Multivariate Juggling Probabilities

We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities. The normalization factor in one case can be explicitly written as a homogeneous symmetric polynomial. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in bounded time.


Introduction
Although juggling as a human endeavour has been around since time immemorial, it is fairly recently that mathematicians have taken an active interest in exploring the field.Combinatorialists became interested in juggling towards the end of the last century after an article in the Amer.Math.Monthly by Buhler, Eisenbud, Graham and Wright [BEGW94], where they enumerate what they call juggling sequences and relate it to other known combinatorial structures.Since then, their results have been q-ified [ER96] and further refined in various ways [Sta97,Sta02,CG07,CG08,BG10].Other connections between juggling and mathematics appear for instance in algebraic geometry [DM,KLS13].A mathematical history of juggling is given in the fascinating book by Polster [Pol03].
Probabilists, on the other hand, are newer to this game, possibly because no one is able to perform totally random juggling.Coincidentally, it was another popular article in the Amer.Math.Monthly by Warrington [War05] on natural stochastic models inspired by juggling that got the attention of a few other members of the community.Exact combinatorial formulas for the stationary distribution of these models were found, and proved by a beautiful argument involving an auxiliary model defined on a larger state space whose elements may be viewed as set partitions, for which the stationary distribution is the uniform distribution.
The simplest model considered by Warrington, where the number of balls is conserved, has been generalized by Leskelä and Varpanen as the so-called Juggler's Exclusion Process (JEP), where the balls can be thrown arbitrarily high so that the state space is infinite [LV12].These authors showed that the JEP converges to a (unique) stationary distribution under very mild assumptions, could obtain an exact expression for this stationary distribution in the particular case of a "memoryless" (geometric) height distribution, and noted an intriguing phenomenon of "ultrafast convergence" in this case.More recently, a q-deformation of Warrington's original finite model was considered by Engström, Leskelä and Varpanen [ELV13], who also obtained exact expressions for the stationary distribution via an enriched chain formulated in terms of rook placements (in bijection with some set partitions).
In this paper, we provide multivariate deformations of all the models introduced in [War05], namely juggling, add-drop juggling and annihilation juggling.Furthermore, in the case of the juggling model with a conserved number of balls, we investigate the limiting case where balls can be thrown arbitrarily high, which corresponds to the so-called infinite juggling model suggested by Warrington.We also consider the limiting case where the number of balls tends to infinity.In all these cases, we obtain an exact formula for the stationary distribution (in particular, in the case where balls can be thrown arbitrarily high, we find an "exactly solvable" instance of the general JEP of [LV12] with countably many parameters).Our proofs were obtained using two approaches.The direct approach involves guessing the general formulas (for instance by a computation for small system sizes) and then proving the results in a straightforward manner by considering the juggling process itself.The other approach is more combinatorial, and consists in introducing an enriched chain whose stationary distribution is simpler, and which yields the original chain by a projection or "lumping" procedure.
The rest of the paper is organized as follows.In Section 2, we concentrate on the simplest version of our model, which we call the Multivariate Juggling Markov Chain (MJMC).The model is defined in Section 2.1, and we also discuss the uniqueness of the stationary distribution (a technicality deemed obvious in [War05]).The expression for the stationary distribution of the Markov chain is stated in Section 2.2.For pedagogical purposes, we decide to provide two independent proofs by the direct and the combinatorial approaches.The direct proof is given in Section 2.3, via an interesting reformulation of the MJMC in terms of integer partitions restricted to lie within a rectangle.The combinatorial proof comes in Section 2.4 and involves set partitions with a prescribed number of elements and blocks.We then turn to extended models.Extensions to infinite state spaces are considered in Section 3: the case of a finite number of balls but unbounded heights (UMJMC) in discussed in Section 3.1, while the case of an infinite number of balls (IMJMC) is considered in Section 3.2.In both cases, we find the stationary measure by the direct approach.Extensions to a fluctuating number of balls (but with a finite state space) are considered in Section 4: we provide the multivariate extension of the add-drop and the annihilation models introduced in [War05], in the respective Sections 4.1 and 4.2.These models have the same transition graph, only the transitions probabilities differ.In both cases we find the stationary distribution by the combinatorial approach.In the case of the annihilation model (precisely defined in Section 4.2.1),we further observe the interesting property that the stationary distribution is attained in finite time (Section 4.2.2), and we then discuss the remarkable algebraic structure of the model (Section 4.2.3).We end with some remarks and questions for future study in Section 5.
The claims of this paper can be verified by downloading the Maple TM program RandomJuggling either from the arXiv source or the first author's (A.A.) homepage.
2. The finite Multiparameter Juggling Markov Chain 2.1.Definition.In this section, we introduce our juggling model in the simplest setting, i.e. a Markov chain on a finite state space.We start by explaining the model in colloquial terms, and refer to [War05] for further motivation.Consider a person, called Magnus with no loss of generality, who is juggling with a fixed finite number of balls.Time is discretized in, say, steps of one second and we assume that, at each second, Magnus is able to catch at most one ball, and then throws it back immediately.Besides this limitation Magnus juggles perfectly, i.e. in such a way that the ball will always return to him after some time (an imperfect juggler could drop a ball or throw it in a wrong direction, for instance).Magnus controls the velocity at which he sends the ball, which determines how long it will take for the ball to return to him.We suppose for now that the launch velocity is bounded or, in other words, that the number of seconds before the ball returns to Magnus is bounded by an integer h.
Ignoring further spatial constraints, a simplified description of the state of the balls at a given time consists in associating to each ball the (integer) number of seconds remaining before it is caught by Magnus.This is now known as the siteswap notation.Of course, to be able to juggle for an indefinite amount of time, Magnus must choose the successive launch velocities in such a way that no two balls arrive to him at the same time.Thus the numbers associated to different balls shall be distinct and, treating the balls as indistinguishable, there are h possible ball states.It is convenient to think of a state as a configuration of non overlapping particles on a one-dimensional lattice with h sites, where the i-th site (read from the left) is occupied if and only if a ball is scheduled to arrive i seconds in the future.However, beware that sites do no correspond to actual spatial positions, but to the "remaining flight times" of the balls.We denote by k = h − the number of empty (unoccupied) sites.
In this language, the time evolution of a state is easy to describe: at each time step, all particles are moved one site to the left.If there is no particle on the first site (i.e.Magnus catches no ball), then nothing else has to be done.Otherwise the particle on the first site, which would exit the lattice if moved to the left, is instead taken away and reinserted at one of the k + 1 available (empty) sites on the lattice (determined by the launch velocity chosen by Magnus).This defines the transition graph of our model, illustrated on Figure 1 for h = 4 and = k = 2 (ignoring edge labels for now).
