Dynamics of the ultra-discrete Toda lattice via Pitman's transformation
aa r X i v : . [ m a t h - ph ] M a y DYNAMICS OF THE ULTRA-DISCRETE TODA LATTICEVIA PITMAN’S TRANSFORMATION
DAVID A. CROYDON, MAKIKO SASADA, AND SATOSHI TSUJIMOTOA
BSTRACT . By encoding configurations of the ultra-discrete Toda lattice by piecewise linearpaths whose gradient alternates between − ± Date : May 2, 2019.2010
Mathematics Subject Classification.
Key words and phrases. ultra-discrete Toda lattice, Pitman’s transformation, box-ball system on R .
1. I
NTRODUCTION
To introduce and motivate this study of the dynamics of the ultra-discrete Toda lattice, whichwere first defined in [9] (another version of the ultra-discrete Toda lattice was described in [7]),we start by recalling the related box-ball system (BBS) of [11]. We note that the link betweenthe ultra-discrete Toda lattice and the BBS was initially presented in [8]. In particular, states ofthe BBS are particle configurations η = ( η n ) n ∈ Z ∈ { , } Z , where we write η n = n , and η n = ∑ n ∈ Z η n < ∞ . The system evolution is then described by meansof a particle ‘carrier’, which moves along Z from left to right (that is, from negative to positive),being initially empty until it meets the first particle, picking up a particle when it crosses one, anddropping off a particle when it is holding at least one particle and sees a space (see the left-handside of Figure 1). Writing T η for the new configuration after one step of the dynamics, thisdescription can be formalised in terms of the ultra-discrete Korteweg-de Vries (KdV) equation:(1.1) ( T η ) n = min ( − η n , n − ∑ m = − ∞ ( η m − ( T η ) m ) ) , where we suppose ( T η ) n = n ≤ inf { m : η m = } , so that the sums in the above definitionare well-defined. An alternative characterisation of these dynamics can be given in terms of thevector (( Q n ) Nn = , ( E n ) N − n = ) , where Q n ∈ N is the length of the n th string of consecutive particles,and E n ∈ N is the length of string of empty spaces between the n th and ( n + ) st particle string(see the right-hand side of Figure 1). It is then possible to check that after one time step of theBBS dynamics, the lengths of strings of particles and empty spaces in T η is given by a vector ((( T Q ) n ) Nn = , (( T E ) n ) N − n = ) ∈ N N − of the same length as (( Q n ) Nn = , ( E n ) N − n = ) that arises as thesolution of the ultra-discrete Toda lattice equation: ( T Q ) n = min ( n ∑ k = Q k − n − ∑ k = ( T Q ) k , E n ) , ( T E ) n = Q n + + E n − ( T Q ) n , (1.2)where for the purposes of these equations we suppose that E N = ∞ . Whilst the BBS naturallygives rise to vectors with integer-valued entries, the dynamics given by (1.2) makes sense forany vector (( Q n ) Nn = , ( E n ) N − n = ) ∈ ( , ∞ ) N − ; it is this system that we call the ultra-discrete Todalattice, and which will be the focus here.Towards studying the BBS from a probabilistic viewpoint, including its invariant measures,in [1] a systematic extension of the BBS dynamics to configurations consisting of an infinitenumber of particles (in both the positive and negative directions) was presented. (Other recentprobabilistic studies that have also considered examples of two-sided infinite configurations are[4,5].) At the heart of [1] was the observation that the BBS dynamics could be expressed in termsof a certain transformation of an associated path encoding, which we now describe. Firstly, let η ∈ { , } Z be a particle configuration (now no longer restricted to a finite number of particles),and then define a path encoding of η , denoted S : Z → Z , by setting S : =
0, and(1.3) S n − S n − : = − η n , ∀ n ∈ Z . LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 3 F IGURE
1. Initial configuration and three time evolutions of the BBS (left-handside), as well as the ultra-discrete Toda lattice representation (right-hand side).We then introduce the transformation of reflection in the past maximum
