The Speed of a Second Class Particle in the ASEP Axel Saenz - PowerPoint PPT Presentation

The Speed of a Second Class Particle in the ASEP Axel Saenz University of Virginia April 11, 2019 Joint work with Promit Ghosal and Ethan Zell CIRM Integrability and Randomness in Mathematical Physics and Geometry Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Outline The Model and The Result 1 Background 2 Proof 3 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Section 1 The Model and The Result Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

The ASEP Figure: ASEP on a line. A jumping rates asymmetry p � = q S particles only move up to one position to the left or right at each instance E particles may not occupy the same position P the model is a Markov process Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

The Second Class Particle Figure: ASEP with Second class particles. Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Step Initial Conditions Figure: Step Initial Conditions Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Simulations ℙ t ( x * 0 ( t ) < s ) 1.0 0.8 0.6 0.4 0.2 s - 1000 - 500 500 1000 Figure: t = 1 , 000 and 10 , 000 trails Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Simulations ℙ t ( x * 1 ( t ) < s ) 1.0 0.8 0.6 0.4 0.2 s - 400 - 200 200 400 Figure: t = 500 and 10 , 000 trails Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

The Result Theorem (Ghosal, S., Zell (2019)) Given the asymmetry parameter γ := p − q for p ∈ ( 1 2 , 1 ] and q = 1 − p , we have x ∗ L ( t ) d → U L , as t → ∞ (1) t with U L , a random variable supported on [ − γ, γ ] , and � 1 − γ − 1 s � L + 1 � � U L ≥ s = , ∀ s ∈ [ − γ, γ ] . (2) P 2 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Section 2 Background Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Hydrodynamic Limit Introduce the occupation variable � 1 , x is occupied at time t η ( x , t ) = (3) 0 , x is not occupied at time t for x ∈ Z and t ∈ R ≥ 0 . Then, for the scaling limit with τ = ǫ t , χ = ǫ x , and ǫ → 0 , (4) we have the inviscid Burgers equation ∂ u ( τ, χ ) + ( p − q ) ∂ [ u ( τ, χ )( 1 − u ( τ, χ ))] = 0 . (5) ∂τ ∂χ for u ( χ, τ ) = E ( η ( τ, χ )) . Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Characteristic Lines A characteristic line χ ( τ ) is defined so that � � u χ ( τ ) , τ = constant , (6) In fact, the characteristic line are straight lines χ ( τ ) = v τ + χ ( 0 ) (7) with � � v = 1 − 2 η χ ( 0 ) , 0 . (8) Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Shocks Take the (partial) step initial conditions, or shock initial conditions , � ρ, χ < 0 u ( χ, 0 ) = (9) λ, χ > 0 with ρ < λ . u ( 0, χ ) 1.0 0.8 λ 0.6 0.4 ρ 0.2 1.0 χ - 1.0 - 0.5 0.0 0.5 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Shocks Then, the speed of the characteristic line is given by � 1 − 2 ρ, χ < 0 � � v = 1 − 2 η χ ( 0 ) , 0 = (10) 1 − 2 λ, χ > 0 with ρ < λ . t 1.0 0.8 0.6 0.4 0.2 1.0 χ - 1.0 - 0.5 0.0 0.5 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Pre-scaling Shocks In the pre-scaling regime, the shock initial conditions are given by independent occupation variables { η ( 0 , x ) } x ∈ Z with � ρ, x > 0 P ( η ( x , 0 ) = 1 ) = . (11) λ, x < 0 with ρ < λ . Moreover, we introduce a second class particle at the origin P ( η ( x , 0 ) = 2 ) = 1 , (12) and denote the location of the second class particle by X ( t ) . Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

