The Ghost-Box-Ball System: A Unified Perspective on Soliton Cellular Automata, the RSK Algorithm and Phase Shifts
aa r X i v : . [ n li n . S I] F e b The Ghost-Box-Ball System: A Unified Perspective onSoliton Cellular Automata, the RSK Algorithm andPhase Shifts
Nicholas M. Ercolani , Jonathan Ramalheira-Tsu Abstract
In this paper, we introduce the ghost-box-ball system, which is an extendedversion of the classical soliton cellular automaton. It is initially motivated as amechanism for making precise a connection between the Schensted insertion (ofthe Robinson-Schensted-Knuth correspondence) and the dynamical process ofthe box-ball system. In addition to this motivation, we explore generalisationsof classical notions of the box-ball system, including the solitonic phenomenon,the asymptotic sorting property, and the invariant shape construction.We analyse the ghost-box-ball system beyond its initial relevance to theRobinson-Schensted-Knuth correspondence, unpacking its relationship to its un-derlying dynamical evolution on a coordinatisation and using a mechanism foraugmenting a regular box-ball configuration to study the classical ultradiscretephase shift phenomenon.
Keywords: box-ball system, RSK correspondence, cellular automata, soliton,ultradiscretization, phase shift
Contents Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089([email protected]). Supported by NSF grant DMS-1615921. Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089([email protected]). Supported by NSF grant DMS-1615921. .1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Box-Ball System and the Robinson-Schensted-Knuth Corre-spondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Box-Ball Systems . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 The Box-Ball Evolution . . . . . . . . . . . . . . . . . . . 41.2.3 The Robinson-Schensted-Knuth Correspondence . . . . . 51.3 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . 6
Intrinsic Ghost-Box-Ball Dynamics and the Phase Shift 46
1. Introduction
The analogy between computational algorithms and dynamical systems isa natural one that received a concrete realization in Conway’s seminal workon cellular automata (CA) [Ga70], [Wo02]. Within this setting it is perhapsnot surprising that, in time, CA corresponding to integrable dynamical systemswould be identified; and, indeed, this was realized in the pioneering work ofTakahashi and Satsuma [TS90] [TTS96] on so-called box-ball systems (BBS)identifying the striking solitary wave character of these systems. Over the pastthree decades there has been an explosion of interest in this system and its manyvariations. For more on this we refer the reader to the very good survey papers[To04] and [IKT12]. A most remarkable outgrowth of this circle of ideas hasbeen the discovery of deep connections with combinatorics and representationtheory. One of the most notable examples of this is the analogy between box-ball systems and the Robinson-Schensted-Knuth (RSK) algorithm, one of thefundamental combinatorial tools of modern representation theory, [Ki00] and[NY04]. The focus of this paper is to make this analogy more precise throughan extension to what we refer to as ghost box-ball systems (GBBS). A strongmotivation for doing this comes from recent developments related to integrablestochastic processes ([O13], [BBO09]) motivated, in turn, by integrable systemsapproaches to random matrix theory. 3 .2. The Box-Ball System and theRobinson-Schensted-Knuth Correspondence
In order to provide an overview of the results in this paper we first give, inthis subsection, a brief summary of BBS and the RSK algorithm.
A cellular automaton is a special type of discrete dynamical system withboth discrete time steps and a discrete (in fact finite) number of states. Ofparticular interest is the box-ball system (BBS) which was introduced in 1990by Takahashi and Satsuma [TS90].
Definition 1.1.
The (basic) box-ball system consists of a one-dimensional in-finite array of boxes with a finite number of the boxes filled with balls, and nomore than one ball in each box (see, for example, Figure 1).More formally, the phase space of this system, which we denote by BBS, can beidentified with the space of binary sequences { , } Z , with all but finitely many entries equal to zero, so that ’s correspond to filledboxes and ’s to empty boxes. · · ·· · · Figure 1: A Box-Ball State
A simple evolution rule is provided for the box-ball dynamics:(1) Take the left-most ball that has not been moved and move it to the left-mostempty box to its right.(2) Repeat (1) until all balls have been moved precisely once.Since the algorithm requires one to know which balls have been moved, we can,without technically changing the algorithm, introduce a colour-coding based onwhether balls have moved or not. Balls will be blue until they have moved, after4hich they will become red. When all balls are red, the colours should be resetto blue, ready for the next time step. Or, equivalently, a 0-th step of colouringall balls blue should be prescribed. We will use the latter for a minor benefitin brevity. Below is an example of the evolution with this colour-coding, witheach ball move separated into a sub-step: · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · ·
Figure 2: A box-ball system time evolution (one time step).
RSK is an algorithm for the direct sum decomposition of tensor products ofrepresentations of the unitary groups U ( k ). This general area is referred to asSchur-Weyl theory [Ku02]. It has important applications for quantum-many-body theory and quantum field theory.The fundamental building block of RSK is Schensted insertion . Its descriptionis a bit involved (and will be more fully developed in Section 3) but for ourintroductory purposes here it will suffice to point out that Schensted insertioncan be reduced to a discrete evolution on R := [ n ∈ N R n , where R n := N n × N n and N := N ∪{ } . The foundation for Schensted insertionis a prescription for taking an input pair of sequences( a , x ) = (( a , . . . , a n ) , ( x , . . . , x n )) , { , . . . , n } , and transforming the input sequences into an output pairof sequences ( b , y ) = (( b , . . . , b n ) , ( y , . . . , y n ))which encode the result of performing Schensted insertion. This encoding isintroduced in full detail in Section 3.1.1.This dynamic evolution on R (pairs of n -tuples) is what, going forward, we willrefer to as RSK : R → R . (1.1) There is a natural connection between BBS and RSK given by a processknown as tropicalization (or Maslov dequantization) that will be described inSection 3.2. However, this connection is not a precise correspondence. Themain point of this paper is to address that problem. We do this by means ofintroducing a (ghost) background, or environment, against which the basic box-ball system moves and interacts. Once this is developed, our main result mayindeed be summarized as showing that the following diagram (from Figure 14)commutes: GBBS GBBS B B R R ˆ ̺ RSK C − Cφ RSK → BBS φ BBS → RSK ̺ is the dy-namics for our extended ghost-box-ball system , which is constructed in Section4. Finally, B represents the precise coordinatisation linking the dynamics ofGBBS to that of the algorithm of RSK. This is developed in Section 4.2.In fact, we will see that much more is true: our construction yields a detailedstage-by-stage correspondence between the fine structure of the respective sys-tems.Prior to this work, a relationship between RSK and an advanced version ofthe box-ball system was developed by Fukuda [Fu04]. However, this advancedbox-ball system requires various extra features not automatically possessed bythe dequantisation mentioned above. As such, our system is simpler, and thissimplicity may open the way to connections with continuous space - continuoustime systems that have a deeper relation to Lie theory and classical solitonicstructures. Some discussion of that will be presented in Sections 5 - 6. We willalso discuss compare our work with that of [Fu04] in Section 4.7.In the remainder of this paper, Sections 2 - 3 provide the detailed background forthe main ingredients we have just surveyed. The precise statement of our resultsand their proofs is carried out in Section 4. Natural extensions of those resultsare presented in Section 5, where we discuss the intrinsic dynamics of the ghost-box-ball system, as well as its phase shift phenomenon. Further motivation forwhat has been done here is presented in Section 6.
2. The Box-Ball System
The box-ball system is sometimes referred to as a soliton cellular automaton. Toappreciate this reference, we think of an entire box-ball configuration as beingthe soliton. In the classical setting, a soliton is thought of as being composed7f masses (or pulses) that are nonlinearly related. In the box-ball setting, a“mass” corresponds to a consecutive sequence of balls. One may observe (seebelow) that such a block travels with velocity equal to the number of balls in it,so that larger blocks have velocity greater than smaller blocks. As with classicalsoliton masses, during the course of its evolution, the blocks may collide, andtemporarily change their sizes. However, asymptotically in both forward andbackward (discrete) time ( t ), the sizes of blocks comprising the soliton are thesame. We will therefore refer to such a configuration as an N -soliton , if thetotal number of blocks asymptotically is N .After blocks collide, they come out of the collision ordered with the longer blocksahead of smaller blocks, but having a phase shift due to the nonlinearity. Byphase shift here, we mean the difference between where the block ends up afterthe collision and where the block would have been if it were not for the colli-sion. For now, we take this for granted. A more detailed analysis will be givenin Section 5.2.1, along with a conjectured, explicit formula for the phase shift.In the following figure, we illustrate how the blocks become ordered after suf-ficiently many time evolutions. Once sorted, they travel with their respectivevelocities, never to collide again. · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · Figure 3: The sorting property of the box-ball system.
Invariants of the box-ball system may be expressed in terms of combinatorialstructures known as
Young diagrams . Definition 2.1.
Let n ∈ N and let λ = ( λ , λ , . . . , λ k ) be a partition of n , i.e. λ i ∈ N for each i , and λ ≥ λ ≥ · · · ≥ λ k and P λ i = n . Associated to λ is the oung diagram (or Ferrers diagram) of shape λ which is composed of λ boxesin the first row, λ boxes in the second row, . . . , and λ k boxes in the k -th row.The boxes are of equal size and aligned in a grid, justified to the left. Example 2.2.
The partition λ = (5 , , , , , which partitions n = 14 , hasYoung diagram: A Young diagram for a box-ball state can be described in terms of balls andboxes, but it is slightly more convenient to represent the box-ball system as asequence of 0’s (for empty boxes) and 1’s (for boxes with balls), which is madeexplicit in [TTS96] and [To04]. The construction goes as follows: • Let p be the number of 10’s in the sequence. • Eliminate all of these 10’s, and let p be the number of 10’s in the resultingsequence. • Repeat this process until no 10’s remain. • The sequence ( p , p , . . . ) is weakly decreasing, hence a partition of thenumber of balls. It can be represented by a Young diagram by taking the j th column to have p j boxes.For example, representing the first box-ball system in Figure 3 as a sequence of1’s and 0’s, as described in Definition 1.1, we have · · · · · · In the above, there are four instances of 10’s in the sequence, so p = 4. Re-moving those instances of 10’s, we obtain · · · · · ·
9o see that p = 2. Next, we have · · · · · · there is only one 10 here, so p = 1. The last removal of the 10 above yields asequence with no 1’s: · · · · · · Thus, the process terminates and we have the sequence (4 , , Figure 4: The invariant shape of the box-ball system(s) in Figure 3
Theorem 2.3. [TTS96] The sequence of p i ’s are time invariants of the box-ball evolution. Equivalently, the Young diagram is invariant under the box-ballevolution. A proof of Theorem 2.3 is given in Section 5 of [TTS96], using combinatorialtechniques.