T S : Z → Z via therelation(1.4) ( T S ) n : = M n − S n − M , ∀ n ∈ Z , where M n : = sup m ≤ n S m is the past maximum of S ; this operation of reflection in the pastmaximum is well-known in the probabilistic literature as Pitman’s transformation (after [10]).Clearly, for T S to be well-defined, we require M < ∞ . If this is the case, then we let T η ∈{ , } Z be the configuration given by ( T η ) n : = { ( T S ) n − ( TS ) n − = − } , ∀ n ∈ Z , (so that T S is the path encoding of T η ). It is possible to check that the map η T η coincideswith the original definition of the BBS dynamics η T η in the finite particle case [1, Lemma2.3], and moreover is consistent with an extension to the case of a bi-infinite particle configura-tion satisfying M < ∞ from a natural limiting procedure [1, Lemma 2.4]. Thus this descriptionof the BBS could be seen as a natural generalisation of (1.1).With a view to studying invariant measures for the ultra-discrete Toda lattice and its periodicvariant, which is done in the sister article [3], the principal aim of this work is to show thatPitman’s transformation is also a suitable means by which to extend the definition of this modelto infinite configurations. The one distinction that should be highlighted, however, is that for theultra-discrete Toda lattice, we incorporate a spatial shift into the transformation of the associatedpath encoding. We note that the reason for this shift being needed relates to the fact that theultra-discrete Toda lattice picture does not retain the information about the spatial location ofparticles that the BBS system does.To explain the relevant transformation in detail, we first generalise the notion of a configura-tion for the ultra-discrete Toda lattice. Specifically, we will now consider states to be elements LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 4 of the form (( Q n ) N n = N , ( E n ) N − n = N ) ∈ ( , ∞ ) ( N − N )+ , where N , N ∈ Z ∪ {− ∞ } ∪ { ∞ } satisfy N ≤ ≤ N . By convention, in the case N is finite, we set E N − = ∞ , and in the case N isfinite, we set E N = ∞ . Furthermore, we suppose that(1.5) ∑ n = N ( Q n + E n − ) = ∞ , N ∑ n = ( Q n + E n ) = ∞ . The collection of such elements (( Q n ) N n = N , ( E n ) N − n = N ) will be denoted by C . Given a memberof C , the corresponding path encoding is then obtained by concatenating linear segments oflength Q N , E N , Q N + , E N + , . . . , E N − , Q N whose gradient alternates between − Q starts at 0 and has gradient −
1. More precisely, for n ∈ { N , . . . , N } , let(1.6) O n : = " n − ∑ k = ( Q k + E k ) , n − ∑ k = ( Q k + E k ) + Q n , (where we apply the convention that ∑ j = is zero, and ∑ − nj = = − ∑ j = − n + for n ∈ N ,) and set O = ∪ N n = N O n . We then define S ∈ C ( R , R ) by setting, analogously to (1.3),(1.7) S x : = Z x (cid:0) − { y ∈ O } (cid:1) dy , ∀ x ∈ R ;the set of path encodings will be denoted S . (Note that the conditions set out at (1.5) are notnecessary to ensure the function S is well-defined on the whole of R , but do mean that there isno accumulation of local maxima/minima.) It is easy to see that the operation of mapping anelement of C to its path encoding is a bijection. As at (1.4), we can define T S : R → R by setting(1.8) ( T S ) x : = M x − S x − M , ∀ x ∈ R , where M x : = sup y ≤ x S x is the past maximum of S , and again we require M < ∞ for this operationto be well-defined. We will denote by C T the set of configurations in C whose path encodingssatisfy the latter condition, and S T the associated set of path encodings. Now, although T S isagain an piecewise linear element of C ( R , R ) with gradient either − S . To obtain a path that is in S , weintroduce a shift operator. Indeed, for x ∈ R and S ∈ C ( R , R ) , we characterise θ x S by ( θ x S ) y = S x + y − S x , ∀ y ∈ R , let τ ( S ) : = inf { x ≥ x ∈ LM ( S ) } , where LM ( S ) is the set of local maxima of S (for the elementsof C ( R , R ) that will be considered in this article, τ ( S ) is always well-defined and finite), and set θ τ ( S ) : = θ τ ( S ) ( S ) . We then define an operator T on S T by the composition of T and θ τ , that is(1.9) T S : = θ τ ( T S ) , ∀ S ∈ S T . It is possible to check from the definitions that for S ∈ S T , τ ( T S ) = Q < ∞ , which means T S is well-defined, and moreover T S ∈ S with the values of N and N preserved (see Lemma2.1). Hence we can define an associated configuration ((( T Q ) n ) N n = N , (( T E ) n ) N − n = N ) ∈ C (seeFigure 2 for an example). The following result demonstrates that this procedure is an extensionof the dynamics of the ultra-discrete Toda lattice for finite configurations, as given by (1.2).In Subsection 2.2, we moreover show that the transformation of the path encoding we have LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 5 (cid:0) ( Q n ) Nn = , ( E n ) N − n = (cid:1) ST S T S (cid:0) ( T Q ) Nn = , (( T E ) n ) N − n = (cid:1) F IGURE
2. Graphical representation of the dynamics of the ultra-discrete Todalattice in terms of the associated path encodings. NB. The red line in the graphsfor S and T S shows the path of M .described also yields the dynamics of the periodic ultra-discrete Toda lattice, see Theorem 2.3in particular. Theorem 1.1.