We now assume that Magnus juggles at random: each reinsertion is made randomly at one of the k + 1 available sites.In our model, we assume that the reinsertion is made at the i-th available site (read from the left) with probability x i−1 , independently of the past, so that our model is a Markov chain.Here, x 0 , . . ., x k are fixed nonnegative real numbers such that x 0 + • • • + x k = 1.This defines the Multivariate Juggling Markov Chain (MJMC), which generalizes the model considered in [War05], obtained by taking ), but is a particular case of the general "Juggling Exclusion Process" defined in [LV12]: beyond the extension to infinitely many empty sites which we will discuss in the next section, the main difference is that, in the model of Leskelä and Varpanen, the x i are allowed to depend on the current state.We now provide a more formal mathematical definition of the MJMC.Following Warrington's notation, let St h denote the set of words of length h on the alphabet {•, •}, and let St h,k ⊂ St h be the subset of words containing exactly k occurrences of •.For A ∈ St h,k+1 and i ∈ {0, . . ., k}, we let T i (A) ∈ St h,k be the word obtained by replacing the (i + 1)-th occurrence of • in A by •.
Definition 2.1.Given h, k nonnegative integers such that h ≥ k, and x 0 , . . ., x k nonnegative real numbers such that x 0 + • • • + x k = 1, the Multivariate Juggling Markov Chain is the Markov chain on the state space St h,k for which the transition Example 2.2. Figure 1 illustrates the Markov chain in the case h = 4, k = 2, and the transition matrix in the basis 2 ) is a left eigenvector with eigenvalue 1.
Remark 2.3.To compare our model with that of [LV12], note that St h,k may be canonically identified with the set of subsets of {1, . . ., h} with = h − k elements (corresponding to the positions of the occupied sites).For two such subsets S and S , the transition probability from S to S reads (2.3) where S − 1 is the set whose entries are those of S with 1 subtracted.
Proposition 2.4.If x 0 > 0, then the MJMC has a unique closed communicating class, whose all states are aperiodic.If furthermore x k > 0, then the MJMC is irreducible.
Proof.Assume that x 0 > 0 and let E = • • k be the "lowest" state.It is clear that, starting from any initial state, we may obtain E by a sequence of transitions of probabilities x 0 or 1, simply by doing any reinsertion at the first available site.Thus there is a unique closed communicating class, containing E. Since E jumps to itself with probability x 0 , all recurrent states are aperiodic.Assume now that x k > 0: to prove irreducibility we simply need to show that any state A = a 1 . . .a h ∈ St h,k may be obtained from E. For i ∈ {0, . . ., h}, we let Thus A may be obtained from E by a sequence of possible transitions.
Remark 2.5.When x 0 = 0, the chain may have several closed communicating classes.For example, see Figure 1 and let x 0 = x 1 = 0.

Stationary distribution.
From now on we assume x 0 > 0. By Proposition 2.4, the MJMC admits a unique stationary probability distribution.Our main result is an explicit form for it, given as follows.
Theorem 2.6.The stationary distribution π of the MJMC is given by and where Z h,k ≡ Z h,k (x 0 , . . ., x k ) is determined by the condition that π be a probability distribution.
We will provide two proofs of this theorem, as they both lead to interesting generalizations to be studied later in Sections 3 and 4. The first proof, presented in Section 2.3, is a rather direct computational check that (2.5) indeed defines a stationary measure of the MJMC (but we find it clearer to first reformulate the chain in terms of integer partitions).The second proof, presented in Section 2.4, is more combinatorial and generalizes the approach of [War05]: it consists in defining an "enriched" Markov chain on a larger state space, where the stationary probability of a given state will be given by a monomial in x 0 , . . ., x k .The MJMC is then recovered by lumping together states of the enriched chain, which will explain the form of (2.5).
Before proceeding to the proofs, let us discuss a few interesting particular cases, and properties of the normalization factor Z h,k .
Example 2.8.If the reinsertion site is chosen according to a geometric distribution of parameter q conditionned to be smaller than or equal to k + 1, i.e. if we take x i = (1 − q)q i /(1 − q k+1 ), i = 0, . . ., k, then we recover the so-called bounded geometric JEP considered in [ELV13].It is simple exercise to check that we recover [ELV13, Theorem 2.1] from Theorem 2.6 (see also Corollary 2.12 below for the normalization factor).
Example 2.9.A slight variant of the previous example consists in picking the reinsertion site according to a "truncated" geometric distribution of pararameter q, i.e. taking x i = (1 − q)q i for i = 0, . . ., k − 1 and x k = q k .In that case, the stationary probability of (see again Corollary 2.12 below for the normalization factor).
Regarding Z h,k , observe first that the expression allows to define it for any x 0 , . . ., x k (not necessarily positive or subject to the condition x 0 + • • • + x k = 1) and h, k ≥ 0. It vanishes for h < k and is otherwise a homogeneous polynomial of degree = h − k.
Proposition 2.10.We have with h the complete homogeneous symmetric polynomial of degree = h − k, and with the shorthand notation (2.9) x j , m = 0, . . ., k.
Again, there are several ways to establish this result.Perhaps the most combinatorial explanation comes from the integer partition approach of Section 2.3, see Remark 2.17 below.But, starting from the definition (2.7), we may easily check (2.8) by induction, using the following: Lemma 2.11.For h, k ≥ 0, we have Proof.Obtained immediately by distinguishing whether B ∈ St h,k starts with a • or a • (for h < k all terms vanish as wanted).
We readily verify by induction and homogeneity some explicit specializations: Corollary 2.12.We have where • • denotes Stirling numbers of the second kind, • • q their q-analogues as defined in [Gou61] and • • q q-binomial coefficients.Using homogeneity, we deduce the normalization factors for Examples 2.7 and 2.8.
For completeness, we mention another recursion relation for Z h,k of a different nature, as we remove x k instead of x 0 .Proposition 2.13.For h, k ≥ 0, we have where, by convention, Proposition 2.14.For λ = (λ 1 , λ 2 , . . ., λ ) and µ two partitions in Par k, , the transition probability from λ to µ reads The proof of this proposition is left to the reader.The interest of the reformulation in terms of integer partitions is that the stationary distribution of the MJMC takes a particularly simple form: indeed, Theorem 2.6 is equivalent to Theorem 2.15.The stationary distribution π of the MJMC is given by where λ = (λ 1 , . . ., λ ) is an element of Par k, and where the y m are as in (2.9).