Remark 2.4.
The row lengths then act as a signature for the system: if thecolumn lengths of the Young diagram are the p i ’s, then the row lengths give theasymptotic lengths of the blocks. One can see this heuristically by noting thatas time goes to ±∞ , the blocks will be sufficiently separated by empty boxes sothat each block provides precisely one “ ” for each particle comprising it. Suppose at time t , one has N blocks in the soliton. Let Q t , Q t , . . . , Q tN de-note the lengths of these blocks, taken from left to right. Let W t , W t , . . . , W tN − denote the lengths of the sets of empty boxes between the N sets of filled boxes,again taken from left to right. Lastly, let W t and W tN be formally defined to be ∞ , reflecting the fact that the empty boxes continue infinitely in both directions.The following theorem gives evolution equations for these coordinates. Theycan be found, for example, in [To04]. 10 heorem 2.5. ([To04]) The coordinates ( W t , Q t , W t , . . . , Q tN , W tN ) evolve un-der the box ball dynamics according to W t +10 = W t +1 N = ∞ (2.1) W t +1 i = Q ti +1 + W ti − Q t +1 i , i = 1 , . . . , N − Q t +1 i = min W ti , i X j =1 Q tj − i − X j =1 Q t +1 j , i = 1 , . . . , N, (2.3) Remark 2.6.
By Theorem 2.3, p is invariant under the box-ball evolution.Since p is the number of blocks of a box-ball state, it follows that the number ofblocks is invariant under this evolution. Pairing this with Theorem 2.5, we seeeach W ti > for each i and for all time. Furthermore, since there are always N blocks, each Q ti > (by definition of a block). Example 2.7.
Take the initial state in Figure 2: · · ·· · · W t Q t W t Q t W t Q t W t Q t W t · · ·· · · W t +10 Q t +11 W t +11 Q t +12 W t +12 Q t +13 W t +13 Q t +14 W t +14 Figure 5: The box-ball coordinates on a box-ball system and its time evolution.
Under the time evolution, the coordinates evolve as ( ∞ , , , , , , , , ∞ ) ( ∞ , , , , , , , , ∞ ) (2.4)We now introduce some fundamental definitions, which are key to the way wewill later extend the classical box-ball system, that serve to distinguish betweenthe box-ball evolution and the induced coordinate evolution. Definition 2.8.
The BBS phase space of n -soliton states is coordinatised bysequences of the form ( ∞ , Q , W , . . . , W n − , Q n , ∞ ) ∈ {∞} × N n − × {∞} =: B n . The full BBS phase space is coordinatised by B := [ n ∈ N B n . Let C : BBS → B denote the map taking a box-ball system state to its coor-dinates. We further define ̺ : BBS → BBS to be the box-ball evolution and χ : B → B to be the corresponding evolution on coordinates given in Theorem2.5.
Corollary 2.9.
The following diagram commutes
BBS BBS
B B ̺C χ C .
3. RSK and gRSK
In this section, we provide the background and some basic motivation behindthe Robinson-Schensted-Knuth correspondence and Schensted insertion. Webegin with a review of some of the combinatorial objects of interest, the RSKequations describing Schensted word insertion, and Kirillov’s geometric lifting ofthe (tropical) RSK equations to the geometric RSK (gRSK) equations. We willbe following the papers [AD99] by Aldous and Diaconis and [NY04] by Noumiand Yamada, and the book [Ai07] by Aigner.The coverage of this background is fairly in-depth, with examples provided toaid in following the rather algorithmic constructions presented here. However,what is most pertinent to this paper are the equations for Schensted insertionwhich are described in Corollary 3.12.
We describe in this section, following the treatment and notation given by[NY04], the process known as Schensted insertion, but only to the extent neededfor understanding the RSK equations: Schensted insertion into a word . Definition 3.1.
A (weakly increasing) word in an alphabet { , , . . . , n } is asequence l , l , . . . , l k with l i ∈ { , , . . . , n } for each i and for which l i ≤ l i +1 for each ≤ i < k . emark 3.2. For convenience of expression, when it is appropriate to do soand when it creates no confusion, we will drop the commas when writing words.So, for example, the word , , , , , , , will simply be written as .Additionally, all words we work with will be weakly increasing, so that is what“word” will be taken to mean from now on. The process of Schensted word insertion will be an evolution on pairs of words:one will start with an initial word , into which another word (the insertion word )will be inserted. This process will result in a new word and a byproduct of theinsertion, in the form of a bumped word . The following diagram is typical forrepresenting this structure:Initial Word Insertion Word New WordBumped Word
Figure 6: Pictorial representation of Schensted insertion of a word into a word.
Notation 3.3.
Let n ∈ N be fixed (giving the bound on the alphabet for thewords). Each word in this alphabet can be represented by (and identified with)an n -tuple in N n , where N = N ∪ { } , by letting the i -th entry be the number ofinstances of i in the word. Since words are weakly increasing, there is no loss ofinformation from doing this. For example, with n = 5 , the 5-tuple (3 , , , , represents the word w = 11122444 = 1 .For the words involved in Schensted insertion, we adopt this notation as follows: • x = ( x , . . . , x n ) : n -tuple for the initial word, • a = ( a , . . . , a n ) : n -tuple for the insertion word, • y = ( y , . . . , y n ) : n -tuple for the new word, • b = ( b , . . . , b n ) : n -tuple for the bumped word. We begin by prescribing the rules for inserting a letter (a number) into a word.Word insertion is then the iterative application of letter insertion.
Definition 3.4.
The process of
Schensted insertion of a letter (number)into a word is given as follows. To insert a number a ∈ { , . . . , n } into a word w = 1 w w · · · n w n :1. Look for the left-most number in w that is strictly greater than a , if itexists, then replace that number by a . The replaced number, which hasbeen removed from the word, is now bumped. . If no such number exists in w , then append a to the end of w . Remark 3.5.
In performing this procedure, the inserted number, a , will alwaysfind a place in the word, whilst keeping the word weakly increasing. It will eitherdo so by bumping something out of the word or it will do so by growing the word. Example 3.6.
Here are some examples with the initial word given by x =1112234 :1. Inserting creates a new word of , bumping out .2. Inserting creates a new word of , bumping out
3. Inserting creates a new word of , bumping nothing.4. Inserting creates a new word of , bumping nothing. Definition 3.7.
The process of
Schensted insertion of a word into an-other word is defined as follows. Perform letter insertion (Definition 3.4)iteratively, reading in the insertion word from left to right.
Remark 3.8.
This process does indeed produce a new pair of weakly increasingwords:1. Since the word being inserted is weakly increasing, so too must the se-quence of bumped letters when concatenated. This concatenation formsthe bumped word.2. By the nature of letter insertion, the new word is always weakly increasing.
Example 3.9.
We insert the word into :1. We insert into , resulting in (bumping ),2. we insert into , resulting in (bumping ),3. we insert into , resulting in (bumping ),4. we insert into , resulting in (bumping ),5. we insert into , resulting in (bumping ),6. we insert into , resulting in (nothing is bumped).The new word is then and the bumped word is . Or, using thepictorial representation in Figure 6, we summarise this insertion as (2,0,0,2,3) (3,0,2,1,0) (5,0,2,1,0)(0,0,0,2,3)14 .1.2. Explicit Formulæ for the RSK Dynamics Write x = 1 x x · · · n x n for the word into which we wish to insert the word a = 1 a a · · · n a n , writing y = 1 y y · · · n y n for the result, and write b =1 b b · · · n b n . Thus, we want to know how ( y , . . . , y n ) and ( b , . . . , b n ) arisefrom ( x , . . . , x n ) and ( a , . . . , a n ).In order to simplify the calculations, we introduce new variables by taking partialsums: ξ j = x + · · · + x j , η j = y + · · · + y j (3.1)for j = 1 , . . . , n . These will serve to reformulate the information in a “max-plus”form (Lemma 3.10), which will be crucial for detropicalisation, discussed in thesubsequent section.The y ’s can then be recovered from the η ’s as y = η and y j = η j − η j − for j >
1. The b ’s can be obtained once the y ’s are known. For example, b j representsthe number of j ’s bumped from w . Since w started with x j consecutive j ’s,and we introduced a j consecutive j ’s, and ended up with y j consecutive j ’s, thenumber of bumped j ’s must equal b j = x j + a j − y j = a j + ξ j − ξ j − − η j + η j − for j >
1. For j = 1, note that 1 cannot be bumped, so y = x + a , so that b = 0 always. Lemma 3.10. ([NY04]) When v = k a and w = 1 x x · · · n x n , we get η j = ξ j if j < kξ k + a if j = k max { η k , ξ j } if j > k . (3.2) Proof. If ξ j > η k , then some of the j ’s ‘survived’ the bumping. This means thebumping did not get past the last j , and so the number of boxes with numbers ≤ j is still given by ξ j , hence η j = ξ j . If ξ j = η k , then the bumping got tothe very last j , which still gives the same conclusion of η j = ξ j . However, if ξ j < η k , then this means that the bumping eradicated all instances of j (hence,by the nature of the insertion algorithm, no numbers k < l ≤ j remain) and so η j = η k , hence η j = max { η k , ξ j } for all j > k .15 orollary 3.11. ([NY04]) Inserting v = 1 a a · · · n a n into w = 1 x x · · · n x n ,one has η j = max ≤ k ≤ j { x + · · · + x k + a k + · · · + a j } (3.3) for all j .Proof. Applying Lemma 3.10 recursively, one obtains: ( ξ , ξ , ξ , . . . ) a −−−−→ ( ξ + a = η , max { η , ξ } , max { η , ξ } , . . . , max { η , ξ j } , . . . ) a −−−−→ ( η , max { η , ξ } + a = η , max { η , η , ξ } , . . . , max { η , η , ξ j } , . . . ) a −−−−→ ( η , η , max { η , η , ξ } + a , . . . , max { η , η , η , ξ j } , . . . ) . Thus, η j = max { η , η , . . . , η j − , ξ j } + a j for all j > η = ξ + a .Since the η ’s are weakly-increasing, we have η j = max { η j − , ξ j } + a j = max { η j − + a j , ξ j + a j } (3.4)for all j > η l ’s, for l < j ), we get η j = max { max { η j − + a j − , ξ j − + a j − } + a j , ξ j + a j } (3.5)= max { η j − + a j − + a j , ξ j − + a j − + a j , ξ j + a j } (3.6)= · · · = max ≤ k ≤ j { ξ k + a k + a k +1 + . . . + a j } (3.7)= max ≤ k ≤ j { x + · · · + x k + a k + . . . + a j } (3.8)which also covers j = 1.To summarise, one now has y j = x + a if j = 1max ≤ k ≤ j { x + · · · + x k + a k + · · · + a j } − max ≤ k ≤ j − { x + · · · + x k + a k + · · · + a j − } if j > and b j = x j + a j − y j for all j .However, for the purpose of convenient calculation, dividing the computationinto phases, using Equation 3.8, one has Corollary 3.12. ([NY04]) Given input coordinates ( a , . . . , a n ) and ( x , . . . , x n ) ,one obtains the output coordinates ( b , . . . , b n ) and ( y , . . . , y n ) as follows:1. compute ξ j = x + · · · + x j for j = 1 , . . . , n ,2. compute η j = max { η j − , ξ j } + a j recursively for j = 1 , . . . , n , initialisingwith η = ξ + a , . the y -coordinates are obtained by taking y = η and y j = η j − η j − for j = 2 , . . . , n ,4. the b -coordinates are obtained by taking b = 0 and b j = a j + x j − y j for j = 2 , . . . , n . The above description of Schensted insertion as a dynamical evolution on pairsof n -tuples motivates the following definition that will be key in relating RSKand the box-ball system. Definition 3.13.