Let N ∈ N , (( Q n ) Nn = , ( E n ) N − n = ) ∈ ( , ∞ ) N − , and S represent the associated pathencoding, as defined by (1.7) . It then holds that ((( T Q ) n ) Nn = , (( T E ) n ) N − n = ) , as given by (1.2) ,has path encoding T S, as defined by (1.9) . In the final section of the article, Section 3, we generalise the above result to a version ofthe box-ball system on R , as described in [1], where states of the system are described by acertain class of continuous functions, and the dynamics are given by Pitman’s transformation(1.8). Perhaps most significantly, the result we establish (see Proposition 3.1 below) allows usto handle functions S whose gradient is not restricted to ±
1, which is potentially relevant in thestudy of the scaling limits of box-ball systems where the box capacity is not restricted to one,see [2] for background on path encodings for such a model.2. D
YNAMICS OF THE PATH ENCODING
To check the claims of the introduction as we do in this section, it will be convenient to con-sider a decomposition of C T into subsets. To this end, for N , N ∈ Z ∪ {− ∞ } ∪ { ∞ } with N ≤ ≤ N , let C N , N be the subset of C consisting of elements of the form (( Q n ) N n = N , ( E n ) N − n = N ) ,and let S N , N be the set of associated path encodings. It is easy to see that the latter set is givenby functions S : R → R of the form S x = Z x (cid:18) − { y ∈∪ N n = N [ a n , b n ] } (cid:19) dy , where ([ a n , b n ]) N n = N is a collection of disjoint intervals such that: a = a n increases with n ; a n < b n for each n ; and only a finite number of intervals intersect any compact set. Althoughthis description is a little cumbersome, it is useful to introduce notation for the points ( a n ) N n = N , LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 6 which correspond to the local maxima of S , and ( b n ) N n = N , which correspond to the local minimaof S ; we will sometimes write a n = a n ( S ) and b n = b n ( S ) when we wish to stress which path isbeing considered. We moreover note that if S is the path encoding of (( Q n ) N n = N , ( E n ) N − n = N ) , then [ a n , b n ] is the interval O n , as defined at (1.6). We further define C T N , N : = C T ∩ C N , N , which corresponds to the set of path encodings given by S T N , N : = S T ∩ S N , N . The key step towards showing that the path transformation S T S yields the dynamics of theultra-discrete Toda lattice is the following lemma. Lemma 2.1.
