � �� � :=d P [ ρ ] Li-Cheng Tsai Lower-tail LDs of KPZ

Laplace’s method / Varadhan’s lemma — a general picture � i =1 ψ t,z ( λ i ) � � � �� e − � ∞ E ≈ exp − min ψ t,z ( ρ ) + penalty ( ρ ) ρ Li-Cheng Tsai Lower-tail LDs of KPZ

Laplace’s method / Varadhan’s lemma — a general picture � i =1 ψ t,z ( λ i ) � � � �� e − � ∞ E ≈ exp − min ψ t,z ( ρ ) + penalty ( ρ ) ρ Examples [Corwin Ghosal 18] P [ ρ ≈ ρ sq ] ≈ 1 , but R ψ t,z ( λ ) ρ sq ( λ )d λ ≈ e − t 2 b 1 ( z ) . � e − Li-Cheng Tsai Lower-tail LDs of KPZ

Laplace’s method / Varadhan’s lemma — a general picture � i =1 ψ t,z ( λ i ) � � � �� e − � ∞ E ≈ exp − min ψ t,z ( ρ ) + penalty ( ρ ) ρ Examples [Corwin Ghosal 18] R ψ t,z ( λ ) ρ push ( λ )d λ ≈ 1 , but � P [ ρ ≈ ρ sq ] ≈ 1 , but e − P [ ρ ≈ ρ push ] ≈ e − t 2 b 2 ( z ) . R ψ t,z ( λ ) ρ sq ( λ )d λ ≈ e − t 2 b 1 ( z ) . � e − Li-Cheng Tsai Lower-tail LDs of KPZ

Stochastic Airy Operator Theorem (Ramirez Rider Virag 06) The Stochastic Airy Operator √ A := − d 2 2 W ′ ( x ) d x 2 + x + acting on Dom( A ) ⊂ L 2 ( R + ) has spectrum { λ 1 < λ 2 < . . . } , where W := standard BM. Li-Cheng Tsai Lower-tail LDs of KPZ

Large deviations controlled by W ′ � � �� E [ e − � ∞ i =1 ψ t,z ( λ i ) ] ≈ exp − min ψ t,z ( ρ ) + penalty ( ρ ) ρ ρ = eigenvalues distribution of √ A = − d 2 2 W ′ ( x ) d x 2 + x + Li-Cheng Tsai Lower-tail LDs of KPZ

Large deviations controlled by W ′ and then by v � � �� E [ e − � ∞ i =1 ψ t,z ( λ i ) ] ≈ exp − min ψ t,z ( ρ ) + penalty ( ρ ) ρ ρ = eigenvalues distribution of √ A = − d 2 2 W ′ ( x ) d x 2 + x + 2 3 v ( t − 2 Postulate: relevant deviations controlled by W ′ ( x ) ≈ t 3 x ) √ − f ′′ ( x ) + xf ( x ) + 2 W ′ ( x ) f ( x ) = λf ( x ) (eigen prob) 2 λ of order t 3 Li-Cheng Tsai Lower-tail LDs of KPZ

Large deviations controlled by W ′ and then by v � � �� E [ e − � ∞ i =1 ψ t,z ( λ i ) ] ≈ exp − min ψ t,z ( ρ ( v )) + penalty ( v ) v ρ ( v ) = eigenvalues distribution of √ 2 3 v ( t − 2 A v = − d 2 3 x ) d x 2 + x + 2 t 2 3 v ( t − 2 Postulate: relevant deviations controlled by W ′ ( x ) ≈ t 3 x ) penalty( v ) = ρ ( v ) = Li-Cheng Tsai Lower-tail LDs of KPZ