Define R n = ( N n ) which is the set of pairs of n -tuples. Schensted insertion defines a map RSK n : R n → R n by taking the first (respectively, last) n coordinates of ( a , x ) ∈ R n to be the insertion (respectively, initial) word in the RSK algorithm, and lettingRSK n ( a , x ) = ( b , y ) . Define R = S n ∈ N R n , and let RSK : R → R be definednaturally.
Example 3.14.
Take w = 1 = 1 and v = 1 , i.e. x = (2 , , , , and a = (1 , , , , , where emboldened letters denote thevectors of the corresponding variables. According to Equation 3.8, we shouldhave η = 3 , η = max { , } = 7 , η = max { , , } = 7 ,η = max { , , , } = 8 , η = max { , , , , } = 9 . Thus, y = 3 , y = 7 − , y = 7 − , y = 8 − , y = 9 − . So, b = 1 + 2 − , b = 2 + 3 − , b = 0 + 1 − ,b = 1 + 0 − , b = 1 + 2 − . To check this, let us perform the word insertion, using the notation v ← w forthe result of Schensted inserting a word w into a word v : ← ← (bumped 2) = 11122255 ← (bumped 3) = 11122225 ← (bumped 5) = 11122224 ← (bumped 5) = 111222245 (no bumps)So, the word w ′ = 1 is left, and v ′ = 2 is bumped. This gives y = (3 , , , , and b = (0 , , , , agreeing with the results of the formulæ. sing the notation of Figure 6, we can represent this evolution as x = (2 , , , , a = (1 , , , , y = (3 , , , , b = (0 , , , , Figure 7: Schensted word insertion represented as in Figure 6
Tropical mathematics ([LMRS11], [Li07], [Vi01]) is the study of the max-plussemiring, which we will now define. In this section, we follow the presentationgiven by Maslov [LMRS11]. The structure of the semiring ( R ≥ , + , × ) is carriedover to the set S = R ∪ {−∞} by a family of bijections D ~ , for ~ >
0, given by D ~ ( x ) = ~ ln x if x = 0 −∞ if x = 0 . (3.9)This induces a family of semirings, parametrised by ~ >
0, ( S, ⊕ ~ , ⊗ ~ ) withoperations given by a ⊕ ~ b = D ~ ( D − ~ ( a ) + D − ~ ( b )) = ~ ln( e a/ ~ + e b/ ~ ) if a, b = −∞ max( a, b ) otherwise (3.10) a ⊗ ~ b = D ~ ( D − ~ ( a ) D − ~ ( b )) = a + b. (3.11)In the limit, ~ →
0, Maslov ‘dequantises’ ( R ≥ , + , × ) to obtain the tropicalsemiring ( R ∪ {−∞} , max , +), where its addition is the usual max operationand its multiplication operation is usual addition, hence the name “max-plussemiring”.Maslov views this construction as an analogue of the correspondence principlefrom quantum mechanics, with ( R ≥ , + , × ) as the quantum object and ( R ∪{−∞} , max , +) as its classical counterpart.18 .2.1. Ultradiscretisation and Subtraction-Free Rational Functions In general, one can apply the above process to rational maps. However, whensubtraction is present, one may encounter the so-called minus-sign problem (see,for example, [KKNT09]). If a rational map is subtraction-free, then we maysafely apply the above construction of Maslov, to perform what is often referredto as ultradiscretisation . We will see this in action in Section 4.1.
The formulæ in the previous section involve only the operations max andaddition, hence the formulæ live in the tropical max-plus algebra.We will make the change of operations:(max , +) → (+ , · )to the formulæ in the Corollary 3.12, making the necessary algebraic analoguefor the ‘additive’ identities (0 →
1) to go from ξ j = x + · · · + x j ∀ j = 1 , . . . , nη = ξ + a η j = max { η j − , ξ j } + a j ∀ j = 2 , . . . , n and y = η y j = η j − η j − ∀ j = 2 , . . . , nb = 0 b j = a j + x j − y j = a j + ξ j − ξ j − − η j + η j − ∀ j = 2 , . . . , n. to the (de)tropicalised analogue: ξ j = x · · · x j ∀ j = 1 , . . . , nη = ξ a η j = ( η j − + ξ j ) a j ∀ j = 2 , . . . , n y = η y j = η j η j − ∀ j = 2 , . . . , nb = 1 b j = a j x j y j = a j ξ j η j − ξ j − η j ∀ j = 2 , . . . , n. Lemma 3.15. ([NY04]) Returning to the original variables of x , . . . , x n , a , . . . , a n , y , . . . , y n , b , . . . , b n , the above formulæ reduce to the following system ( x , a ) ( y , b ) : b = 1 a x = y a j x j = y j b j ∀ j = 2 , . . . , n a + 1 x = 1 b a j + 1 x j +1 = 1 y j + 1 b j +1 ∀ j = 2 , . . . , n. (3.12) Proof.
The first two formulæ are by virtue of η = y and ξ = x . For theother formulæ, take η = ξ a and η j = ( η j − + ξ j ) a j and rearrange to get η ξ a = 1 ( ∗ ) η j − η j − a j ξ j a j = 1 ∀ j = 2 , . . . , n ( ∗ j )Equating ( ∗ ) and ( ∗ ) yields η − η a ξ a = η ξ a ⇒ y x a − x = 1 a ⇒ b − x = 1 a resulting in the third formula. 20quating ( ∗ j +1 ) and ( ∗ j ) for j = 2 , . . . , n yields η j +1 − η j a j +1 ξ j +1 a j +1 = η j − η j − a j ξ j a j ⇒ η j ξ j (cid:18) y j +1 x j +1 a j +1 − x j +1 (cid:19) = η j ξ j (cid:18) a j − y j (cid:19) ⇒ y j +1 x j +1 a j +1 + 1 y j = 1 a j + 1 x j +1 ⇒ b j +1 + 1 y j = 1 a j + 1 x j +1 . Returning to the system of equations presented in Lemma 3.15, and lettingbars denote reciprocals (i.e. ¯ x := x ), one obtains the following equations:¯ a ¯ x = ¯ y ¯ a j ¯ x j = ¯ y j ¯ b j ∀ j = 2 , . . . , n ¯ a + ¯ x = ¯ b ¯ a j + ¯ x j +1 = ¯ y j + ¯ b j +1 ∀ j = 2 , . . . , n. This can be represented in the following form: ¯ a ¯ a a n − ¯ a n ¯ x ¯ x x n − ¯ x n = ¯ y ¯ y y n − ¯ y n ¯ b b n − ¯ b n . (3.13)
4. The Ghost-Box-Ball System
In Section 3.3, we demonstrated how one can pass from the RSK equations( cf.
Corollary 3.12) to gRSK (as summarised in Lemma 3.15). We begin byshowing that, as one would hope, the ultradiscretisation of the gRSK equations21ndeed results in the original RSK equations. The purpose of this exercise, how-ever, goes far beyond simply recovering the RSK equations: by performing theultradiscretisation method on gRSK, we obtain a representation of the RSKequations in a form that lends itself to comparison with the box-ball coordinateevolution ( cf.
Theorem 2.5).We find that our comparison results in the need to be able to interpret thebox-ball system when some of its coordinates are zero. This leads us to a newcellular automaton, extending the box-ball system by two new types of object,and we call this cellular automaton the ghost-box-ball system (GBBS). We provesome key results about our ghost-box-ball system, including its reduction to theoriginal box-ball system under an operation we call exorcism . The operationof exorcism, along with the properties of the GBBS, allows us to extend theclassical Young diagram conserved quantity ( cf.
Section 2.1.1) of the box-ballsystem to the ghost-box-ball system.
We begin with an application of the ultradiscretisation process to the geomet-ric RSK equations. Since we are just (re)tropicalising the detropicalised RSKequations, one should not be surprised to recover RSK. However, the processrewrites the RSK equations in a way that is essential for seeing the connectionbetween RSK and the box-ball system.
Lemma 4.1.
The ultradiscretisation of the geometric RSK equations (Equa-tions 3.12) results in the (tropical) RSK equations for Schensted insertion.Proof.