Example 2.16.We may now revisit Examples 2.7, 2.8 and 2.9 in the language of integer partitions.The corresponding stationary probabilities of λ ∈ Par k, read Remark 2.17.one can immediately recover Proposition 2.10 from Theorem 2.15, as where the m i correspond to the part multiplicities.A slight extension of this reasoning consists in restricting the sum to partitions λ such that, say, λ j = n (with 1 ≤ j ≤ and 0 ≤ n ≤ k), so as to obtain the probability that the j-th part in the stationary distribution π is n: More generally, the joint distribution for a fixed number of parts reads 2.4.Enriched Markov chain on set partitions.We now provide another, more combinatorial, proof of Theorem 2.6 whose rough idea goes as follows: consider the stationary probability (2.5) and expand the product in the right hand side as a sum of monomials in the x i .We will interpret each of these monomials as the stationary probability of an "enriched" state belonging to the larger state space of another Markov chain.The MJMC will then be obtained as a projection of this enriched chain (see e.g.[LPW09, Section 2.3.1] for a definition of this notion).
2.4.1.Definitions and basic properties.Our construction is a generalization of that of Warrington's.Rather than working with "landing/throwing-states", we prefer to work directly in the language of set partitions, see [War05, Lemma 2].Recalling that h stands for the total number of sites and k for the number of empty sites in the particle picture of the MJMC, let us introduce the shorthand notations (2.26) An enriched state will then be a partition of the set {1, 2, . . ., H} into K subsets or blocks.We denote by S(H, K) the set of enriched states, and recall that H K = |S(H, K)| is a Stirling number of the second kind.To each enriched state σ, we associate a word ψ(σ) = a 1 . . .a h by setting, for all i between 1 and h, a i = • if i is a block maximum of σ (i.e. the largest element of its block in σ), and We now define the enriched Markov chain on S(H, K), which requires some notations.For an enriched state σ, we denote by σ ↓ the partition of the set {1, . . ., h} obtained by removing 1 from σ (i.e.removing 1 from its block in σ, and removing this block from σ if it becomes empty), and shifting all the remaining elements of all blocks down by 1.Note that σ → σ ↓ is a mapping from S(H, K) to S(h, k)∪S(h, K) (which is surjective).For τ ∈ S(h, K) and i ∈ {0, . . ., k}, we denote by I i (τ ) the set partition of {1, . . ., H} obtained by inserting H into the (i + 1)-th block of τ , where the blocks are numbered by ascending order of their maxima (i.e. the first block has the smallest maximum among all blocks, etc. -this differs from the so-called standard form which consists in ordering blocks by ascending order of their minima).Note that I i (τ ) ∈ S(H, K) and that the mapping (τ, i) → I i (τ ) is injective.
Definition 2.18.The enriched chain is the Markov chain on S(H, K) for which the transition probability from σ to τ is given by Proposition 2.21.For x 0 > 0, the enriched chain has a unique closed communicating class, whose all states are aperiodic.The chain is irreducible if and only if all x i s are nonzero.
Proof.The first assertion is proved in the same way as that of Proposition 2.4, the "lowest" state being now the set partition (2.28) Ẽ = {{1, . . ., H} ∩ (j + KN)} j=1,...,K (precisely, starting from any initial state, we eventually obtain Ẽ by applying only transitions of probabilities x 0 and 1).
For the second assertion, suppose that x i > 0 for all i: we want to prove that the transition graph is strongly connected.We first replace the transitions of probability 1 by K transitions of probabilities x 0 , . . ., x k , so that each state has K outgoing transitions, counting transitions to itself.It is then not difficult to check from the definition of the enriched chain that each state also has K incoming transitions.It follows that for any set of states, the number of incoming and outgoing transitions are the same, thus all communicating classes are closed.As there is a unique closed communicating class, irreducibility follows.Conversely, if x i = 0 for some i, then clearly some set partitions, e.g.
2.4.2.Projection, stationary distribution and second proof of Theorem 2.6.From the surjection ψ introduced above, we define a rectangular matrix Ψ with rows indexed by elements of S(H, K) and columns indexed by elements of St h,k in the obvious manner (namely a coefficient of Ψ is 1 if its column index is the image by ψ of its row index, and 0 otherwise).Enriched states (set partitions) are represented via "arches".Red, green and grey arrows represent transitions of respective probability x 0 , x 1 and 1. Arches covering 1 or 2 blocks are represented in green or red respectively.The reader may easily check that attaching a weight x 0 (resp.x 1 ) to each red (resp.green) arch yields the stationary measure of the enriched chain.Enriched states in a same column map to the same MJMC state (displayed below) via ψ.
For clarity, the proofs of all lemmas in this section are postponed to Section 2.4.3.Note that, if we view the preimages by ψ of elements of St h,k as equivalences classes for an equivalence relation on S(H, K), then (2.29) shows that the MJMC is the projection of the enriched chain [LPW09, Lemma 2.5].Noting that the sum of each row of Ψ is 1, we immediately deduce: Corollary 2.23.We have where π is the stationary probability distribution of the enriched chain (viewed as a row vector), and π is that of the MJMC.
Our route to Theorem 2.6 is now clear.We want to find an explicit expression for π, then compute πΨ.We first need to introduce some definitions and notations.Definition 2.24.Let σ be an enriched state and s, t two integers such that 1 ≤ s < t ≤ H.We say that the pair (s, t) is an arch of σ if s and t belong to the same block β of σ, while no integer strictly between s and t belongs to β.
Note that σ ∈ S(H, K) has exactly = H − K arches.
Notation 2.25.For 1 ≤ s < t ≤ H, we denote by C σ (s, t) the number of blocks containing at least one element in {s, s + 1, . . ., t − 1, t} (when (s, t) is an arch, then we say that these blocks are those covered by (s, t)).
We are now ready to express the stationary distribution of the enriched chain, see again Figure 3 for an illustration in the case H = 4, K = 2.
Lemma 2.26.For σ ∈ S(H, K), the monomial defines an unnormalized stationary measure of the enriched chain.
Remark 2.27.It is here natural to introduce the notation z i = x K−i , i = 1, . . ., K, so that z i is simply the weight per arch covering i blocks.In the juggling language, z i is the probability of doing an insertion at the i-th available site starting from the right.This notation will be useful for the "add-drop" and "annihilation" generalizations of the MJMC, see Sections 4.1 and 4.2.However, the x i notation is more convenient to study the k → ∞ limit, as we do in Sections 3.1 and 3.2.
