An interacting particle system with geometric jump rates near a partially reflecting boundary

This paper constructs a new interacting particle system on a two--dimensional lattice with geometric jumps near a boundary which partially reflects the particles. The projection to each horizontal level is Markov, and on every level the dynamics match stochastic matrices constructed from pure alpha characters of $Sp(\infty)$, while on every other level they match an interacting particle system from Pieri formulas for $Sp(2r)$. Using a previously discovered correlation kernel, asymptotics are shown to be the Discrete Jacobi and Symmetric Pearcey processes.


Introduction
To motivate this paper, first review some previous results. In [8], the authors construct a continuous-time interacting particle system on N × Z+ using the representation theory of symplectic Lie groups. One distinguishing feature of these dynamics is a wall at {0} × Z+ which suppresses jumps of particles into the wall. In [4], the Pieri formulas from the representation theory of the symplectic Lie groups Sp(2r) are used to construct discrete-time dynamics with geometric jumps, and again jumps into the wall are suppressed. In [2], there is a construction of continuous-time dynamics using Plancherel characters of the infinite-dimensional symplectic group Sp(∞), and again there is a suppressing wall.
Some previous work had been done with the orthogonal groups as well. In [1], Plancherel characters of the infinite-dimensional orthogonal group O(∞) led to continuous-time dynamics with a reflecting wall, and in [3], Pieri rules for the orthogonal groups O(2r), O(2r + 1) led to interacting particles with discrete-time geometric jumps, again with a reflecting wall. In [5], it was shown that the dynamics of [3] on each level N × {k} fit into the general framework of [1] with pure alpha characters of O(∞). Therefore, it is reasonable to expect that the dynamics of [4] might also fit into the framework of [2] with pure alpha characters of Sp(∞). However, it turns out that the dynamics only match on the even levels N × {2r}.
In order to create a physically meaningful interacting particle system which matches that of [2] on every level, we will slightly modify the Pieri formulas of [3], [4]. The result is a wall which is partially reflecting. Mathematically, this means that a jump to −x is reflected to x − 1, rather than being totally reflected to x or totally suppressed at 0. Observe that after the usual scaling limit of discrete-time geometric jumps to continuous-time jumps with exponential waiting times, the particles only jump one step, so the partially reflecting boundary becomes a suppressing boundary.
Note that there may be an algebraic intuition for the discrepancy between the dynamics of [4] and [2]. The odd symplectic groups Sp(2r + 1) of [7] are not simple, in contrast to Sp(2r), O(2r), O(2r + 1). In a sense, the odd symplectic groups are less canonical, which may explain why the two dynamics only match at the levels corresponding to Sp(2r).
The paper is outlined as follows. In section 2, the interacting particle system is defined. In section 3, new formulas for the stochastic matrices from [2] are written. In section 4, it is shown that the projection to each horizontal level is still Markov, and the resulting transition probabilities are precisely the ones from section 3. Using the explicit expression for the correlation kernel in [2], section 5 finds the asymptotics for our particle system.
Note that despite the algebraic motivation and background, the body of the paper is written with minimal reference to representation theory.
Acknowledgments. Financial support was available through NSF grant DMS-1502665.

