Invariant measures for KdV and Toda-type discrete integrable - PowerPoint PPT Presentation

Invariant measures for KdV and Toda-type discrete integrable systems Online Open Probability School 12 June 2020 David Croydon (Kyoto) joint with Makiko Sasada (Tokyo) and Satoshi Tsujimoto (Kyoto)

1. KDV AND TODA-TYPE DISCRETE INTEGRABLE SYSTEMS

KDV AND TODA LATTICE EQUATIONS Korteweg-de Vries (KdV) equation: ∂x + ∂ 3 u ∂u ∂t + 6 u∂u ∂x 3 = 0 , where u = ( u ( x, t )) x,t ∈ R . Source: Shnir Toda lattice equation:  = e − ( q n − q n − 1 ) − e − ( q n +1 − q n ) , d dt p n  d = p n , dt q n  where p n = ( p n ( t )) t ∈ R , q n = ( q n ( t )) t ∈ R .

KDV AND TODA LATTICE EQUATIONS Korteweg-de Vries (KdV) equation: ∂x + ∂ 3 u ∂u ∂t + 6 u∂u ∂x 3 = 0 , where u = ( u ( x, t )) x,t ∈ R . Source: Brunelli Toda lattice equation:  = e − ( q n − q n − 1 ) − e − ( q n +1 − q n ) , d dt p n  d = p n , dt q n  Source: Singer et al where p n = ( p n ( t )) t ∈ R , q n = ( q n ( t )) t ∈ R .

t = 1 ・・・ t = 0 １ 2 3 4 5 6 7 8 9 10 11 12 13 14 １ 2 3 4 5 6 7 8 9 10 11 12 13 14 ・・・ BOX-BALL SYSTEM (BBS) Discrete time deterministic dynamical system (cellular automa- ton) introduced in 1990 by Takahashi and Satsuma. In original work, configurations ( η x ) x ∈ Z with a finite number of balls were considered. (NB. Empty box: η x = 0; ball η x = 1.) • Every ball moves exactly once in each evolution time step • The leftmost ball moves first and the next leftmost ball moves next and so on... • Each ball moves to its nearest right vacant box Dynamics T : { 0 , 1 } Z → { 0 , 1 } Z :   n − 1   � ( Tη ) n = min  1 − η n , ( η m − ( Tη ) m )  , m = −∞ where ( Tη ) n = 0 to left of particles.

BBS CARRIER • Carrier moves left to right • Picks up a ball if it finds one • Puts down a ball if it comes to an empty box when it carries at least one ball Set U n to be number of balls carried from n to n + 1, then  U n − 1 + 1 , if η n = 1 ,   U n = U n − 1 , if η n = 0 , U n − 1 = 0 , U n − 1 − 1 , if η n = 0 , U n − 1 > 0 ,   and ( Tη ) n = min { 1 − η n , U n − 1 } .

� � � LATTICE EQUATIONS The local dynamics of the BBS are described via a system of lattice equations: η t +1 η t +1 n � n +1 F (1 , ∞ ) F (1 , ∞ ) · · · U t U t U t n +1 · · · , n n − 1 udK udK η t η t n n +1 . . . . . . where F (1 , ∞ ) is an involution, as given by: udK F (1 , ∞ ) ( η, u ) := (min { 1 − η, u } , η + u − min { 1 − η, u } ) . udK This is (a version of) the ultra-discrete KdV equation (udKdV) . Can generalise to box capacity J ∈ N ∪ {∞} and carrier capacity K ∈ N ∪ {∞} .

BASIC QUESTIONS In today’s talk, I will address two main topics for the BBS (and related systems): • Existence and uniqueness of solutions to initial value problem for (udKdV) with infinite configurations? • I.i.d. invariant measures on initial configurations? Other recent developments in the study of the BBS that I will not talk about: • Invariant measures based on solitons, e.g. [Ferrari, Nguyen, Rolla, Wang]. See also [Levine, Lyu, Pike], etc. ✐②②②✐✐✐✐✐✐②✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ✐✐✐✐②②②✐✐✐✐②✐✐✐✐✐✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐②②②✐✐②✐✐✐✐✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐✐✐②②✐②②✐✐✐✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐✐✐✐✐②✐✐②②②✐✐✐✐✐✐✐ ✐✐✐✐✐✐✐✐✐✐✐✐✐②✐✐✐✐②②②✐✐✐✐ • Generalized hydrodynamic limits, e.g. [C., Sasada], [Kuniba, Misguich, Pasquier].

INTEGRABLE SYSTEMS DERIVED FROM THE KDV AND TODA EQUATIONS

� � ULTRA-DISCRETE KDV EQUATION (UDKDV) Local dynamics: F ( J,K ) Model Lattice structure udK a +min { J − a,b } η t +1 udKdV n − min { a,K − b } U t � U t � b +min { a,K − b } n n − 1 b − min { J − a,b } η t n a Variables are R -valued. Parameter J represents box capacity, K represents carrier capacity. Multi-coloured version of BBS/ UDKDV also studied [Kondo].

� � DISCRETE KDV EQUATION (DKDV) Local dynamics: F ( α,β ) Model Lattice structure dK ω t +1 b (1+ βab ) dKdV n (1+ αab ) U t � U t � a (1+ αab ) n n − 1 b (1+ βab ) ω t n a Variables are (0 , ∞ )-valued. UDKDV is obtained as ultra-discrete/ zero-temperature limit by making change of variables: α = e − J/ε , β = e − K/ε , a = e a/ε , b = e b/ε .