Noting that, for Ẽ as in (2.28), we have w( Ẽ) = x 0 , we find that is positive whenever x 0 > 0, so that π = w/Z h,k is the stationary probability distribution of the enriched chain.For B ∈ St h,k , we set that is w = wΨ.This implies that π = w/Z h,k is the stationary probability distribution of the MJMC by Corollary 2.23.Theorem 2.6 then follows immediately from: Lemma 2.28. where Example 2.29.Returning again to the case H = 4, K = 2 (i.e.h = 3, k = 1) illustrated on Figure 3, we find Remark 2.30.The unnormalized weights w(σ) are well-defined for arbitrary x 0 , . . ., x k (not necessarily positive or subject to the condition , we recover Warrington's combinatorial proof of the identity (2.11).For x i = q k−i , we have w(σ) = q N (σ) with (2.36) The identity (2.12) shows that N (σ) is a so-called Mahonian statistic on set partitions, which seems different from the inversion number and the major index [Sag91].
(since inserting H into the (i + 1)-th block of σ ↓ transforms its maximum into a non-maximum, which means replacing the , then, by the definition of the transition matrices and (2.37), (2.39) Otherwise, if {1} / ∈ σ (i.e. a 1 = •), we may write where we use (2.38) to go from the first to the second line.In both cases, we have the wanted relation.
Proof of Lemma 2.26: We need to show that (2.41) which is done by considering the possible predecessors of σ in the enriched chain.
If {H} ∈ σ then σ has a unique precedessor τ such that σ = τ ↓ ∪ {H}.The arches of σ and τ are clearly in one-to-one correspondence, and their weights are unchanged, thus w(σ) = w(τ ) as wanted.Otherwise, there exists a unique pair (ρ, i) ∈ S(h, K) × {0, . . ., k} such that σ = I i (ρ) and it is easily seen that w(σ) = x i w(ρ) (since inserting H in the (i + 1)-th block of ρ amounts to creating an arch covering K − i blocks, the other arch weights being unaffected).The predecessors of σ are the τ such that τ ↓ = ρ and {1} / ∈ τ , and we have Pτ,σ = x i regardless of τ .There are exactly K predecessors, whose weights are x j w(ρ) with j ∈ {0, . . ., k} (indeed all these predecessors are obtained by shifting all elements of all blocks of ρ up by one, which preserves the arch weights, then inserting 1 into one of the K blocks, which creates a new arch covering an arbitrary number of blocks between 1 and K).From the condition x 0 + . . .+ x k = 1 we conclude that (2.41) holds as wanted.
Proof of Lemma 2.28: We proceed by induction on = h − k = H − K.For = 0, the statement is true since B = • h has only one preimage, namely the set partition consisting of singletons {{1}, {2}, . . ., {H}} which has no arch, thus has a weight 1 consistent with (2.34).Let us now assume that > 0 and consider the smallest i such that b Consider τ ∈ ψ −1 (B): it is easily seen from the definition of ψ that, for all j < i, the singleton {j} is necessarily a block of τ , but the block β containing i contains at least another (larger) element, thus τ has an arch of the form (i, i ) with weight wi,i = x n for some n ≥ E i (B).

Extensions to infinite state spaces
In this section, we discuss extensions of the Multivariate Juggling Markov Chain to an infinite setting.More precisely, we first let the number of available sites k tend to infinity, keeping the number of balls fixed.This is the Unbounded MJMC discussed in Section 3.1.Further, we consider the case where the number of balls tends to infinity (with k infinite or not), which corresponds to the Infinite MJMC discussed in Section 3.2.
3.1.Unbounded heights.As suggested in the conclusion of [War05], a first natural extension is to allow Magnus to throw balls arbitrarily high, so that the ball flight times are unbounded.This corresponds to taking the limit h → ∞ of the MJMC, keeping the number of balls fixed.In the particle picture, the sites are now labelled by the set of positive integers, and exactly sites are occupied by a particle.The time evolution is essentially unchanged: at each time step, all particles are moved one site to the left, and if there was a particle on the first site, it is reinserted at an available site anywhere on the lattice.We keep the MJMC prescription of choosing the i-th available site with probability x i−1 , but since there are now infinitely many available sites, we have an infinite sequence (x i ) i≥0 of nonnegative real numbers such that ∞ i=0 x i = 1.Formally, states can be viewed as infinite words on the alphabet {•, •} containing exactly occurrences of •.We denote by St ( ) the set of such states (which can be viewed as the direct limit of the set sequence (St h,h− ) h≥ ).For A ∈ St ( −1) and i a nonnegative integer, we let T i (A) ∈ St ( ) be the word obtained by replacing the (i + 1)-th occurrence of • in A by •.Definition 3.1.Given a nonnegative integer and a sequence (x i ) i≥0 of nonnegative real numbers such that ∞ i=0 x i = 1, the Unbounded Multivariate Juggling Markov Chain (UMJMC) is the Markov chain on the state space St ( ) for which the transition probability from Again, we might identify St ( ) with the set of -element subsets of N (corresponding to the positions of the occupied sites).For two such subsets S and S , the UMJMC transition probability from S to S is still given by (2.3).Our model is thus a particular case of the general model considered in [LV12].The following proposition is an immediate extension of Proposition 2.4.Proposition 3.2.The UMJMC is irreducible and aperiodic whenever x 0 and infinitely many x i are nonzero.
Note that, if k = sup{i : x i > 0} is finite, then no insertion is ever made at a position larger than k.Thus, the states with particles at positions larger than k are transient, and upon removing them we recover the finite MJMC on St +k,k .From now on, we assume that x 0 > 0 and sup{i : x i > 0} = +∞.We are again interested in the stationary distribution, but the fact that the state space is now countably infinite requires a bit of care.Still, we might extend Theorem 2.6 in the following form.
Theorem 3.3.The unique invariant measure (up to constant of proportionality) of the UMJMC is given by where We prove this theorem by a straightforward extension of the discussion of Section 2.3 (it seems more tedious to extend that of Section 2.4).In the integer partition language, the extension simply consists in lifting the bound on part sizes.We denote by Par ∞, the set of integer partitions with at most parts and unbounded part sizes (which we can view as nonincreasing sequences of nonnegative integers).This set is naturally identified with St ( ) : given a state in St ( ) , we denote by s 1 < s 2 < . . .< s −1 < s the positions of •, so that the corresponding integer partition is (s − , s −1 − ( − 1), . . ., s 2 − 2, s 1 − 1).We let the reader verify that, in the integer partition language, the UMJMC transitions are still given by (2.18).Theorem 3.3 may then be reformulated as follows.