We begin with the geometric RSK equations: y = a x (4.1) y i b i = a i x i i = 2 , . . . , n (4.2) b = a + x (4.3) y i + b i +1 = a i + x i +1 i = 2 , . . . , n − . (4.4)22sing Equations 4.2 and 4.4, one obtains b i +1 = x i +1 + a i − y i (4.5)= x i +1 + a i b i ( b i − x i ) (4.6)= x i +1 + a i b i ( a i − − y i − ) (4.7)= · · · = x i +1 + Q ij =1 a j Q ij =2 b j (4.8)for i = 2 , . . . , n − b = 1, as it should be, the geometric RSK can be expressed as b = 1 (4.9) y i = a i x i b i i = 1 , . . . , n (4.10) b i +1 = x i +1 + Q ij =1 a j Q ij =2 b j i = 1 , . . . , n − cf. Section 3.2.1),we begin ultradiscretisation, first by changing variables a i → e − a i ( ε ) /ε , x i → e − x i ( ε ) /ε , y i → e − y i ( ε ) /ε , b i → e − b i ( ε ) /ε . The change of variables, applied to Equations 4.9 - 4.11, yields e − ε b ( ε ) = 1 (4.12) e − ε y i ( ε ) = e − ε ( a i ( ε )+ x i ( ε ) − b i ( ε )) i = 1 , . . . , n (4.13) e − ε b i +1 ( ε ) = e − ε x i +1 ( ε ) + e − ε ( P ij =1 a j ( ε ) − P ij =2 b j ( ε ) ) i = 1 , . . . , n − , (4.14)where the empty sum is taken to be zero.Solving for the exponentiated variables on the left-hand side, we get the following b ( ε ) = 0 (4.15) y i ( ε ) = a i ( ε ) + x i ( ε ) − b i ( ε ) i = 1 , . . . , n (4.16) b i +1 ( ε ) = − ε log (cid:16) e − ε x i +1 ( ε ) + e − ε ( P ij =1 a j ( ε ) − P ij =2 b j ( ε ) ) (cid:17) i = 1 , . . . , n − . (4.17)The final step in ultradiscretisation is taking the limit as ε → + . If we abusenotation by recycling the original RSK variables in the ultradiscrete equations by23etting, for example, b i = lim ε → + b i ( ε ), we obtain the following ultradiscretisationof the geometric RSK equations b = 0 (4.18) y i = a i + x i − b i i = 1 , . . . , n (4.19) b i +1 = min x i +1 , i X j =1 a j − i X j =2 b j i = 1 , . . . , n − . (4.20)It remains to show that the solution to this system of equations solves the RSKequations (Corollary 3.12). Equations 4.18 and 4.19 are already part of the RSKequations. What is left of the RSK equations is for the following to hold η = ξ + a (4.21) η j = max { η j − , ξ j } + a j , j = 2 , . . . , n (4.22)where η i = y + · · · + y i and ξ i = x + · · · + x i for i = 1 , . . . , n . Since both theRSK equations and the ultradiscrete geometric RSK equations have a uniquesolution, this will complete the proof. We proceed directly:Equation 4.21 is equivalent to y = x + a , which clearly holds.For Equation 4.22, we use the defining equations for the ultradiscrete geometricRSK equations in the following computation for i ≥ y i +1 = a i +1 + x i +1 − b i +1 = a i +1 + x i +1 − min x i +1 , i X j =1 a j − i X j =2 b j = a i +1 + x i +1 − min x i +1 , i X j =1 a j − i X j =2 ( a j + x j − y j ) = a i +1 + x i +1 − min x i +1 , a + i X j =2 ( y j − x j ) = a i +1 + x i +1 − min x i +1 , a − y + x + i X j =1 ( y j − x j ) = a i +1 + x i +1 − min ( x i +1 , ( b + η i − ξ i ))= a i +1 + x i +1 + max ( − x i +1 , ξ i − η i )= a i +1 + max (0 , x i +1 + ξ i − η i )= a i +1 + max (0 , ξ i +1 − η i )= a i +1 + max ( η i , ξ i +1 ) − η i . η i +1 = y i +1 + η i = a i +1 + max ( η i , ξ i +1 ) , (4.23)which completes the proof. The key point of the derivation in Section 4.1 is that we have now obtained theRSK equations in the following form: b = 0 (4.24) y i = a i + x i − b i i = 1 , . . . , n (4.25) b i +1 = min x i +1 , i X j =1 a j − i X j =2 b j i = 1 , . . . , n − . (4.26)which lends itself to comparison with the box-ball system equations for an n +1-soliton: W t +1 i = Q ti +1 + W ti − Q t +1 i i = 1 , . . . , n (4.27) Q t +1 i = min W ti , i X j =1 Q tj − i − X j =1 Q t +1 j i = 1 , . . . , n + 1 . (4.28)In the box-ball system equations (Equations 4.27 and 4.28), we perform thefollowing change of variables: Q ti +1 = a i , W ti = x i , W t +1 i = y i , Q t +1 i = b i , (4.29)producing the following y i = a i + x i − b i i = 1 , . . . , n (4.30) b i = min x i , i X j =1 a j − − i − X j =1 b j i = 1 , . . . , n + 1 . (4.31)Since b = min( x , a ) = min( W t , Q t ), to obtain the RSK condition b = 0, wetake a = Q t = 0. Under this condition, the box-ball equations now take the25orm b = 0 (4.32) y i = a i + x i − b i i = 1 , . . . , n (4.33) b i +1 = min x i +1 , i X j =0 a j − i X j =1 b j i = 1 , . . . , n. (4.34)Finally, we make sense of b n +1 in the above system: b n +1 = min x n +1 , n X j =0 a j − n X j =1 b j = a + · · · + a n − b − · · · − b n (4.35)since x n +1 = W tn +1 = ∞ .Although the box-ball coordinate evolution was defined on B n (as defined inDefinition 2.8), in which all coordinates are positive integers, these equationsnaturally extend to coordinates which may contain zeroes and for which thedynamics satisfies Q t +11 = 0 if Q t = 0.With this extension in mind, we introduce a corresponding modification to the n -soliton phase space B n with the following definition (where, recall, N denotesthe natural numbers augmented by 0): Definition 4.2.
Let G n = {∞} × { } × N n − × {∞}G = [ n ∈ N G n . The dynamics χ : B → B defined in Definition 2.8 naturally extends to a dy-namics χ : G → G . Remark 4.3.
Note that G n and B n each have n − finite coordinates, thedifference is that the first is zero for G n with the remaining allowed to be anynon-negative integers, whereas all must be positive integers for B n . Recall the RSK phase space, R n , introduced in Definition 3.13. This phasespace for RSK is seen to correspond naturally to the phase space G n +1 in thatthe former is a pair of n -tuples and the latter is a corresponding n -tuple of pairs.26 efinition 4.4. For a pair of sequences a = ( a , . . . , a n ) , x = ( x , . . . , x n ) ∈ N n , define a map φ RSK → BBS : R n → G n +1 by φ n RSK → BBS ( a , x ) = ( ∞ , , x , a , x , a , . . . , x n , a n , ∞ ) . (4.36) Conversely, for a sequence z ∈ G n +1 , define a map φ BBS → RSK : G n +1 → R n by φ n BBS → RSK ( ∞ , , z , . . . , z n , ∞ ) = ((0 , z , . . . , z n − ) , ( z , z , . . . , z n − )) . (4.37) Let φ RSK → BBS : R → G and φ BBS → RSK : G → R be their natural extensions. Remark 4.5.
Note that φ RSK → BBS is a bijective mapping from R n to G n +1 .However φ BBS → RSK is not its inverse, in fact φ BBS → RSK is neither injectivenor surjective. This property of φ BBS → RSK is a consequence of the shift in thefirst equation of 4.29.
We summarise the calculations of this section in the following theorem:
Theorem 4.6.
Under the box-ball evolution χ : G n +1 → G n +1 , RSK insertionis captured as the following χ ( ∞ , , x , a , x , . . . , a n − , x n , a n , ∞ ) = ( ∞ , b , y , b , y , . . . , b n , y n , b n +1 , ∞ ) , (4.38) noting that b = 0 . Corollary 4.7.
One hasRSK = φ BBS → RSK ◦ χ ◦ φ RSK → BBS . (4.39) We now introduce the ghost-box-ball system which is designed to be thecellular automaton realisation of χ , given in Definition 4.2. This amounts tomodifying the original box-ball system to reflect the zeroes that we are allowinginto the box-ball coordinates. Ultimately, what we want is a modified box-ballsystem into which one can encode an RSK pair, and from whose evolution onecan read off the RSK output. Definition 4.8. A ghost-box-ball system consists of a one-dimensional infi-nite array of boxes with a finite number of boxes designated precisely one of thefollowing three states (the rest of the boxes are empty):1. filled (with a ball),2. filled ghost, . empty ghost,and subject to the following constraints:1. a filled ghost may not be adjacent to another filled ghost, nor to a filledbox, and2. an empty ghost may not be adjacent to another empty ghost, nor to anempty box.We let GBBS denote the set of all ghost-box-ball states. Remark 4.9.
For the purpose of this section, we will only be interested in aparticular class of ghost-box-ball states: the set of ghost-box-ball states for whichthe left-most box that isn’t an empty box is a filled ghost. For the rest of thissection, until our conclusions, we will use the term “ghost-box-ball” and thenotation GBBS to refer to this particular subset of interest. We will discussextensions in Section 5, where the full set of ghost-box-ball states will be studied.
For a graphical representation of the ghost-box-ball states, we employ the fol-lowing key:1. An empty box shall be represented by2. A filled box shall be represented by3. A filled ghost shall be represented by4. An empty ghost shall be represented by
Example 4.10.
As an example, the following is a ghost-box-ball state: · · ·· · ·
Note that no filled ghost is neighbours either a ball or another filled ghost, nordoes any empty ghost neighbour an empty box or another empty ghost.
Remark 4.11.
Although ghost-box-ball states extend infinitely right by emptyboxes, we will often truncate them in our depiction with the understanding thatthey still extend infinitely to the right. Similarly, since all ghost-box-ball statesappearing in this section have as their left-most non-empty box a filled ghost, wewill always truncate at the filled ghost in our graphical representations, knowingthat there are infinitely many empty boxes to the left of the first filled ghost.For example, the above ghost-box-ball state might be depicted as follows:
28e now define the following evolution rule on GBBS.
Definition 4.12. (The Ghost-Box-Ball Algorithm)
1. Move each ball exactly once.2. Move the leftmost unmoved ball to its nearest right empty box.3. If a ball’s new position has a filled ghost to its immediate right, ma-terialise (create) an empty ghost between them.4. If a ball’s new position has a filled ghost to its immediate left, exorcise(delete) the ghost.5. If a ball is moved from a position with an empty ghost right-adjacentof it, insert a filled ghost between the box vacated by the ball and theempty ghost.6. If a ball is moved from a position with an empty ghost left-adjacentof it, exorcise that ghost.7. Repeat (2)-(6) until all balls have been moved.
Remark 4.13.
Performing the whole ghost-box-ball algorithm constitutes one time-step of the evolution of a ghost-box-ball system, with each ball movement(together with any resulting materialisations and exorcisms) constituting a stage of the overall time-step.
Example 4.14.
We demonstrate the ghost-box-ball evolution, stage-by-stagebelow:
Figure 8: A single time-step of the ghost-box-ball evolution, split into the stages that makeit up.