� � � � ULTRA-DISCRETE TODA EQUATION (UDTODA) Model Lattice structure Local dynamics: F udT Q t +1 E t +1 a + b udToda min { b, c } n n − min { b,c } U t � U t � a + c n c n +1 − min { b,c } E t Q t b a n n +1 Variables are R -valued. For BBS(1, ∞ ), can understand ( Q t n , E t n ) n ∈ Z as the lengths of consequence ball/empty box sequences.

� � � � DISCRETE TODA EQUATION (DTODA) Model Lattice structure Local dynamics: F dT I t +1 J t +1 ab dToda b + c n n b + c U t � U t � ac c n n +1 b + c b a J t I t n n +1 Variables are (0 , ∞ )-valued. UDTODA is obtained as ultra- discrete/ zero-temperature limit by making change of variables: a = e − a/ε , b = e − b/ε , c = e − b/ε .

INTEGRABLE SYSTEMS DERIVED FROM THE KDV AND TODA EQUATIONS NB. [Quastel, Remenik 2019] connected the KPZ fixed point to the Kadomtsev-Petviashvili (KP) equation. Both dKdV and dToda can be obtained from the discrete KP equation.

2. GLOBAL SOLUTIONS BASED ON PATH ENCODINGS

PATH ENCODING FOR BBS AND CARRIER Let η be a finite configuration. Define ( S n ) n ∈ Z by S 0 = 0 and S n − S n − 1 = 1 − 2 η n . Let U n = M n − S n , where M n = max m ≤ n S m . Can check ( U n ) n ∈ Z is a carrier process, and the path encoding of Tη is TS n = 2 M n − S n − 2 M 0 .

PITMAN’S TRANSFORMATION The transformation S �→ 2 M − S is well-known as Pitman’s transformation. (It transforms one- sided Brownian motion to a Bessel process [Pitman 1975].) Given the relationship between η and S , and U = M − S , the relation TS = 2 M − S − 2 M 0 is equivalent to: ( Tη ) n + U n = η n + U n − 1 , i.e. conservation of mass.

‘PAST MAXIMUM’ OPERATORS Above corresponds to udKdV( J , ∞ ) and dKdV( α ,0); parameters appear in path encoding. More novel ‘past maximum’ operators for udKdV( J , K ), J ≤ K [C., Sasada]. Spatial shift θ needed for Toda systems.

‘PAST MAXIMUM’ OPERATORS � ∗ =dToda. T ∨ =udKdV, T =dKdV, T ∨∗ =udToda, T �

GENERAL APPROACH Aim to change variables a t n := A n ( η t n ), b t n := B n ( u t n ) so that ( a t +1 n − m , b t n ) = K n ( a t n , b t n − 1 ) satisfies K (1) ( a, b ) − 2 K (2) ( a, b ) = a − 2 b. n n Path encoding given by S n − S n − 1 = a n . Existence of carrier ( b n ) n ∈ Z equivalent to existence of ‘past max- imum’ satisfying � + S n . M n = K (2) � S n − S n − 1 , M n − 1 − S n − 1 n Dynamics then given by S �→ T M S := 2 M − S − 2 M 0 . Advantage: M equation can be solved in examples. Moreover, can determine uniquely a choice of M for which the procedure can be iterated. Gives existence and uniqueness of solutions.

APPLICATION TO BBS( J , ∞ ) Given η = ( η n ) n ∈ Z ∈ { 0 , 1 , . . . , J } Z , let S be the path given by setting S 0 = 0 and S n − S n − 1 = J − 2 η n for n ∈ Z . If S satisfies S n S n lim n > 0 , lim n > 0 n →∞ n →−∞ then there is a unique solution ( η t n , U t n ) n,t ∈ Z to udKdV that sat- isfies the initial condition η 0 = η . This solution is given by n := J − S t n + S t n + J n − 1 η t U t n := M ∨ ( S t ) n − S t , 2 , ∀ n, t ∈ Z , 2 where S t := ( T ∨ ) t ( S ) for all t ∈ Z . [Essentially similar results hold for other systems.]

APPLICATION TO BBS( J , ∞ ) [Simulation with J = 1. For configurations, time runs upwards.]

3. INVARIANT MEASURES VIA DETAILED BALANCE

APPROACHES TO INVARIANCE 1. Ferrari, Nguyen, Rolla, Wang: BBS soliton decomposition . 2. C., Kato, Tsujimoto, Sasada - Three conditions theorem for BBS (later generalized). Any two of the three following conditions imply the third: d U d Tη d ← − ¯ η = η, = U, = η, where ← − is the reversed configuration, and ¯ η U is the reversed carrier process given as ← − η n = η − ( n − 1) , ¯ U n = U − n . 3. C., Sasada - Detailed balance for locally-defined dynamics .

� DETAILED BALANCE (HOMOGENEOUS CASE) Consider homogenous lattice system η t +1 n � · · · U t U t F n . . . , n − 1 η t n . . . Suppose µ is a probability measure such that µ Z ( X ∗ ) = 1, where X ∗ are those configurations for which there exists a unique global solution. It is then the case that µ Z ◦ T − 1 = µ Z if and only if there exists a probability measure ν such that ( µ × ν ) ◦ F − 1 = µ × ν. Moreover, when this holds, U t n ∼ ν (under µ Z ).