Let N , N ∈ Z ∪ {− ∞ } ∪ { ∞ } be such that N ≤ ≤ N . For any S ∈ S T N , N , itholds that T S ∈ S N , N . Moreover, the elements of ( a n ( T S ) , b n ( T S )) N n = N are given bya n ( T S ) = b n ( S ) − b ( S ) , b n ( T S ) = min (cid:8) b n ( S ) + M b n ( S ) − S b n ( S ) , a n + ( S ) (cid:9) − b ( S ) , where we define a N + : = ∞ in the case N < ∞ .Proof. In the proof, we will denote a n ( S ) , b n ( S ) by a n , b n for simplicity. For S ∈ S T N , N , it holdsthat(2.1) M x = M a n for a n ≤ x ≤ b n for each n . It is also the case that, for each n < N , there exists a unique c n ∈ ( b n , a n + ] such that(2.2) M x = (cid:26) M b n = M a n , for b n ≤ x ≤ c n , S x , for c n ≤ x < a n + . In the case N < ∞ , the previous claim is also true for n = N in the sense that there exists aunique c n ∈ ( b n , ∞ ) such that (2.2) holds with a N + = ∞ . Finally, if N > − ∞ , observe that(2.3) M x = x + S a N − a N for x ≤ a N . From (2.1), (2.2) and (2.3), we obtain that
T S x − T S y = M a n − S x − M − ( M a n − S y + M ) = S y − S x = x − y for a n ≤ x , y ≤ b n , T S x − T S y = M a n − S x − M − ( M a n − S y + M ) = S y − S x = y − x for b n ≤ x , y ≤ c n , and T S x − T S y = S x − S x − M − ( S y − S y + M ) = S x − S y = x − y for c n ≤ x , y < a n + , for each n , where we again interpret a N + = ∞ in the case N < ∞ . Moreover, in the case when N > − ∞ , we have that T S x − T S y = ( x + S a N − a N ) − S x − M − ( ( y + S a N − a N ) − S y − M )= x − S x − y + S y = x − y for x , y ≤ a N . LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 7
In particular,
T S is a continuous function of gradient − ([ b n , c n ]) N n = N . It follows from this description that τ ( T S ) = b , and so we obtain T S is a functionin S N , N with [ a n ( T S ) , b n ( T S )] = [ b n − b , c n − b ] for each n .To complete the proof, it remains to check that c n = min { b n + M b n − S b n , a n + } , with a N + = ∞ in the case N < ∞ . For n < N , we have that S a n + = S b n + a n + − b n , and somin { b n + M b n − S b n , a n + } = min (cid:8) b n + M b n − S a n + + a n + − b n , a n + (cid:9) = a n + + min { M b n − S a n + , } . Now, if M b n ≥ S a n + , then the above expression is equal to a n + . By the definition of c n at (2.2),it is clear that c n = a n + in this case as well. On the other hand, if M b n < S a n + , then the aboveexpression is equal to a n + + M b n − S a n + . Moreover, we also have that a n + − c n = S a n + − S c n = S a n + − M b n , i.e. c n = a n + + M b n − S a n + , as desired. Finally, when N < ∞ , the argument for n = N is similar. (cid:3) Finite configurations.
We are now ready to prove Theorem 1.1, which we recall concernsthe dynamics of the ultra-discrete Toda lattice started from a finite initial configuration.
Proof of Theorem 1.1.
Let N ∈ N , (( Q n ) Nn = , ( E n ) N − n = ) ∈ ( , ∞ ) N − , and S be the correspondingpath encoding, which is clearly an element of S T , N . In particular, this allows us to apply Lemma2.1 to deduce that T S ∈ S , N , and it remains to prove that ( T Q ) n = b n ( T S ) − a n ( T S ) , ( T E ) n = a n + ( T S ) − b n ( T S ) . Denoting a n ( S ) , b n ( S ) by a n , b n for simplicity, Lemma 2.1 further implies that the above equa-tions are equivalent to ( T Q ) n = c n − b n , (2.4) ( T E ) n = b n + − c n , (2.5)where c n : = min { b n + M b n − S b n , a n + } . Since the dynamics of the ultra-discrete Toda lattice aregiven by ( T Q ) n = min ( n ∑ k = Q k − n − ∑ k = ( T Q ) k , E n ) = min ( b n + n ∑ k = Q k − n − ∑ k = ( T Q ) k , a n + ) − b n , ( T E ) n = Q n + + E n − ( T Q ) n = b n + − b n − ( T Q ) n , we only need to prove that n ∑ k = Q k − n − ∑ k = ( T Q ) k = M b n − S b n (2.6)for each n = , , . . . , N . We will do this by induction. For n =
1, (2.6) holds since ∑ k = Q k − ∑ k = ( T Q ) k = Q = b − a = b = − ( − b ) = M b − S b . LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 8
Next, suppose that (2.6) holds for some n ≤ N −
1, and note that this implies that we also have(2.4) and (2.5) for indices up to this value. In particular, since (2.4) also holds for n , n + ∑ k = Q k − n ∑ k = ( T Q ) k = M b n − S b n + Q n + − ( T Q ) n = M b n − S b n + b n + − a n + − c n + b n . Hence, to prove (2.6) for n +
1, it is enough to show that M b n + − S b n + = M b n − S b n + b n + − a n + − c n + b n . Since S b n + − S b n = ( a n + − b n ) − ( b n + − a n + ) , this is equivalent to showing M b n + − M b n = a n + − c n . If S a n + ≤ M b n , then M b n + = M b n and a n + = c n holds (cf. the proof of Lemma 2.1). Also, if S a n + > M b n , then M b n + − M b n = S a n + − M b n = S a n + − S c n = a n + − c n . So, the desired relationholds. (cid:3) To complete this subsection, we show, as a simple corollary of Theorem 1.1, that T preservesmass. Corollary 2.2.