As in Section 1.2.2, we employ the blue-red colouring for unmoved-moved balls toaid in the visual tracking of the algorithm. Since the Ghost-Box-Ball Algorithmis more involved than the classical Box-Ball Algorithm, we also provide somesupplementary discussion on the first few stages in the above example.1. From (a) to (b): we move the left-most blue ball to the nearest empty boxto its right. The ball was not neighbouring a ghost initially, so its removalfrom the initial position does not in itself prompt a materialisation or ex-orcism of a ghost. However, its new position contains a left -neighbouringfilled ghost. By 4.12(4), we must exorcise that ghost.2. From (b) to (c): the next ball to move does so to a new position not neigh-boured by a ghost, however its initial position has a right -neighbouringempty ghost. By 4.12(5), an filled ghost must materialise .3. From (c) to (d): we have ghosts neighbouring both sides of the initialposition of the next ball to move. Since we have discussed initial right-neighbouring in the previous step, we focus on the left -neighbouring emptyghost to our moving ball. By 4.12(6), the empty ghost to the left must be exorcised .As a mnemonic device: • left -adjacency leads to exorcism : where a movement would violate a con-straint due to a left -adjacent ghost, that ghost must be exorcised . • right -adjacency leads to materialisation : where a constraint would beviolated due to a right -adjacent ghost, a ghost must materialise to re-store order. uppressing the intermediate stages (b)-(h), the single time-step of the ghost boxball evolution corresponding to Example 4.14 is summarised in the following: t : t + 1 : Figure 9: A single time-step of the ghost-box-ball evolution (without the intermediate stages).
Lemma 4.15.
The result of performing this algorithm on a ghost-box-ball sys-tem is again a ghost-box-ball system.Proof.
This is just a consequence of the construction: (3)-(6) serves the pur-pose of keeping the constraints of the ghost-box-ball definition (Definition 4.8)satisfied.
Definition 4.16.
The map ˆ ̺ : GBBS → GBBS will be defined to be the resultof applying this algorithm.
Definition 4.17.
In a ghost-box-ball state, a filled block will be either of thefollowing:1. a maximal sequence of adjacent balls, or2. a single filled ghost.Similarly, an empty block will be either of the following:1. a maximal sequence of adjacent empty boxes, or2. a single empty ghost.
Remark 4.18.
Since filled ghosts cannot neighbour other filled ghosts or filledboxes, and empty ghosts cannot neighbour other empty ghosts or empty boxes,we make the following observations of the general structure of a ghost-box-ballstate: • The infinite sequences of empty boxes at the beginning and end constitutesthe infinite empty block, with all other blocks (both filled and empty) beingfinite. • A ghost-box-ball state consists of blocks, alternating between the finite filledblocks and empty blocks, terminating in the infinite empty block.
For reference, consider the time t state in Figure 9: t :There are fourteen “blocks”, the initial sequence of which we list in order below:31. Infinitely many empty boxes on the left (suppressed) in the depictionabove.2. Filled block consisting of one filled ghost.3. Empty block consisting of two empty boxes.4. Filled block consisting of two balls.5. Empty block consisting of one empty ghost.6. Filled block consisting of one ball.7. Empty block consisting of one empty ghost.8. Filled block consisting of one filled ghost. etc. , terminating in the empty block consisting of infinitely many empty boxes.Between Definition 4.17 and Remark 4.18, we now have a natural way of ex-tending the earlier coordinate map C : BBS → B as follows: Definition 4.19.
We define a coordinatisation C : GBBS → G by mapping aghost-box-ball system to a tuple ( W , Q , W , . . . , Q N , W N ) , where Q i = (cid:26) if the i -th filled block is a filled ghost length of the i -th filled block otherwise W i = (cid:26) if the i -th empty block is an empty ghost length of the i -th empty block otherwise , with W = W N = ∞ . For each ghost-box-ball system state, its coordinates lie in G n for some n ∈ N .Therefore, the full set of ghost-box-ball states is identified with G . In the ghost-box-ball algorithm, ghosts can be exorcised as a result of themovement of balls (either by moving a ball to the right of an empty ghost or from the right of an empty ghost). There is a natural map z : GBBS → BBS given32y exorcising all ghosts in a ghost-box-ball system, while shifting the remainingballs and boxes to fill in the newly created voids (i.e., create a sequence of emptyand filled boxes, scanning the ghost-box-ball system from left to right, ignoringall ghosts). For example, see the following ghost-box-ball state and box-ballstate: · · ·↓ z · · ·· · · Figure 10: Exorcising ghosts to produce a box-ball state.
The map z : GBBS → BBS shall be referred to as global exorcism . Lemma 4.20.
The following diagram commutes
GBBS GBBSBBS BBSˆ ̺ z ̺ z Proof.
The ghost-box-ball algorithm and the box-ball algorithm only differ inhow the ghosts are materialised and exorcised: the new position of a movingball in the ghost-box-ball algorithm agrees with that of the box-ball algorithm,relative to just the empty boxes and balls. Therefore, the result of globallyexorcising the ghosts and then evolving according to the box-ball dynamics co-incides with evolving according to the ghost-box-ball dynamics and then globallyexorcising.In light of this result, it also makes sense to ask about how the soliton struc-ture of the box-ball-system translates to the ghost-box-ball system. It is nothard to see that the GBBS algorithm preserves the existence and location ofblocks of consecutive ghosts; such a block can grow or shrink, but never dis-appear altogether. We can view such ghost blocks as single zero length blocks33n the soliton structure (see Figure 10). Along with the previous lemma, thisobservation yileds the following.
Lemma 4.21.
The ghost-box-ball dynamics exhibits the same soliton behaviouras the box-ball dynamics, subject to the following augmentation of the traditionalnotion of an n -soliton box-ball state: we add the construct of a ghost soliton,which is any configuration of ghost blocks. This construct does not move – it haszero velocity. Asymptotically, all other blocks comprising the soliton state travelwith speed equal to their respective lengths. Hence, the ghost-box-ball system stillexhibits the sorting property of the box-ball system. Our final analogue of a classical box-ball system property is its conserved shape,seen in Section 2.1.1.
Corollary 4.22. If G ∈ GBBS, define B = z ( G ) . Representing B as a se-quence of ’s and ’s, • let p be the number of ’s in the sequence. • Eliminate all of these ’s, and let p be the number of ’s in the resultingsequence. • Repeat this process until no ’s remain.We associate the weakly decreasing sequence ( p , p , . . . ) , or, equivalently, theYoung diagram whose j th column has p j boxes is the shape associated to G .This Young diagram is the same for ˆ ̺ k G for every k ∈ N .Proof. This follows from Lemma 4.20 and Section 2.1.1. It is already known thatthis Young diagram is conserved for all ̺ k ( B ) = ̺ k ◦ z ( G ). Since ̺ ◦ z = z ◦ ˆ ̺ ,it follows that z (ˆ ̺ k ( G )) = ̺ k ◦ z ( G ) = ̺ k ( B ) . Example 4.23.
Returning to the ghost-box ball system in Example 4.14, beloware the initial state and the subsequent three evolutions. · · ·· · ·· · ·· · ·
Figure 11: Three iterations of the ghost-box-ball algorithm.
Applying global exorcism to these four ghost-box-ball states yields · ·· · · · · ·· · · · · ·· · · · · ·· · · Figure 12: Three iterations of the ghost-box-ball evolution (after global exorcism). where the blue lines indicate the locations previously occupied by ghosts. Theinvariant shape (recall Section 2.1.1) for this sequence (and all future states) is
Figure 13: The invariant shape of the ghost-box-ball system(s) in Figure 11
In Definition 4.4, we constructed maps φ RSK → BBS : R → G and φ BBS → RSK : G → R to represent Schensted insertion in terms of the natural extension of thecoordinatised box-ball evolution to the setting in which some coordinates mayvanish. In this section, we demonstrate how to lift the correspondence betweenRSK and χ to one between RSK and the ghost-box-ball evolution, using thecoordinate map on GBBS.The main tool will be the coordinate map C : GBBS → G in Definition 4.19.One can encode an RSK pair ( a , x ) ∈ R in a ghost-box-ball system by compos-ing the maps φ RSK → BBS : R → G and C − : G → GBBS.The main result of this section will be to establish the commutativity of thefollowing diagram: 35BBS GBBS G G R R ˆ ̺ RSK C − Cφ RSK → BBS φ BBS → RSK
Figure 14: Commutative diagram relating the ghost-box-ball evolution to the RSK dynamics or, more succinctly,RSK = φ BBS → RSK ◦ C ◦ ˆ ̺ ◦ C − ◦ φ RSK → BBS . (4.40)In establishing this result, we will in fact prove something stronger: if an RSKinput pair is encoded in a ghost-box-ball system, the number of stages in theRSK insertion (see, for example, 3.9) is equal to the number of stages in theghost-box-ball evolution (see, for example, Figure 8), and the data from eachstage of the RSK insertion is fully recoverable from the corresponding stage ofthe ghost-box-ball evolution.The stages of the ghost-box-ball evolution alluded to here are just the stagesgiven by Definition 4.12. To describe the corresponding stages for the RSKinsertion ( cf. Section 3.1.1), we introduce the following:
Definition 4.24.
For a pair ( a , x ) ∈ R n , define a sequence of triples r i ( a , x ) := ( a i , x i , b i ) ∈ ( N n ) (4.41) for i = 0 , , . . . , n P k =1 a k , where . r ( a , x ) = ( a , x , (0 , , . . . , .2. If j = min k { a ik = 0 } , then x i +1 is the tuple resulting from (Schensted)inserting j into the word with tuple x i , a i +1 is obtained from a i by sub-tracting 1 from the j -th entry, and b i +1 is the result of adding 1 to the k -th entry of b i if a k is bumped from x i to obtain x i +1 , or b i +1 = b i ifnothing is bumped. We write r i for r i ( a , x ) if it is unambiguous to do so.This construction clearly encodes the steps of Schensted insertion. In particular,if RSK( a , x ) = ( b , y ) and m = n P k =1 a k , then one has r m = ((0 , , . . . , , y , b ). We now introduce a bookkeeping device, referred to as wall placement , thatwill help to establish the stage-by-stage correspondence alluded to at the end ofthe previous section.The key observation underlying this is that there is a conservation law acrossthe stages of RSK insertion: the total number of instances of a given numberis conserved, i.e. a ij + x ij + b ij is a function of just j (it is constant in i ). Inparticular, taking i = 0, this quantity is a j + x j + b j = a j + x j =: w j which is expressed solely in terms of the input pair. This conserved quantitywill be called the j -th width .Moreover, this conservation for the RSK stages may be visualised in terms ofwall placements in the initial ghost-box-ball configuration using the conservedquantities, the w j ’s. Subsequent to the definition, we will then describe howthe wall placement evolves during the successive stages of the ghost-box-ballevolution. To represent the wall placement, we will superimpose red zigzags37ver the ghost-box-ball configurations.The following is how we initialise the wall placement on the GBBS associatedto an RSK pair: Definition 4.25.