We have exhibited an invariant measure of the UMJMC which is clearly σ-finite.We might wonder whether it is actually finite, so that it may be normalized into a probability distribution.
Proposition 3.5.The invariant measure w of the UMJMC is finite if and only if in which case its total mass reads (3.6) Z ( ) = h (y 0 , y 1 , y 2 , . ..).
Proof.Let Z ( ) ∈ [0, ∞] be the total mass of w.Observe that (3.7) thus the first assertion is obviously true for = 1.It remains true for general as we have the inequalities since Par ∞, contains the set of partitions with at most one nonzero part (which has mass Z (1) by (3.3) and the fact that y 0 = 1), but may be viewed as a subset of N , for which we have λ∈N w(λ) = Z (1) by (3.3).The identity (3.6) is obtained by letting k → ∞ in (2.23), since (Par k, ) k≥0 forms an increasing family of sets with union Par ∞, so that Z k+ ,k Z ( ) .Note that, when y m < ∞, the mapping f → f (y 0 , y 1 , y 2 , . ..) forms a well-defined real specialization of the ring of symmetric functions, as seen for instance by noting that the power sums are specialized to finite real numbers.
By standard results from the theory of Markov chains, see e.g.[LG06, Chapter 13] or [LPW09, Chapter 21], we deduce: Corollary 3.6.The UMJMC is positive recurrent if and only if (3.5) holds.In that case, there is a unique stationary probability distribution, and the chain started from any initial state converges to it in total variation as time tends to infinity.
Remark 3.8.It is interesting to note that (3.5) is a necessary and sufficient condition for the UMJMC to be uniformly integrable in the sense of [LV12].When (3.5) does not hold, the chain may either be null recurrent or transient.For = 1, the state with the first site occupied is clearly recurrent, thus the chain is null recurrent.Figuring out the situation for > 1 is an intriguing question.By somewhat heuristic arguments inspired by Polyá's theorem on recurrence of random walks, we expect that, for a sequence x i decaying asymptotically as i −1−α with α ∈ (0, 1) (so that A formal proof of this statement is beyond the scope of this paper. 3.2.Infinitely many balls.We now consider another limit in which Magnus juggles with infinitely many balls ( → ∞).Let us first heuristically discuss this limit in the particle picture: since the particles tend to accumulate on the left side of the lattice (i.e.balls tend to have low remaining flight times), we expect that, as becomes large, all sites at a finite distance from the first site will be occupied with high probability (in particular, Magnus receives a ball at every time step), and the first available site will then be typically at a position of order .It is convenient to relabel the sites by arbitrary (non necessarily positive) integers, so that the leftmost site has label − , and the lowest state corresponds to having particles on sites − , − + 1, . . ., −1.In the limit → ∞, the leftmost site is sent to −∞, and the lowest state corresponds to having all negative sites filled and all nonnegative sites empty.A general state is obtain by moving finitely many particles from occupied to empty sites, in other words it is a bi-infinite word (such a word is sometimes called a "Maya diagram").The time evolution is now easy to describe: at each time step, all particles are moved one site to the left and one particle is reinserted at an available site (so that the condition (3.9) is preserved).Again, we keep the MJMC prescription of picking the i-th available site from the left with probability x i−1 (this is well-defined since the set of available positions is bounded from below).Rather than writing down the transitions formally, we prefer to work directly in the language of integer partitions.It is well-known that bi-infinite words subject to (3.9) are in one-to-one correspondence with arbitrary integer partitions (i.e.nonincreasing sequences of integers that vanish eventually).We denote by Par the set of integer partitions and, for λ = (λ j ) j≥1 ∈ Par and i ≥ 0, we set (3.10) with j the smallest index such that λ j ≤ i.This is the Par equivalent of the above time evolution with insertion at the (i + 1)-th available site.
Definition 3.9.Given a sequence (x i ) i≥0 of nonnegative real numbers such that ∞ i=0 x i = 1, the Infinite Multivariate Juggling Markov Chain (IMJMC) is the Markov chain on the state space Par for which the transition probability from λ to µ reads (3.11) The existence and uniqueness (up to normalization) of an invariant measure of the IMJMC is ensured by the following: Proposition 3.10.If x 0 > 0, then the IMJMC has a unique closed communicating class, whose all states are aperiodic.They are precisely the integer partitions whose parts are smaller than or equal to sup{i : x i > 0}.In particular, if infinitely many x i are nonzero, then the IMJMC is irreducible.
Proof.It is easily seen that, applying repeatedly the transition x 0 , we end up with the empty partition starting from any initial state (which is a fixed point).This proves the first statement.For the second statement, let us prove that, for any k such that x k > 0, we may obtain any partition µ with µ 1 ≤ k from the empty partition using only the transitions x 0 and x k .We proceed by double induction on the number of parts p of µ and on k − µ 1 .If p = 0, then µ is already the empty partition.Otherwise, if k − µ 1 = 0, then µ is obtained from (µ 2 , µ 3 , . ..) (which has one less part) by the transition x k .Finally, if k − µ 1 > 0, then µ is obtained from µ = (µ 1 + 1, . . ., µ p + 1, 0, 0, . ..) (which still has p parts and k − µ 1 < k − µ 1 ) by the transition x 0 .
The expression (3.13) follows from standard considerations on integer partitions, and is finite if and only if y m < ∞, i.e. if (3.5) holds.
Again, when (3.5) does not hold, the IMJMC is either null recurrent or transient, and it would be interesting to know which situation occurs.
Example 3.12.Consider again the geometric case x i = (1 − q)q i , q ∈ (0, 1).Then, the stationary distribution is nothing but the "q size " measure over arbitrary integer partitions.Note that, contrary to the case of finitely many balls (Example 3.7), the phenomenon of ultrafast convergence to stationarity observed in [LV12] cannot occur: the stationary distribution is supported on the set of all integer partitions with arbitrarily many parts, but since we may create at most one new part at each time step, the distribution at a finite time starting from a given initial partition is supported on a strictly smaller set.

Extensions with a fluctuating number of balls
We now consider extensions of the MJMC where the number of balls is not fixed but is allowed to fluctuate.These extensions are the natural multivariate generalizations of the so-called add-drop and annihilation models introduced in [War05, Section 4] (to which we refer for motivations), and we thus keep the same denomination here.Both models are defined on the same state space and have the same transition graph, only the transitions probabilities differ.In both cases, we will provide an exact expression for the stationary distribution, whose validity will be proved by considering the enriched version of the chain.