Let N ∈ N and (( Q n ) Nn = , ( E n ) N − n = ) ∈ ( , ∞ ) N − . It then holds that N ∑ n = Q n = N ∑ n = ( T Q ) n . Proof.
For x ≥ b N , it holds that(2.7) S x = x − N ∑ n = ( b n − a n ) = x − N ∑ n = Q n , where S is the path encoding of (( Q n ) Nn = , ( E n ) N − n = ) . Moreover, for x ≥ c N , as defined in theproof of Lemma 2.1, it holds that S x = M x . In particular, for x ≥ c N , we have ( T S ) x = S x . By thedefinition of T S , it follows that, for x ≥ c N − b , ( T S ) x = ( T S ) x + b − ( T S ) b = S x + b − b = x − N ∑ n = Q n . Since we also have, analogously to (2.7), that, for x ≥ b N ( T S ) = c N − b , ( T S ) x = x − N ∑ n = ( T Q ) n , the result follows. (cid:3) Periodic configurations.
In this subsection, we introduce the path encoding of the ultra-discrete periodic Toda lattice. For this model, which was first presented in [6], we describe thecurrent state by a vector of the form (( Q n ) Nn = , ( E n ) Nn = ) ∈ ( , ∞ ) N for some N ∈ N . Althoughit appears we have an extra variable to the non-periodic finite configuration case, this is notso, because we assume that L = ∑ Nn = Q n + ∑ Nn = E n for some fixed L >
0. Moreover, in orderto define the dynamics, we suppose that ∑ Nn = Q n < L . The collection of such configurationswill be denoted by C Per ( L ) , N , and we further associate with (( Q n ) Nn = , ( E n ) Nn = ) ∈ C Per ( L ) , N an element LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 9 F IGURE
3. First 25 configurations for an example realisation of the periodicultra-discrete Toda lattice; the top part of the figure shows the spatial positionof the configuration, as determined by the unshifted Pitman’s transformation,and the bottom part of the figure shows the configuration shifted so that theinterval of length Q (shown in red in both depictions) is placed first. (( Q n ) n ∈ Z , ( E n ) n ∈ Z ) ∈ C T by extending the sequences ( Q n ) Nn = and ( E n ) Nn = periodically. Intro-ducing the additional notation ( D n ) Nn = for convenience, the dynamics of the system are givenby the following adaptation of (1.2): ( T Q ) n : = min { Q n − D n , E n } , ( T E ) n : = E n + Q n + − ( T Q ) n , D n : = min ≤ k ≤ N − k ∑ ℓ = ( E n − ℓ − Q n − ℓ ) . (2.8)(In this definition, we make use of the periodic extension of (( Q n ) Nn = , ( E n ) Nn = ) .) Given a statevector (( Q n ) Nn = , ( E n ) Nn = ) ∈ C Per ( L ) , N , we define an associated path encoding S to be the elementof S T associated with its periodic extension (( Q n ) n ∈ Z , ( E n ) n ∈ Z ) . We then have the followingadaptation of Theorem 1.1 to the periodic case. See Figure 3 for an example realisation of thedynamics, as described by the unshifted Pitman’s transformation, and in the original coordinates. Theorem 2.3.