Let ( a , x ) ∈ R n and G = C − ◦ φ RSK → BBS ( a , x ) be the ghost-box-ball configuration associated to the RSK pair ( a , x ) . The walls of G , whichwe will represent by red zigzags, will be placed as follows:1. Take the initial filled ghost and the following w boxes of G , and separatethem from the rest of the subsequent boxes by a wall (a red zigzag). Note:If x = 0 , include the corresponding empty ghost. Similarly, If a = 0 ,include the corresponding filled ghost.2. Take the next w boxes of G and place a wall at the end of them. Again,if a = 0 , include the corresponding filled ghost before the zigzag.3. Continue in this manner until the boxes are separated into n + 1 regions( n finite and one infinite). Example 4.26.
Take the RSK input to be a = (3 , , , , and x = (2 , , , , ,so that the associated ghost-box-ball system has coordinates φ RSK → BBS ( a , x ) = ( ∞ , , , , , , , , , , , , ∞ ) . In the above, we embolden the odd positions, which, by definition of C , corre-spond to filled blocks.The widths are calculated as follows: w = a + x = 3 + 2 = 5 w = a + x = 0 + 0 = 0 w = a + x = 2 + 0 = 2 w = a + x = 1 + 2 = 3 w = a + x = 0 + 3 = 3 We put a wall before the first filled ghost, enclose the two subsequent empty boxesand three filled boxes by another wall (this is now “Region 1”). In “Region 2”,we include the filled ghost before the wall because a = 0 . Continuing in thismanner, we obtain the resulting picture: · · · Figure 15: The initial ghost-box-ball system with its finite regions labelled.
Example 4.27.
Below is the evolution of the ghost-box-ball system in Figure15, with the walls included at each step. · ·· · ·· · ·· · ·· · ·· · ·· · · Figure 16: The evolution of a ghost-box-ball system with walls.
The initial ghost-box-ball system came from ( a , x ) = ((3 , , , , , (2 , , , , x = (2 , , , , a = (3 , , , , y = (5 , , , , b = (0 , , , , Figure 17: The Schensted evolution encoded by Figure 16
In the final stage of the above ghost-box-ball evolution, we can construct twosequences: the number of empty boxes in each region and the number of (red)balls in each region. These two sequences are (5 , , , ,
0) and (0 , , , , y and b , respectively. We will prove that this holds in general, byproving a stronger result. Theorem 4.28.
Let ( a , x ) ∈ R n and let G = C − ◦ φ RSK → BBS ( a , x ) . Let ( r i ) i be as defined in Definition 4.24. At the i -th stage of the ghost-box-ball evolutionof G , the following holds for each j ∈ [ n ] :(1) a ij is equal to the number of blue (unmoved) balls in the j -th region,(2) b ij is equal to the number of red (moved) balls in the j -th region, x ij is equal to the number of empty boxes in the j -th region.Proof. We prove this by induction on i .The base case ( i = 0) is satisfied by construction: the initial ghost-box-ball stateand wall structure were built out of the RSK input variables so that (1) and (3)are satisfied, and (2) holds trivially because nothing has been bumped yet (so b j = 0 for each j ) and no balls have been moved yet (so all balls, if any, are blue).Let us now suppose that, at some stage, say the k -th stage, we have for each j ∈ [ n ]:(1) a kj is equal to the number of blue (unmoved) balls in the j -th region of the k -th step in the GBBS algorithm,(2) b kj is equal to the number of red (moved) balls in the j -th region of the k -thstep in the GBBS algorithm,(3) x kj is equal to the number of empty boxes in the j -th region of the k -th stepin the GBBS algorithm.We now consider the ( k + 1)-st stage of the ghost-box-ball evolution. We needto show that the equalities hold for the region the ball was in and the regionthe ball moves to (unless it moves to the infinite region).Suppose the ball we moved was in Region j . We first make the observation thatthe ball cannot move within Region j : the walls were placed so that each regioneither has no blue balls or the block of blue balls is precisely the last part ofthe region (if there are blue balls, the wall comes right after the last of them).Therefore, the ball must move to Region l , for some l > j (which may be theinfinite region).With this observation, we can immediately check the counts for Region j afterthe ball is moved: the number of red balls has not changed (by the observation),the number of blue balls has decreased by 1, and the number of empty boxeshas increased by 1. We therefore need to show: a k +1 j = a kj − b k +1 j = b kj x k +1 j = x kj + 1 . By the induction hypothesis, for us to be able to move a ball from Region j ,we must have had a kj ≥ a kj ′ = 0 for all j ′ < j . Thus, in terms of RSK, this means we are inserting a j into the current row for x k . This reduces a kj by one (since the j is insertedinto x k ), increases x kj by one (because the j finds a place in the row, and doesnot alter b kj , since a number can only bump a number greater than itself). Wesee, therefore, the validity of the counts (in terms of the GBBS and the RSK41nsertion) agree for Region j .Now we split into two cases (based on the destination of the ball): • (Case 1): The ball lands in some finite region, say Region l , where n ≥ l > k • (Case 2): The ball lands in the infinite region. (Case 1): Since the ball moves to the left-most empty box to its right, allspaces between the ball’s origin and its new box must be full or ghosts. Inparticular, at the k -th stage, there were no empty boxes in the regions betweenRegion j and Region l . By the induction hypothesis, x km = 0 for all j < m < l and x kl ≥ j to be inserted bumps an l ). In terms of Region l , we loseone empty box and gain a red ball (there is no change to the number of blueballs here). We therefore need to show: a k +1 l = a kl b k +1 l = b kl + 1 x k +1 l = x kl − . This is clearly the case, since a j is bumping an l . (Case 2): By the same reasoning in Case 1, there must be no empty boxesin any of the finite regions beyond the j -th, so x km = 0 for all j < m ≤ n .By the induction hypothesis, there are no numbers in the row that are strictlygreater than j . Therefore, in this RSK insertion step, we do not bump anything.Instead, we extend the row by a box and fill it with the j . Since the theoremdoes not contain any conditions relating the current RSK step and the infiniteregion of the ghost-box-ball system, we have nothing more to check here. Simplyby having the only changes be a k +1 j = a kj − b k +1 j = b kj x k +1 j = x kj + 1 . establishes that nothing has been bumped and that the row has been extendedby a j ; only the variables/counts for Region j are affected in this case.We now have the following immediate corollary: Corollary 4.29.
One hasRSK = φ BBS → RSK ◦ C ◦ ˆ ̺ ◦ C − ◦ φ RSK → BBS . Proof.
Since, at each stage, the RSK triple r i is encoded in the i -th stage of theghost-box-ball evolution, and no blue balls remain at the end of the ghost-box-ball evolution, only empty boxes, empty ghosts, filled ghosts and red balls areleft. One can simply apply the coordinate mapping C : GBBS → G and readoff the RSK variables (using φ BBS → RSK ) to find the output pair ( b , y ) for theRSK insertion a → x . 42 .6. Ghost-Box-Ball Dynamics Beyond RSK Throughout this section so far, we have focused on connections to the RSKalgorithm. We showed a precise correspondence between RSK and the ghost-box-ball algorithm (Corollary 4.29)), which was our original goal. A key con-struct in doing this was the implementation of a coordinate dynamics for theghost-box-ball system. These coordinates ( cf.
Definition 4.19) are given interms of lengths of filled and empty blocks of boxes of various types. The dy-namics of the ghost-box-ball system (denoted ˆ ̺ ) and the coordinate dynamics(denoted χ ) have a direct relation, independent of RSK, which is of interest inits own right. That is what we will now discuss. More precisely, we show theanalogue of Corollary 2.9 holds for the ghost-box-ball system by proving thatthe following diagram commutes:GBBS GBBS G G ˆ ̺C χ C .In this section, we provide a partial answer to this question simply by utilisingthe results of the previous sections.From Corollary 4.7, we haveRSK = φ BBS → RSK ◦ χ ◦ φ RSK → BBS (4.42)and Corollary 4.29 establishesRSK = φ BBS → RSK ◦ C ◦ ˆ ̺ ◦ C − ◦ φ RSK → BBS . (4.43)Since φ RSK → BBS is bijective, one obtains from the above φ BBS → RSK ◦ χ = φ BBS → RSK ◦ C ◦ ˆ ̺ ◦ C − . (4.44)In what follows, we show that, although φ BBS → RSK is not injective, we caneliminate this map from both sides of the above equation by a local analysis.43 efinition 4.30.
For z = ( ∞ , Q = 0 , W , Q , W , . . . , Q n − , W n − , Q n , ∞ ) ∈G n , define ζ n : G n → N by ζ n ( z ) = n X j =1 Q j . (4.45)Since this counts the number of balls in the system with coordinates z , thisshould be conserved. To see that this is the case, consider the Q t +1 n +1 : Q t +1 n = min W tn , n X j =1 Q tj − n − X j =1 Q t +1 j (4.46)= min ∞ , n X j =1 Q tj − n − X j =1 Q t +1 j (4.47)= n X j =1 Q tj − n − X j =1 Q t +1 j , (4.48)Rearranging this shows that ζ n is invariant under the coordinate dynamics on G . Theorem 4.31.
For each m ∈ N , let G ,mn := ζ − n ( m ) . On this level set of ζ n , φ BBS → RSK is injective, and one has χ = C ◦ ˆ ̺ ◦ C − (4.49) when restricted to G ,mn .Proof. For z = ( ∞ , Q = 0 , W , Q , W , . . . , Q n − , W n − , Q n , ∞ ), recall thatthe explicit form of the mapping φ BBS → RSK ( cf. Definition 4.4) gives φ BBS → RSK ( z ) = ((0 , Q , . . . , Q n − ) , ( W , . . . , W n − )) . (4.50)Clearly the obstruction to injectivity is in losing the data of Q n . However, on G ,mn , one has Q n = m − m + Q n = m − n X j =1 Q j + Q n = m − n − X j =1 Q j . (4.51)Thus, when restricted to G ,mn , φ BBS → RSK is injective and therefore has a leftinverse. We post compose Equation 4.44 by this left inverse to complete theproof.
Remark 4.32.