Since many definitions and notations will be common to both models, we factorize their discussion here.The basic state space of the model will be St h = {•, •} h , with h a fixed nonnegative integer.As hinted in Remark 2.27, it will be convenient here to read a word from right to left.For A ∈ St h and i a nonnegative integer, we let S i (A) be the word obtained by replacing the i-th occurrence of • in A by •, upon reading the word from the right (if i is equal to 0 or larger than the number of occurrences of • in A then we set S i (A) = A by convention).Note that S i (A) = T k−i (A) for A ∈ St h,k and 1 ≤ i ≤ k, with T as in Section 2.1.The state space of the enriched model will be the set S(H) of all set partitions of {1, . . ., H}, with H = h + 1, and we recall the notations from Section 2.4.1:ψ is a surjection from S(H) onto St h and σ → σ ↓ is a mapping from S(H) to S(h).For τ ∈ S(h) and i a nonnegative integer, we let J i (τ ) ∈ S(H) be the set partition obtained by inserting H into the i-th block of τ , now numbered by decreasing order of maxima (if i is equal to 0 or larger than the number of blocks of τ then we set J i (τ ) = τ ∪ {H} by convention).Note that J i (τ ) = I K−i (τ ) for τ ∈ S(h, K) and 1 ≤ i ≤ K. Observe that we have the fundamental "intertwining" relation which is nothing but a compact rewriting of (2.37) and (2.38).
The basic transition graph is defined as follows: for any A = a 1 a 2 . . .a h ∈ St h and i ≥ 0, we have an oriented edge from A to S i (a 2 . . .a h •) (we ignore edge multiplicities).See figure 4 for h = 2. Similarly, the enriched transition graph is obtained by connecting each σ ∈ S(H) to J i (σ ↓ ) for all i ≥ 0. It is not difficult to check that both transition graphs are strongly connected using Proposition 2.4 and Proposition 2.21.Definition 4.1.Given h a nonnegative integer and a = z 0 , z 1 , . . ., z h nonnegative real numbers, the (multivariate) add-drop model is the Markov chain on the state space St h for which the transition probability from A = a 1 a 2 . . .a h to B reads (4.2) P A,B = with k the number of occurrences of • in a 2 . . .a h •.

Warrington's add-drop model is recovered by taking
It is easily seen that, in general, the chain is aperiodic with a unique communicating class whenever a > 0, so that the stationary distribution is unique.
Theorem 4.2.The stationary distribution of the add-drop model is given by where h is the complete homogeneous symmetric polynomial of degree .
Example 4.3.The transition matrix for h = 2 in the lexicographically-ordered basis , and the stationary probabilities are given by the normalized coordinates of the row eigenvector with eigenvalue 1, namely where Theorem 4.2 is the natural multivariate generalization of [War05, Theorem 3, item 1].The parameter a has the physical interpretation of the fugacity for a •.In other words the distribution Π is the grand-canonical version of the stationary distribution π of Theorem 2.6.We prove Theorem 4.2 by a straightforward extension of the construction of Section 2.4, which we now detail.
Definition 4.4.The (multivariate) enriched add-drop model is the Markov chain on the state space S(H) for which the transition probability from σ to τ is given by (4.7) Pσ,τ = with k the number of blocks of σ ↓ .
By (4.1) it is immediate that the add-drop model is indeed the projection of the enriched chain, and furthermore it is easily seen that the enriched chain is aperiodic with a unique communicating class for z 0 = a > 0.
Lemma 4.5.For σ ∈ S(H) with, say, K blocks, the monomial where C σ (s, t) is as in Definition 2.25, defines an unnormalized stationary distribution of the enriched chain.
Proof.We want to show that W (τ ) = σ∈S(H) W (σ) Pσ,τ for any τ .Let τ ↑ be the set partition obtained by removing the element H from τ , so that the possible predecessors of τ are the σ such that σ ↓ = τ ↑ .More precisely, any predecessor is obtained by shifting all elements of τ ↑ up by 1, then inserting the element 1 either as a singleton or into a preexisting block.Hence, there are k + 1 predecessors where k is the number of blocks of τ ↑ , and we readily see that their weights are z i W (τ ↑ ) with i = 0, . . ., k.On the other hand, let j be equal to 0 if {H} is a singleton of τ , or equal to C τ (r, H) where r is such that (r, H) is an arch of τ .We have W (τ ) = z j W (τ ↑ ) and Pσ,τ = z j /(z 0 + • • • + z k ) for any predecessor σ.We deduce that, as wanted, (4.9) Example 4.6.The transition matrix for H = 3 in the ordered basis (123, 1|23, 2|13, 12|3, 1|2|3) is given by (4.10) , and admits the row vector z 2 1 , az 1 , az 2 , az 1 , a 2 as a left eigenvector of eigenvalue 1. Compare this with the stationary distribution in Example 4.3.
Proof of Theorem 4.2: By Lemma 2.28 and Remark 2.27 (note that x Ei(B) = z ψi(B)+1 ), we readily deduce (4.12) The total mass of W is clearly equal to h k=0 a k Z h,k (z k+1 , z k , . . ., z 1 ) with Z h,k the normalization factor of the MJMC defined in Proposition 2.10.

Annihilation juggling.
4.2.1.Definition and stationary distribution.As the reader is by now familiar with our approach, we define the basic model and its enriched version at the same time.
Definition 4.7.Given h a nonnegative integer and z 1 , . . ., z h , z h+1 = a nonnegative real numbers such that z 1 + • • • + z h + a = 1, the (multivariate) annihilation model is the Markov chain on the state space St h for which the transition probability from A = a 1 a 2 . . .a h to B reads (4.13) with k the number of occurrences of • in a 2 . . .a h •.Similarly, the (multivariate) enriched annihilation model is the Markov chain on the state space S(H) for which the transition probability from σ to τ is given by with k the number of blocks of σ ↓ .
Remark 4.8.By our convention for S i (A) (resp.J i (τ )) when i is larger than the number of occurrences of • in A (resp. the number of blocks of τ ), we have the more compact expression P A,B = z i (resp.Pσ,τ = z i ) where the sum runs over all i ∈ {1, . . ., h + 1} such that B = S i (a 2 . . .a h •) (resp.τ = J i (σ ↓ )).
Remark 4.9.Warrington's annihilation model is recovered by taking z 1 = • • • = z h = a = 1/(h + 1).We still call our multivariate generalization the annihilation model, but this requires some clarification.Indeed, in Warrington's uniform case, one can interpret the dynamics that saying that ball insertions are made at arbitrary (empty or occupied) sites, and a that a ball inserted at an occupied site is annihilated.However, in our multivariate generalization the correct interpretation is to say that we pick an i between 1 and h + 1 and insert the ball at the i-th available site from the right, and that the ball is annihilated if there is no such site on the lattice.