Let L > , N ∈ N , (( Q n ) Nn = , ( E n ) Nn = ) ∈ C Per ( L ) , N , and S be the associated path en-coding, as described above. It then holds that ((( T Q ) n ) Nn = , (( T E ) n ) Nn = ) , as given by (2.8) , haspath encoding T S, as defined by (1.9) . Moreover, it holds that ((( T Q ) n ) Nn = , (( T E ) n ) Nn = ) ∈ C Per ( L ) , N . LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 10
Proof.
In the proof, we again denote a n ( S ) , b n ( S ) by a n , b n for simplicity. Since Lemma 2.1yields that T S ∈ S , to establish the first claim of the theorem we only need to prove that ( T Q ) n = b n ( T S ) − a n ( T S ) , ( T E ) n = a n + ( T S ) − b n ( T S ) . By the definition of T S and Lemma 2.1, this is equivalent to ( T Q ) n = c n − b n , ( T E ) n = b n + − c n , where c n : = min { b n + M b n − S b n , a n + } . Since the dynamics of the ultra-discrete periodic Todalattice is given by ( T Q ) n = min { Q n − D n , E n } = min { b n − a n − D n , a n + } − b n , ( T E ) n = E n + Q n + − ( T Q ) n = b n + − b n − ( T Q ) n , D n = min ≤ k ≤ N − k ∑ ℓ = ( E n − ℓ − Q n − ℓ ) , it will be sufficient to prove that b n − a n − D n = M b n − S b n (2.9)for n ∈ Z . Now, since we assume that ∑ Nn = Q n < L , it holds that, for any x ∈ R ,(2.10) S x + L − S x = N ∑ n = E n − N ∑ n = Q n > . Moreover, since { a n : n ∈ N } is the set of local maxima of S , we have M b n = max m ≤ n S a m = max n − N + ≤ m ≤ n S a m . On the other hand, for any n , k , k ∑ ℓ = ( E n − ℓ − Q n − ℓ ) = k ∑ ℓ = { ( S a n − ℓ + − S b n − ℓ ) + ( S b n − ℓ − S a n − ℓ ) } = k ∑ ℓ = ( S a n − ℓ + − S a n − ℓ )= S a n − S a n − k . Hence we have b n − a n − D n = − S b n + S a n − min ≤ k ≤ N − ( S a n − S a n − k )= − S b n + max ≤ k ≤ N − S a n − k = − S a n + max n − N + ≤ m ≤ n S a m , which establishes (2.9). Finally, to prove the claim that ((( T Q ) n ) Nn = , (( T E ) n ) Nn = ) ∈ C Per ( L ) , N ,we simply observe from Lemma 2.1 that(2.11) a N + ( T S ) = b N + − b = L . (cid:3) LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 11
To conclude this subsection, we demonstrate the periodic version of the mass preservationresult of Corollary 2.2.
Corollary 2.4.
Let L > , N ∈ N and (( Q n ) Nn = , ( E n ) Nn = ) ∈ C Per ( L ) , N . It then holds that N ∑ n = Q n = N ∑ n = ( T Q ) n . Proof.