A Second Class Particle in the Shock Theorem (Ferrari (1991)) For the ASEP with shock initial conditions, the second class particle stays at the shock of the Burgers equation: � ρ, χ < 0 η ( ǫ − 1 τ, ǫ − 1 ( χ + X ( ǫ − 1 τ ))) = 1 � � ǫ → 0 P lim = . (13) λ, χ > 0 Moreover, the speed of the second class particle convergence almost surely X ( t ) → ( p − q )( 1 − λ − ρ ) . (14) t Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Rarefaction Take the (partial) step initial conditions, or rarefaction initial conditions , � ρ, χ < 0 u ( χ, 0 ) = (15) λ, χ > 0 with ρ > λ . u ( 0, χ ) 1.0 ρ 0.8 0.6 0.4 λ 0.2 1.0 χ - 1.0 - 0.5 0.0 0.5 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Rarefaction Then, the speed of the characteristic line is given by � 1 − 2 ρ, χ < 0 � � v = 1 − 2 η χ ( 0 ) , 0 = (16) 1 − 2 λ, χ > 0 with ρ > λ . t 1.0 0.8 0.6 0.4 0.2 1.0 χ - 1.0 - 0.5 0.0 0.5 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Pre-scaling Rarefaction In the pre-scaling regime, the rarefaction initial conditions are given by independent occupation variables { η ( 0 , x ) } x ∈ Z with � ρ, x > 0 P ( η ( x , 0 ) = 1 ) = . (17) λ, x < 0 with ρ > λ . Moreover, we introduce a second class particle at the origin P ( η ( x , 0 ) = 2 ) = 1 , (18) and denote the location of the second class particle by X ( t ) . Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

A Second Class Particle in the Rarefaction Theorem (Ferrari, Kipnis (1995), Ferrari, Goncalves, Martin (2009)) For the ASEP with rarefaction initial conditions, the speed of the second class particle convergence in distribution X ( t ) → U p (19) t with U p , a uniformly distributed random variable on the interval [ − ( p − q ) , ( p − q )] . Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

A Second Class Particle in the Rarefaction Theorem ( Mountford and Guiol (2005)) For the TASEP with rarefaction initial conditions, the speed of the second class particle convergence almost surely X ( t ) → U (20) t with U , a uniformly distributed random variable on the interval [ − 1 , 1 ] . Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Rarefaction Arbitrary Initial Data For TASEP with arbitrary rarefaction initial data, − 1 n 1 � � ρ = lim inf η ( x ) > lim sup η ( x ) = λ, (21) n n →∞ n →∞ x = − n x = 1 Cator and Pimentel gave the law of the speed of the second class particle in 2013. For instance, if we take the initial data  0 , x ≥ 1   η ( x ) = (22) 1 , x ≤ 0 , x � = − L  2 , x = − L  with L ≥ 0, then X ( t ) d → U L , as t → ∞ (23) t with U L , a random variable supported on [ − 1 , 1 ] , and � 1 − s � L + 1 � � U L ≥ s = , ∀ s ∈ [ − 1 , 1 ] . (24) P 2 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

The TASEP Speed Process In 2011, Amir, Angel and Valko consider the TASEP with all different types of classes (colors). In particular, let η ( t , n ) := color of particle at location n (25) X ( t , n ) := location of particle colored n , with the stationary initial condition η ( 0 , n ) = X ( 0 , t ) = n . (26) Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

The TASEP Speed Process Theorem ( Amir, Angel and Valko (2011)) In the TASEP with stationary initial condition η ( 0 , n ) = n , the speed of every particle converges almost surely: X ( t , n ) − n → U n , as t → ∞ (27) t with { U n } n ∈ Z a family of random variables, each uniform on [ − 1 , 1 ] . Definition The process { U n } n ∈ Z is called the TASEP speed process. Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Section 3 Proof Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

The Result Theorem (Ghosal, S., Zell (2019)) Given the asymmetry parameter γ := p − q for p ∈ ( 1 2 , 1 ] and q = 1 − p , we have x ∗ L ( t ) d → U L , as t → ∞ (28) t with U L , a random variable supported on [ − γ, γ ] , and � 1 − γ − 1 s � L + 1 � � U L ≥ s = , ∀ s ∈ [ − γ, γ ] . (29) P 2 Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Strategy Coupling between second class ASEP and multi-colored ASEP. Symmetry between multi-colored and colorblind ASEP. Asymptotic analysis for block probabilities. Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP

Coupling Axel Saenz University of Virginia April 11, 2019 The Speed of a Second Class Particle in the ASEP