Here is the multivariate generalization of [War05, Theorem 3, item 2]: Theorem 4.10.The stationary distribution of the annihilation model is given by Similarly, the stationary distribution of the enriched annihilation model is given by with σ ∈ S(H) and K its number of blocks.There is no normalization factor, as Π and Π are already normalized for a Example 4.11.The transition matrix of the basic annihilation model for h = 2 in the lexicographically-ordered basis , and admits the row vector a (a + z 2 ) , z 1 (a + z 2 ) , (z 1 + z 2 ) (a + z 2 ) , z 2 1 as a left eigenvector of eigenvalue 1.The sum of the coordinates of this row vector is (a + z 1 + z 2 ) 2 = 1.Similarly, the transition matrix of the enriched annihilation model for H = 3 in the same ordered basis as in Example 4.6 is given by (4.18) , and admits the row vector z 2 1 , z 1 (a + z 2 ) , z 2 (a + z 2 ) , z 1 (a + z 2 ) , a (a + z 2 ) as a left eigenvector of eigenvalue 1.
Remark 4.12.Note that the stationary weight Π(σ) is not a monomial.This suggests that there could exist a further refinement of the Markov chain with a monomial stationary distribution.We explore this idea in Section 5.
Proof of Theorem 4.10: Observe first that, as in the add-drop case, (4.1) immediately implies that the basic annihilation model is a projection of the enriched chain.Thus, we simply have to check the stationarity of Π.
Let τ ∈ S(H) be a set partition with, say, K blocks, and let τ ↑ ∈ S(h) still be the set partition obtained by removing the element H from τ .It is not difficult to check that, for any possible predecessor σ of τ , we have Π(τ ) = Pσ,τ Π(τ ↑ ) (indeed, one simply has to consider the two cases τ = τ ↑ ∪ {H} and τ = J i (τ ↑ ) for some i ∈ {1, . . ., K − 1}).On the other hand, a similar analysis shows that the sum of Π(σ) over all possible predecessors σ of τ is equal to Π(τ ↑ ) (here we use that a + z 1 + . . .+ z h = 1).We deduce that (4.19) which proves the stationarity of Π.
In particular, the only eigenvalues of P and P are 1 (with multiplicity 1) and 0.
This result is stronger than a statement about mixing times since we reach the exact stationary state in finite time!A similar "ultrafast convergence to equilibrium" was observed in [LV12] for the model of Example 3.7, but its combinatorial origin seems rather different from that of the annihilation model.
Proof.It is sufficient to prove the statement for the enriched chain and the case of the basic chain follows by projection.Our strategy is to obtain an explicit description of P j for any positive integer j (see also Section 4.2.3 for an algebraic rederivation).
We first reinterpret the formula (4.16).Given τ ∈ S(H, K) written this time in increasing order of the minimal element as τ ≡ t 1 | . . .|t K , define the weight of an integer k ∈ {2, . . ., H} belonging to t i to be (4.21) Then it is not too difficult to show that Suppose {n 1 , . . ., n j } ⊂ [H].Set τ \ {n 1 , . . ., n j } ∈ S(H − j) to be the element obtained by removing the elements n 1 , . . ., n j from τ and standardizing them by subtracting appropriate integers from the remaining elements so that they are distinct and belong to [H − j].τ ↓ is thus the same as τ \ {1}.
We will now prove that for 1 ≤ j ≤ H − 1, (4.23) In other words, the nonzero entries of the matrix P j are constant along columns and are equal to the product of j terms in the stationary distribution of that particular column.
Let us check that this holds for j = 1 first.In that case, (4.23) is equivalent to saying that all incoming arrows to τ have the same probability, namely wt τ (H), which simply follows from the definition of the model.Now let us look at a generic power j < H. Suppose τ ∈ S(H, K) is written as above, (4.24) and σ is a set partition which has at least one path of length j to τ .We will show that the probability of all transitions taking σ → τ is equal to wt τ (H − j + 1).This stronger claim will prove (4.23).
Since possible outcomes of transitions from σ only depend on σ ↓ , we will focus on the latter.Let σ ↓ now be written in increasing order of the maximal element as (4.25) Notice that the insertion of H into σ ↓ at this stage will affect the position of H −j+1 in τ because H will be shifted down by 1 exactly j − 1 times.Let H − j + 1 ∈ t n , where n ∈ [k].We now have to deal with two cases.Suppose H − j + 1 is not the minimum element of t n .Let α be the integer to whom the downward arch from H − j + 1 in t n goes.Then we must necessarily have that α + j − 1 is the maximum entry of one of the s i 's and the unique transition from σ towards τ must insert H into precisely that part.Simply by downward translation, this transition has weight wt τ (H − j + 1), which is what we wanted to prove.
The second case is when H − j + 1 = min(t n ).In this case, there are multiple starting routes leading from σ → τ because H can be placed in any s whose maximum is less that j.After j − 1 steps, H − j + 1 is then guaranteed to be the minimum in its part.In addition H can also be placed in a new part.Suppose max(s i ) < j.Because of the ordering of σ, max(s i−1 ) < j.Let s 1 , . . ., s m be the parts satisfying max(s i ) < j.As we argued before, the probability of adding H to s i is exactly the number of parts covered by the arch linking H to max(s i ), which is equal to z −i+1 .Moreover, the probability of adding the singleton {H} to σ ↓ is given by a + z +1 + • • • + z H−1 .Thus the total one-step probability of all paths from σ to τ is given by To prove that this is wt τ (H − j + 1) is equivalent to showing that the part t n of which H − j + 1 is the minimum, is exactly t −m+1 .To do so, note that σ ↓ \ {1, . . ., j − 1} = τ \ {H − j + 1, . . ., H}.
Since H − j + 1 = min(t n ), all the entries in parts t n , . . ., t k are greater than H − j + 1 by the ordering of τ .Thus the parts s m+1 , . . ., s , after suitable removal of entries less than j and standardization equal t 1 , . . ., t n−1 .Since the maximal entry in each s i is greater than j by assumption, the number of parts in both is the same.Therefore − m = n − 1, or equivalently n = − m + 1. Hence H − j + 1 = min(t −m+1 ) as promised.We have thus shown that (4.23) holds for all j ∈ [H − 1].