By (2.10), there exists an n ∈ , . . . , N such that S a n = M a n . Hence L − N ∑ n = Q n = S a n + N − S a n = ( T S ) a n + N − ( T S ) a n = ( T S ) a n + L − b − ( T S ) a n − b , where b = b ( S ) . Applying the fact that the increments of T S are L -periodic and also (2.11),we further have that ( T S ) a n + L − b − ( T S ) a n − b = ( T S ) b N + − b − ( T S ) b N + − b − L = L − N ∑ n = ( T Q ) n , from which the result follows. (cid:3)
3. R
ELATION TO THE BOX - BALL SYSTEM ON R The BBS on R was introduced in [1] as a means to describing scaling limits of discrete modelsin a high density regime. Specifically, with states of the system being described by continuousfunctions S : R → R with S = M = sup x ≤ S x < ∞ , the collection of which we will denote S T , its dynamics were defined in [1] to be given by Pitman’s transformation, as at (1.8). In thissection, we will show that the link between the dynamics of the ultra-discrete Toda lattice andthe BBS can be extended beyond the simple case discussed in the previous section.We will denote the subset of C ( R , R ) of interest in this section by S N , N , where ( N , N ) =( − ∞ , ∞ ) , ( , ∞ ) , ( − ∞ , ) or ( , N ) for some N ∈ N . NB. This notation conflicts with that usedin the previous section, however the set we now introduce includes the former definition, andso there should not be confusion. Specifically, S N , N will be the class of S ∈ C ( R , R ) with S = ( a n ) N n = N , local minima at ( b n ) N n = N and is otherwise strictlyincreasing/decreasing, where a n < b n < a n + and lim n →± ∞ a n = ± ∞ if N or N is not finite (cf.(1.5)). In the case ( N , N ) = ( − ∞ , ∞ ) , we choose the indices so that a < ≤ a . For S ∈ S N , N ,we let Q n = | S b n − S a n | for N ≤ n ≤ N , E n = | S a n + − S b n | for N ≤ n ≤ N −
1, and k S k be thetotal variation of S , namely k S k x = Z x | dS y | , which by assumption is well-defined and finite for each x ∈ R . Clearly, a path S ∈ S N , N is inone-to-one correspondence with the set of data(3.1) (cid:16) a , (( Q n ) N n = N , ( E n ) N − n = N ) , k S k (cid:17) . Moreover, if S ∈ S TN , N : = S N , N ∩ S T , then it is not difficult to check that T S ∈ S N , N forthe same choice of ( N , N ) . Hence, to analyze the dynamics of BBS for functions in S TN , N , weonly need to study the dynamics of the data at (3.1) under T . In the following proposition, weshow that these dynamics are essentially determined by the ultra-discrete Toda lattice equation. LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 12
Hence, since the ultra-discrete Toda lattice equation is integrable, the corresponding initial valueproblem of BBS on R restricted to elements of S TN , N can be solved explicitly. Proposition 3.1. (i) Suppose ( N , N ) = ( , ∞ ) , ( − ∞ , ) or ( , N ) for some N ∈ N . Then,a ( T S ) = b ( S ) , k T S k = k S k and the dynamics of (( Q n ) N n = N , ( E n ) N − n = N ) is given by the ultra-discrete Toda lattice equation (1.2) . Namely, Q n ( T S ) = T Q n and E n ( T S ) = T E n .(ii) Suppose ( N , N ) = ( − ∞ , ∞ ) . If b ( S ) < , thena ( T S ) = b ( S ) , k T S k = k S k and the dynamics of (( Q n ) ∞ n = − ∞ , ( E n ) ∞ n = − ∞ ) is given by the ultra-discrete Toda lattice equation (1.2) . Namely, Q n ( T S ) = T Q n and E n ( T S ) = T E n . If b ( S ) ≥ , thena ( T S ) = b ( S ) , k T S k = k S k and the dynamics of (( Q n ) ∞ n = − ∞ , ( E n ) ∞ n = − ∞ ) is given by the ultra-discrete Toda lattice equation (1.2) with a spatial shift. Namely, Q n ( T S ) = T Q n − and E n ( T S ) = T E n − .Proof. (i) We will prove the result in the case ( N , N ) = ( , N ) for some N ∈ N ; the remainingcases are dealt with similarly. First, by assumption S is strictly increasing on ( − ∞ , a ( S )) and ( b ( S ) , a ( S )) (where we interpret a ( S ) = ∞ if N = ( a ( S ) , b ( S )) .It readily follows from the definition of Pitman’s transformation at (1.8) that T S is strictly in-creasing on ( − ∞ , b ( S )) , and strictly decreasing on ( b ( S ) , b ( S ) + ε ) for some ε >