Since G n is the union of the level sets of ζ n , Theorem 4.31 gives4.52 on all of G n . Furthermore, since G is itself a disjoint union of the G n sets,Equation 4.52 holds on all of G . As a result, we have the following corollary: orollary 4.33. The equality χ = C ◦ ˆ ̺ ◦ C − (4.52) holds on all of G . Thus, the ghost-box-ball dynamics on ghost-box-ball systems starting with afilled ghost is in agreement with the coordinate evolution on G . There are other works in the literature that describe a direct connection be-tween box-ball systems and the RSK algorithm. We conclude this section with abrief comparison between what we have done and one of the principal and mostcited treatments in this regard due to Fukuda [Fu04]. His work introduces anencoding of Schensted insertion in an advanced box-ball system with carryingcapacities and ball colours . In contrast we emphasise that our work capturesSchensted insertion in what is essentially only a slight deviation from the orig-inal box-ball system (in the sense that it is governed by the original box-ballcoordinate evolution), rather than having to add the complexity of box labelsand capacities to the box-ball system. For a more detailed and self-containeddescription of the advanced box-ball system we refer the reader to [R20].Both the ghost-box-ball and advanced box-ball (with uniform carrying capac-ity 1) systems reproduce Schensted/RSK insertion, but these two systems arefundamentally different. On the one hand, the ghost-box-ball system exhibitswhat we have called ghost solitons: particles with velocity zero, whereas no suchzero soliton exists in the advanced box-ball system. Conversely, prioritisationof certain balls over others (via colouring/labelling) in the advanced box-ballsystem is not something currently in the ghost-box-ball system.It would be interesting to study the applications of a hybrid of the two systems.As far as Schensted insertion alone is concerned, there is an obvious appealto the ghost-box-ball system over the advanced box-ball system: the former45s simply a manifestation of the original (unlabelled, carrying capacity one)box-ball-system, originally introduced by Takahashi and Satsuma [TS90]. Theadvanced box-ball system is not governed by such simple equations. Therefore,purely from the standpoint of simplicity, the attraction of the ghost-box-ballsystem seems clear.
5. Intrinsic Ghost-Box-Ball Dynamics and thePhase Shift
We present constructions for generalising the ghost-box-ball system andstudying its intrinsic dynamical nature in Section 5.1, and we use these con-structions to study the box-ball phase shift phenomenon in Section 5.2. Key toboth sections is the necessity of a backwards time dynamics for the GBBS, andthis is what we explore first.
For the classical box-ball system, it was important to run the time evolu-tion in both forwards and backwards time: in backwards time, asymptotically,the evolution sorts the blocks in descending order. In making a connectionto continuous-time dynamics (see Section 6), unidirectional dynamics is notenough.
Remark 5.1.
A nice property of the classical box-ball dynamics is that onecan perform a backwards time step by reflecting the box-ball system horizontally,performing the usual box-ball algorithm, and then reflecting the system again[TS90].By carefully going through (3)-(6) of the ghost-box-ball algorithm (cf. Definition4.12), on can check that the same principle holds for reversing a time-step ofthe ghost-box-ball algorithm. This is demonstrated below in Example 5.2.
The restricted set of ghost-box-ball configurations studied in Section 4 ( i.e. those with a filled ghost on the left) are not closed under this reverse dynamics.
Example 5.2.
Let us take the following ghost-box-ball state: G := · · ·· · · eflecting this system horizontally, we obtain ¯ G := · · ·· · · We can now apply the ghost-box-ball algorithm to this configuration, which wewill do thrice, for good measure: ˆ ̺ ( ¯ G ) := · · ·· · · ˆ ̺ ( ¯ G ) := · · ·· · · ˆ ̺ ( ¯ G ) := · · ·· · · Finally, reflecting these horizontally, we now obtain the t = − , t = − , t = − time-steps for the t = 0 state given by G : ¯ G := · · ·· · · We can now apply the ghost-box-ball algorithm to this configuration, which wewill do thrice, for good measure: t = − · · ·· · · t = − · · ·· · · t = − · · ·· · · G = · · ·· · · ˆ ̺ → ˆ ̺ → ˆ ̺ → To close the set under this reverse time evolution, one must include ghost-box-ball systems whose left-most non-empty box contains a ball. On this moregeneral set of ghost-box-ball configurations, Definition 4.12 can still be imple-mented, and it is to this general set that we dedicate this section: extendingsome of the results of Section 4.At the heart of this extension will be a simple idea: we will define a map, ϑ ,which embeds the set of general ghost-box-ball configurations into itself, the47mage of which will lie in the (restricted) set of ghost-box-ball configurationsstudied in Section 4 and is invariant under the (forward) time evolution given bythe ghost-box-ball algorithm. We show that this map conjugates the dynamicson the general set to that of the restricted set, and hence lift the key resultsof interest to this general setting. Furthermore, in the next section (Section5.2), these constructions play an important role in proving a soliton phase-shiftformula for the 2-soliton box-ball system. Definition 5.3.
Let GBBS be the set of ghost-box-ball states for which theleft-most non-empty box is a filled ghost. We define the augmentation map ϑ : GBBS → GBBS by taking the empty box that is two spaces to the left ofthis left-most non-empty box and changing the state of this empty box to a filledghost. Example 5.4.
Below is an example of applying the augmentation map: · · ·· · · ↓ ϑ · · ·· · · The augmentation map also applies to ghost-box-ball states that are already inGBBS , for example: · · ·· · · ↓ ϑ · · ·· · · Remark 5.5.
The image of ϑ is an invariant subset ˆ ̺ ( ϑ ( GBBS )) ⊂ ϑ ( GBBS ) .This is just a consequence of the ball dynamics moving to the right, with enoughseparation of the left-most filled ghost from the ball dynamics. Moreover, sincethe augmentation map is clearly injective, it is invertible on its image ϑ ( GBBS ) .Thus, ϑ embeds GBBS into GBBS as an invariant subset. Theorem 5.6.
The following diagram commutes: GBBS ˆ ̺ϑ ˆ ̺ | GBBS ϑ Furthermore, since ϑ is invertible on its image, we have that iterating the ghost-box-ball algorithm on GBBS is equivalent to augmenting, iterating on the result-ing ghost-box-ball state, and then inverting the augmentation. i.e., ˆ ̺ k = ϑ − ◦ (ˆ ̺ | GBBS ) k ◦ ϑ. (5.1) Proof.
The commutation of the diagram is equivalent to showing Equation 5.1for k = 1, which is immediate since this just says that if one introduces a filledghost to the left of the dynamic evolution, applies the ghost-box-ball algorithm,then removes that filled ghost (which was stationary and didn’t interact withthe dynamics), then that is the same as simply applying the ghost-box-ballalgorithm. Thus, we have ˆ ̺ = ϑ − ◦ (ˆ ̺ | GBBS ) ◦ ϑ. (5.2)Equation 5.1 then follows by composition of Equation 5.2 with itself k times. Corollary 5.7.
It now follows that the soliton structure, sorting property andinvariant shape constructions for the subclass GBBS proved in Chapter 4 holdson the entirety of GBBS, the general ghost-box-ball system. We present some extensions of earlier definitions to set us up for the next section.The sets we define offer the generalisations of the sets G n and G for the ghost-box-ball coordinatisations of the ghost-box-balls of Section 4 (those that startwith a filled ghost) to the expected coordinate spaces of the more general ghost-box-ball systems. Definition 5.8.
Let G n = {∞} × N n − × {∞} and G = [ n ∈ N G n . The latter is the domain of the natural extension of the map C in Definition4.19 to the general setting of ghost-box-ball settings. We continue to call thisextension C : GBBS → G . efinition 5.9. Let
Υ :
G → G be the map induced by the maps Υ n : G n →G n +1 : Υ n ( ∞ , z , z , . . . , z n − , ∞ ) = ( ∞ , , , z , z , . . . , z n − , ∞ ) . On the image C ( ϑ ( GBBS )) of ϑ ( GBBS ) under C , one has enough coordinatesto invert Υ , by simply omitting the second and third coordinates. Moreover, thismap extends to all subsequent C ( ϑ k ( GBBS )) for each k ∈ N by still omittingthe second and third coordinates. Call this map ˆΥ . We can now state relations between ϑ and Υ, as well as their inverses on appro-priate sets: Lemma 5.10.
The following two squares commute:
GBBS GBBS G G ϑC Υ C ϑ (GBBS) GBBS C ( ϑ (GBBS)) G ϑ − C ˆΥ C .Ultimately, in studying the general set of ghost-box-ball systems, we would liketo extend Corollary 4.33 to this more general ghost-box-ball setting. For now,with these definitions at our disposal, we demonstrate an application of thesemaps in the next section on the box-ball phase shift. In this section, we make use of these maps ϑ and Υ, as well as their inverseson appropriate sets, to prove a phase-shift formula for the classical box-ballsystem, which comes by means of passing over to the ghost-box-ball setting.The subsequent utility of the map ϑ is in its introduction of a stationary object(an initial filled ghost), relative to which motion is measured. Essentially, wherethe classical box-ball system’s coordinates exhibit left- and right-shift invariance(due to the padding by infinity on each end), the introduced filled ghost plays50he role of an origin point, against which the dynamics is measured.We begin by introducing the notion of the box-ball phase shift phenomenon. We have seen how, as t → + ∞ , the blocks sort themselves by increasinglength. The same holds in reverse time: as t → −∞ , the blocks are orderedby decreasing lengths. The asymptotic sequence of lengths are revealed at anyfinite time using the invariant shape construction (Section 2.1.1). The followingexample demonstrates why one cannot simply just count block lengths (thethird state does not show the soliton structure of blocks of length 1 and 3): · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · Figure 18: A phase shift interaction between two colliding blocks.
Remark 5.11.
In the above example, we can discern the asymptotic ordering inthe first, second, fourth and fifth rows, simply by counting the numbers of ballsin each block of adjacent balls. The middle row (the third) could be misleading,since it reveals a (2 , structure for the blocks (although, the invariant shapeconstruction would reveal the correct soliton structure here). If two blocks arespaced far enough apart, then no such obfuscation occurs. Barring this intricacy ( i.e. when there is enough space between consecutiveblocks), one can take two blocks, evolve sufficiently many times according tothe box-ball evolution, and compare the position of the blocks to where theywould have been if it had not have been for the collision.In the figure below, we replicate Figure 5.2.1. However, we use green ballsto keep track of where the block of three balls would have been without the51ollision, and magenta balls to keep track of where the block of one ball wouldhave been. · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · ·· · ·
When the collision has concluded, we see that the three-block is two positionsahead of where it would have been, and the one-block is two positions behindwhere it would have been. Therefore, we say that the three-block experiences a+2 phase shift, and the one-block experiences a − We want to study the phase shift for 2-soliton box-ball collision. For a collisionto occur, we must have a larger block of balls to the left of a smaller block ofballs. By definition of blocks, the two must be separated by a sequence of emptyboxes, which we refer to as the gap . With this in mind, we shall initialise with aconfiguration of a block of k balls on the left, a gap of g empty boxes, followedby a block of l < k balls: · · ·· · · · · · · · ·· · · k g (gap) l Figure 19: BBS with a ( k, l ) structure, prior to the collision
We assume that g is large enough so that ( k, l ) is representative of the solitonicstructure of this box-ball configuration (as discussed in Remark 5.11). We alsomotivate some terminology in the following: Definition 5.12.