To complete the proof, notice that for any given τ , σ \ {1, . . ., H − 1} = τ \ {2, . . ., H} is satisfied for all σ.Thus, P H−1 = P h is constant along columns, with each entry in column τ equal to Π(τ ).In other words, if e denotes the row vector with all entries equal to one, then P h = e Π, which implies the result.Now, notice that we have the commutation property that J i (τ ↓ ) = J i (τ ) ↓ for any τ, i (intuitively speaking, the mapping τ → τ ↓ amounts to removing the leftmost element while J i amounts to inserting a new element on the right, thus it is not surprizing that these two operations commute).This directly implies that for all H ≥ 1 (note that we may indeed add z H−1 and z H on the r.h.s.since J H and J H−1 act the same on S(H − 2)).It is now easy to check by induction that for all j < H, In particular, when z 1 + • • • + z H = 1, we find that ΠH e H = 1 i.e.ΠH is normalized, as announced in Theorem 4.10.

Conclusion and discussion
We end with speculative ideas for further work along the direction of this paper.The annihilation Markov chains, both on words and on set partitions, have the property that all the eigenvalues of the transition matrices are nonnegative and trivially linear in the parameters z i .Such Markov chains typically arise from an underlying structure which we now describe.If one considers the matrices P i (resp.Pi ) obtained by setting z i = 1 and z j = 0 for j = i in the transition matrix P (resp.P ) in (4.13) (resp.(4.14)) and consider the monoid generated by these matrices, then it has the property that it is an R-trivial monoid.For a recent monograph on the connection between Markov chains and R-trivial monoids, see [ASST14].One could reprove our statement about the eigenvalues in Theorem 4.13 using this connection.The general theory of free tree monoids described there, however, does not directly apply to the annihilation chains because the generators P i (resp.Pi ) do not always square to themselves (i.e. are not idempotents).It should be interesting to study when their results can be applied to situations when the generators are not idempotents.
We explore the idea in Remark 4.12 further.Observe that Π(B) in (4.15) and Π(σ) in (4.16) are homogeneous polynomials of degree h in z 1 , . . ., z h , a, which allows us to extend their definition by relaxing the condition z 1 + • • • + z h + a = 1.Then, the total mass of both measures is equal to (z 1 + • • • + z h + a) h .This reduces to (h + 1) h when z 1 = • • • = z h = a = 1 which, as already noted by Warrington, is the number of rooted Cayley trees with H = h + 1 vertices.This suggests that the expansion of Π(B) and Π(σ) in monomials could be interpreted in terms of such trees.Does there exist a natural Markov chain on Cayley trees that projects to the annihilation model?
This work suggests that it is worth studying a general framework for a class of problems in statistical physics, which we tentatively call "boundary-driven Markov chains".Just as the juggling and set-partition chains (2.1) and (2.27) studied in this paper, they have the property that the motion is deterministic for the most part (that is, in the bulk).Only when a ball (in the juggling context) reaches the boundary, Magnus's hand, something stochastic happens, namely the ball is thrown to a randomly chosen height.Similar things happen with the set-partition Markov chain.
A natural idea is to combine the juggling process with an exclusion process, for instance the Totally Asymmetric Exclusion Process (TASEP) where particles do not all move at the same time but, instead, at each time step one particle is selected randomly and moves if the next site is empty.When a particle on the first site is selected, it is then reinserted, say, uniformly at any available site on the lattice.This model was investigated via a physical hydrodynamic approach in [ABKM13], where a nontrivial phase diagram was found.It would be interesting to obtain more precise results on this model, by looking for instance for a possible exact solution.
Finally, the fact that the stationary distributions of the Markov chains considered in this paper all admit product forms is reminiscent of the Zero-Range Process admitting a "factorized stationary state" [EH05,EMZ06], and one might wonder whether a connection exists.

Figure 3 .
Figure 3. Transition graphs of the enriched chain (top part, above blue horizontal dashed line) and of the MJMC (bottom part) for H = 4 and K = 2 (h = 3 and k = 1).Enriched states (set partitions) are represented via "arches".Red, green and grey arrows represent transitions of respective probability x 0 , x 1 and 1. Arches covering 1 or 2 blocks are represented in green or red respectively.The reader may easily check that attaching a weight x 0 (resp.x 1 ) to each red (resp.green) arch yields the stationary measure of the enriched chain.Enriched states in a same column map to the same MJMC state (displayed below) via ψ.
be the word obtained by replacing the i-th letter of B by •, i.e.B = • i b i+1 . . .b h .
(3.12) w(λ) = ∞ i=1 y λi where λ ∈ Par and y m = ∞ j=m x i .Its total mass reads m and Z is finite (i.e. the IMJMC is positive recurrent) if and only if (3.5) holds.

Figure 4 .
Figure 4.The basic transition graph for h = 2
2.4.3.Technical proofs.We now prove Lemmas 2.22, 2.26 and 2.28.Proof of Lemma 2.22:We check the relation (2.29) coefficientwise.Fix σ ∈ S(H, K) and B ∈ St h,k , and let A = a 1 a 2 . . .a h = ψ(σ), where ψ(σ) is defined in the beginning of Section 2.4.1.Observe that Finally, it remains to check that Π and Π are indeed normalized, which can be done for instance by induction on h (details are left to the reader, who might find it easier to work with words).Alternatively, this is a simple consequence of the algebraic point of view that we present in Section 4.2.3 below.4.2.2.Convergence to the stationary distribution in finite time.The annihilation models have a remarkable property that make it distinct from the generic MJMC and add-drop models.
Theorem 4.13.For any initial probability distribution η over St h or η over S(H), the distribution at time h is equal to the stationary distribution, namely (4.20) 4.2.3.An algebraic point of view.For completeness, let us expound on the rich algebraic structure underlying the enriched annihilation model.It is here convenient to explicitly indicate the size of the state space: we denote by PH the transition matrix on the state space S(H).Let us introduce the rectangular matrix D H whose rows and columns are indexed by respectively S(H) and S(H − 1), defined by the formula(4.26)(DH ) σ,τ = 1 if τ = σ ↓ , 0 otherwise.Similarly, for i a nonnegative integer, we introduce the rectangular matrix U H;i whose rows and columns are indexed by respectively S(H − 1) and S(H), defined by (4.27) (U H;i ) σ,τ = 1 if τ J i (σ),0 otherwise.For z 1 , ..., z H arbitrary parameters, we then set(4.28)UH (z 1 , . .., z H ) = H i=1 z i U H;i .By the definition of the enriched annihilation model and our convention of defining J i (τ ) = τ ∪ {H} when i is larger than the number of elements of τ ∈ S(H − 1), we have (4.29)PH = D H U H (z 1 , . . ., z H ).