0. Hence a ( T S ) = b ( S ) , as claimed. Second, note that on the set { x : S x = M x } (which consists of a finitecollection of closed intervals), we have dT S x = dS x . Similarly, we have on { x : S x < M x } (whichconsists of a finite collection of open intervals) that dT S x = − dS x . Since the set { x : S x > M x } isempty, it follows that k T S k = k S k . Thus it remains to check the claim involving the ultra-discreteToda lattice equation. To this end, we observe that Q n ( T S ) + E n ( T S ) =
T S a n ( TS ) + T S a n + ( T S ) − T S b n ( TS ) = T S b n ( S ) + T S b n + ( S ) − (cid:8) M a n ( S ) , M a n ( S ) − S a n + ( S ) (cid:9) = M a n ( S ) − S b n ( S ) + M a n + ( S ) − S b n + ( S ) − (cid:8) M a n ( S ) , M a n ( S ) − S a n + ( S ) (cid:9) = Q n + ( S ) + E n ( S ) + (cid:8) M a n + ( S ) − S a n + ( S ) , M a n + ( S ) − M a n ( S ) (cid:9) = Q n + ( S ) + E n ( S ) (3.2)for n = , , . . . , N −
1, where we have used that a n ( T S ) = b n ( S ) (which is checked similarlyto the case when n =
1, as described above),
T S b n ( TS ) = max { M a n ( S ) , M a n ( S ) − S a n + ( S ) } − M (which follows from (1.8)), and applied that − S b n ( S ) = E n ( S ) − S a n + ( S ) and − S b n + ( S ) = Q n + − S a n + ( S ) . Moreover, Q n ( T S ) =
T S a n ( TS ) − T S b n ( TS ) = T S b n ( S ) − max (cid:8) M a n ( S ) , M a n ( S ) − S a n + ( S ) (cid:9) = M a n ( S ) − S b n ( S ) − max (cid:8) M a n ( S ) , M a n ( S ) − S a n + ( S ) (cid:9) = min (cid:8) M a n ( S ) − S b n ( S ) , E n ( S ) (cid:9) , (3.3) LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 13 for n = , , . . . , N , where we interpret S a N + ( S ) = ∞ . We next claim that(3.4) M a n ( S ) − S b n ( S ) = n ∑ k = Q k ( S ) − n − ∑ k = Q k ( T S ) , which, similarly to the proof of Theorem 1.1, we prove by induction. The result is obvious bydefinition for n =
1. Next, supposing 1 ≤ n ≤ N −
1, we need to show that(3.5) M a n + ( S ) − S b n + ( S ) = M a n ( S ) − S b n ( S ) + Q n + ( S ) − Q n ( T S ) . The right-hand side here is given by M a n ( S ) − S b n ( S ) + S a n + ( S ) − S b n + ( S ) − min (cid:8) M a n ( S ) − S b n ( S ) , S a n + ( S ) − S b n ( S ) (cid:9) = max (cid:8) S a n + ( S ) , M a n ( S ) (cid:9) − S b n + ( S ) = M a n + ( S ) − S b n + ( S ) , thus establishing (3.5), and consequently (3.4). Returning to (3.3), this yields in turn that Q n ( T S ) = min ( n ∑ k = Q k ( S ) − n − ∑ k = Q k ( T S ) , E n ( S ) ) . Together with (3.2), this establishes that the time evolution of the system is given by ultra-discrete Toda lattice equation (1.2), as desired.(ii) This is proved similarly to (i), with the only difference being that care is needed about theindices depending on whether b ( S ) is < ≥ (cid:3) A CKNOWLEDGEMENTS
DC would like to acknowledge the support of his JSPS Grant-in-Aid for Research ActivityStart-up, 18H05832, MS would like to acknowledge the support of her JSPS Grant-in-Aid forScientific Research (B), 16KT0021, and ST would like to acknowledge the support of his JSPSGrant-in-Aid for Challenging Exploratory Research, 16KT0021.R
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LTRA-DISCRETE TODA LATTICE AND PITMAN’S TRANSFORMATION 14 [10] J. W. Pitman,
One-dimensional Brownian motion and the three-dimensional Bessel process , Advances in Appl.Probability (1975), no. 3, 511–526.[11] D. Takahashi and J. Satsuma, A soliton cellular automaton , J. Phys. Soc. Japan (1990), 3514–3519.R ESEARCH I NSTITUTE FOR M ATHEMATICAL S CIENCES , K
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