For a 2-soliton system, if one cannot simply see the solitonstructure (the asymptotic sizes of blocks) in a box-ball state by counting the blocksizes at a given time, that state will be said to be in the collision phase. We ay that a state is pre-collision (or that we are a time prior to collision) if theblocks are ordered from largest to smallest and not in the collision phase. It is natural to ask how one can say in general whether a box-ball state of thetype in Figure 19 is in its collision phase or is pre-collision. We answer thisusing the invariant shape construction of Section 2.1.1.
Lemma 5.13.
A box-ball configuration of the type shown in Figure 19 is pre-collision if and only if g ≥ l .Proof. Using the “10” construction of Section 2.1.1, and viewing the configura-tion as a sequence of 1’s and 0’s, we recall that we must count all instances of10’s, remove them, and repeat the process to get a sequence of counts.If g ≥ l , then one has enough 0’s between the two blocks of 1’s to get a sequence( p i ) ki =1 : p i = (cid:26) i ≤ l l < i ≤ k The resulting Young diagram has l boxes in the bottom row and ( k − l ) + l = k boxes in the top row. Therefore, if g ≥ l , then the box-ball state in Figure 19 ispre-collision.Conversely, if g < l , one would exhaust the g zeroes in between the two blocksbefore finishing counting off the 1’s in the l -block. At that point, the remaining k − g balls on the left would form a single block with the l − g balls on the right.The resulting sequence would be ( p i ) ki =1 , where p i = (cid:26) i ≤ g g < i ≤ k + l − g The Young diagram here would have g boxes in its bottom row and k + l − g boxes in its top row. Since g < l < k , g is not equal to k or l , so the block sizesin such a state would not be indicative of the soliton structure. Therefore, sucha configuration is in its collision phase.Before proving the 2-soliton result, we characterise the inception of a collisionphase. Lemma 5.14.
Taking the box-ball configuration in Figure 19 to be the time t = 0 state, assuming g ≥ l , the state at time t max := (cid:22) l − gl − k (cid:23) is the last time prior to the collision, any time beyond this is either part of thecollision phase or a time for which the sorting has concluded. Additionally, thisis the unique time for which the gap lies in the interval [ l, k ) . roof. Prior to the collision, the k -block moves forwards (reducing the gap by) k units, and the l -block moves forwards (increasing the gap by) l units. Therefore,for each time-step prior to the collision, the gap becomes g + t ( l − k ) . To then be pre-collision at time t , we require g + t ( l − k ) ≥ l. The maximal such t is t max given in the theorem.At this time, the gap is given by g + t max ( l − k ) = g + ( l − k ) (cid:22) l − gl − k (cid:23) < g + ( l − k ) (cid:18) l − gl − k − (cid:19) = k. The next gap would then be strictly less than k + ( l − k ) = l , hence it wouldnot lie in the correct interval.In proving the main theorem of this section, we assume that we are beginningwith the configuration in Figure 19, with l ≤ g < k , so that we are at the lastpre-collision stage. Theorem 5.15.
Take a box-ball system consisting of just two blocks of adjacentballs, subject to the following:1. The left-most block has k balls.2. The right-most block has l balls.3. k > l
4. The two blocks are separated by at least l empty boxes.After sufficiently many time steps of the box-ball evolution, after the blockshave collided and ordered themselves, the k -block will have experienced a phaseshift of +2 min( k, l ) = 2 l , and the l -block will have experienced a phase shift of − k, l ) = − l .Proof. To begin the proof, we apply the augmentation map to the box-ballsystem to obtain the following ghost-box-ball configuration in G :54 · ·· · · · · · · · · · · · k l ≤ g < k l Figure 20: The augmentation of the canonical representative of a ( k, l ) 2-soliton box-ballsystem.
By Theorem 5.6, studying this augmented ghost-box-ball system for subsequenttime-steps is equivalent to the corresponding study for the original box-ball sys-tem.Coordinatising this ghost-box-ball system yields the coordinates:( ∞ , , , k, g, l, ∞ ) . Since this is in G , we may apply Corollary 4.33: evolving these coordinates byany number of iterations of χ , and then producing the corresponding ghost-box-ball system will yield the same result as iterating the ghost-box-ball algorithmon the GBBS in Figure 20.Lining these up with the ghost-box-ball coordinates for the evolution equations,we have W = ∞ , Q = 0 , W = 1 , Q = k, W = g, Q = l, W = ∞ . We recall the evolution rules below: W t +10 = W t +13 = ∞ W t +1 n = Q tn +1 + W tn − Q t +1 n , n = 1 , . . . , Q t +1 n = min W tn , n X j =1 Q tj − n − X j =1 Q t +1 j , n = 1 , . . . , , Studying the above equations, we make the following observation/simplificationsto the above: • We see that Q t = 0 for all t because Q t +11 = min( ∞ , Q t ) = Q t and Q = 0. • Since Q t +11 = 0, it also follows that W t +11 = Q t + W t . • Since Q t = Q t +11 = 0, we have Q t +12 = min( W t , Q t ). • Since W t = ∞ , we have Q t +13 = Q t + Q t − Q t +12 .Thus, at time t = 1, we have: W = ∞ , Q = 0 , W = k +1 , Q = g, W = l, Q = k + l − g, W = ∞ .
55t time t = 2, we have: W = ∞ , Q = 0 , W = k + g +1 , Q = l, W = k + l − g, Q = k, W = ∞ . At this point, the gap between the two blocks is k + l − g > min( k, l ) = l and theshorter block is behind, so we know that the collision and sorting phenomenonhas resolved.In the absence of collisions, a block travels with velocity equal to its length.From this, we deduce that W t = k + g + 1 + l ( t − , Q t = l, W t = k + l − g + ( t − k − l ) , Q t = k (5.3)for t ≥ th box, with the next box to the right labelled as the 1 st box, and so on. Belowwe demonstrate this by showing the label for the first box in each block afterthe initial filled ghost in Figure 20: · · ·· · · · · · · · · · · · k + 1 k + g + 1 k + g + l + 1 ↓ ↓ ↓ ↓ ↓ Figure 21: The initial ghost-box-ball state with numbering relative to the filled ghost.
In the above, we see that the position of the first ball in the first block is W = 1and the position of the first ball in the second block is W + Q + W = 1 + k + g. This holds in general since W represents the number of spaces between thefilled ghost and the first ball, and the position of the first ball of the secondblock is the number of initial empty boxes ( W ) plus the number of balls in thefirst block ( Q ) plus the number of empty spaces between the first and secondblock ( W ).Since the phase shift pertains to the positions of the blocks, the quantities ofinterest will be W t and W t + Q t + W t . From Equation 5.3 we have for t ≥ W t = k + g + 1 + l ( t − , W t + Q t + W t = 2 l + kt + 1 . k -block overtake the l -block and one would have: ˜ W t = 1 + k + g + lt because the l -block was initially 1 + k + g boxes from the filled ghost. Here, weuse ˜ W t to distinguish between the “would-be” value and W t (the actual value).If t is sufficiently large, the k -block would be 1 + kt spaces away from the filledghost. Therefore, ˜ W t + ˜ Q t + ˜ W t = 1 + kt. We are finally able to reveal the phase shifts. For the k -block, we look at( W t + Q t + W t ) − ( ˜ W t + ˜ Q t + ˜ W t ) = (2 l + kt + 1) − (1 + kt ) = 2 l. For the l -block, we look at W t − ˜ W t = ( k + g + 1 + l ( t − − (1 + k + g + lt ) = − l. This theorem extends to the following conjectured formula for the phase shiftsfor box-ball systems with any number of blocks.
Conjecture 5.16.
If one has a box-ball system with a total of n blocks, with Q k -many balls in the k -th block, and with blocks separated sufficiently so as tobe able to identify the asymptotic soliton structure by simply ordering ( Q k ) nk =1 .After sufficiently many time-steps have passed (i.e. after the blocks have finishedall collisions), the k -th block will have experienced a total phase shift of X j>kQ j 6. Conclusions Our main results, detailed in Theorem 4.28, provide a complete, rigorousand comparatively simple correspondence between Schensted insertion and aparticle system of box-ball type. This opens the door for further connectionsto algorithms, dynamics and integrable systems theory. To be sure some ofthese connections have been noticed in the literature before; however, we be-lieve the simplicity of our correspondence provides avenues for deeper insightsand broader connections. The results in this paper concerned what may becharacterized as discrete time - discrete space systems.The connections we refer to here have to do with passing to continuousversions, going both forwards and back. Some of that is already evident inthe passage from RSK to gRSK (which is a discrete time - continuous spacesystem) described in Section 3.3. gRSK may in fact be reformulated as a matrixfactorization dynamics as described in Section 3.3.1. It is also known [Hi77]that gRSK is related to a discrete-time form of the Toda lattice, a well-knowncontinuous time - continuous space integrable system. In [R20] it was shown howto relate gRSK in terms of matrix factorization to a standard lower-upper matrixfactorization representation for the discrete-time Toda Lattice. The latter suchrepresentation was developed and studied by Symes [Sy80] and Deift-Nanda-Tomei [DNT83]. Our results here make it possible to make all these connectionsprecise. Moreover they provide a means to relate all the solitonic properties ofour ghost box-ball system to those of the integrable Toda lattice. A particularinstance of this will be to deduce the phase shift formulae established in Section5.2 directly from the classical continuous time - continuous space phase shift58ormulae [Mo75].Going in the other direction, from gRSK to the continuous time Toda lattice,we can compare our approach with that of O’Connell and collaborators [O12][O13] [COSZ14]. That work used this direction as a means to push forwardrandom algorithmic structures to the setting of random directed polymers ina way that remarkably connects to semiclassical limits of the quantum Todalattice. 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