Interacting Particle System
First define the state space for the interacting particles. For k ≥ 1, define The state space for the interacting particles will be If ξ1 and ξ2 are two independent geometric random variables with parameter q (i.e. P(ξi = x) = (1 − q)q x for x ≥ 0), then for any x, y ≥ 0 With this in mind, define The particles live on the lattice N × Z+ where N denotes the non-negative integers and Z+ denotes the positive integers. The horizontal line N × {k} is often called the kth level. There are always k+1 2 particles on the kth level, whose positions at time n will be denoted X k The time can take integer or half-integer values. For convenience of notation, X k (n) will denote the element (X k 1 (n), X k 2 (n), X k 3 (n), . . . , X k (k+1)/2 (n)) ∈ N (k+1)/2 . More than one particle may occupy a lattice point. The particles must satisfy the interlacing property for all meaningful values of k and i. This will be denoted X k ≺ X k+1 . With this notation, the state space can be described as the set of all sequences (X 1 ≺ X 2 ≺ . . .) where each X k ∈ N (k+1)/2 . The initial condition is X k i (0) = 0, called the densely packed initial conditions. Now let us describe the dynamics. For n ≥ 0, k ≥ 1 and 1 ≤ i ≤ k+1 2 , define random variables which are independent identically distributed geometric random variables with parameter q. In other words, P(ξ 1 At time n, all the particles except X k (k+1)/2 (n) try to jump to the left one after another in such a way that the interlacing property is preserved. The particles X k (k+1)/2 (n) do not jump on their own. The precise definition is where X k−1 0 (n + 1 2 ) is formally set to +∞. At time n + 1 2 , all the particles except X k (k+1)/2 (n + 1 2 ) try to jump to the right one after another in such a way that the interlacing property is preserved. The particles X k (k+1)/2 (n + 1 2 ) jump according to the law R. The precise definition is when k is odd and where X k−1 0 (n + 1) is formally set to +∞. Let us explain the particle system. The particles preserve the interlacing property in two ways: by pushing particles above it, and being blocked by particles below it. So, for example, in the left jumps, the expression min(X k i (n), X k−1 i−1 (n + 1 2 )) represents the location of the particle after it has been pushed by a particle below and to the right. Then the particle attempts to jump to the left, so the term ξ k i (n + 1 2 ) is subtracted. However, the particle may be blocked a particle below and to the left, so we must take the maximum with X k−1 i (n). While X k i (n) is not simple, applying the shiftX k i (n) = X k i (n) + k+1 2 − i yields a simple process. In other words,X can only have one particle at each location. Figure 1 shows an example ofX.
The following transition probabilities on Wn are from section 5.1 of [2] are where J are the Jacobi polynomials satisfying When φ(x) = e t(x−1) , these are the transition probabilities arising from Plancherel characters of Sp(∞).
In this paper a different φ(x) depending on a parameter α ≥ 0 will be considered. Note that for general φ(x), a priori there is not an obvious physical description of the dynamics.
The following formula is standard (for example, see Lemma 3.3 from [5]) Set Note that the formula for P2r is essentially identical to (2) from [4], and the formula for P2r+1 is also similar to a related formula from [3] with a different definition of R (see Proposition 5.2 and Theorem 7.1). The discussions in [3], [4] describe how to obtain these formulas from Pieri's rule. This next proposition shows that P and T are the same. The proof is similar to the Proposition 3.1 from [5]. The only difference is that R(·, ·) has a different definition here, but it turns out that the only relevant information about R for the proof is that (1) is true. The full proof is still included here for completeness, because much of the notation is different.
Proof. We first prove this for r = 1. Substituting , a = 1/2 and the integral over [−1, 1] becomes an integral over the unit circle, with an extra factor of 1/2 occurring because the map z → x is two-to-one. Thus, the term inside the 1×1 determinant in T φ n can be calculated from the identities for n = 1, 2 respectively. The first line is R(k, l) = P1(k, l). For the second line, note that which shows that P2 = T φ 2 .
Now proceed to higher values of r. By (3), where fs(l) = q l−s ψ(s, l).
By Lemma 2.1 of [1], this equals By identity (4), To see this claim, first notice that P2r+1 = 0 follows immediately from the description of the interacting particle system, or from the fact that {c ∈ N r : c ≺ λ, β} is empty. By (1), in the matrix of T2r+1 the rth column is a multiple of the (r + 1)th column, so that T2r+1 = 0.
Note that while the projections to each level are the same, the multi-level dynamics are different. For the dynamics in this paper, there is zero probability of a jump from (0) ≺ (1) ≺ (1, 0) to (0) ≺ (0) ≺ (0, 0) on the bottom three levels, because X 2 1 prevents X 3 1 from jumping to 0. However, in the dynamics of [2], this probability is nonzero because all of the terms in (60) are nonzero.

Projections to levels
This section will provide a proof that the projection to each level is Markov with an explicit expression for the Markov operator. Note that the method of the proof is very similar to that of [3,4]. The primary difference is that due to the different expression for R and for the branching rule, the identities (11) and (14) are changed.
We consider the subset W and define a Markov kernel S k on W when k = 2r, and when k = 2r − 1. Since the expression for S k does not depend on z, also write it as S k (y, (z , y )). Note that z ∈W (2) k S k ((z, y), (z , y )) = P k (y, y ) Thus Proposition 3.1 implies that S k is a Markov operator.
Theorem 4.1. For each k ≥ 1, the random process (X k (n − 1 2 ), X k (n)) n∈N is a Markov process with transition kernel given by S k . Furthermore, (X k (n)) n∈N is a Markov process with transition kernel given by P k .
Proof. It suffices to prove the first statement, because by (7) the second follows from the first.
The proof will follow from induction on k. For k = 1, Theorem 4.1 is clearly true. If the random process (X k−1 (n − 1 2 ), X k−1 (n)) n∈N is Markov with transition kernel S k−1 , then the random process (X k−1 (n), X k (n − 1 2 ), X k (n)) n∈N is also Markov with some transition kernel Q k , since the evolution of the kth level only depends on the evolution of the (k − 1)th level. Let L k be a Markov projection from the (k − 1)th and kth level onto just the kth level. If we show that then the intertwining property of [6], Theorem 2, will imply that the projection to the kth level is Markov with kernel S k . The explicit expression for L k is not hard to write. Define L k from W ((z0, y0), (x, y, z)) = 1 (z 0 ,y 0 )=(z,y) s k−1 (x) s k (y) 1x≺y By (2), the L k are Markov. In order to prove (8), there also need to be explicit formulas for Q k . In order to write these formulas, first introduce some notation. Let ξ1 and ξ2 be two independent geometric random variables with parameter q. For x ≥ a ≥ 0, let a← P (x, .) denote the law of the random variable max(a, x − ξ1).
With this notation in place, the description of the model implies the following explicit expression for Q k . For (u, z, y), (x, z , y ) ∈ W k−1 × W (2) k such that u ≺ y and x ≺ y Q k ((u, z, y), (x, z , y )) = v∈N r−1 when k = 2r − 1 and when k = 2r. In both cases v0 = ∞ and the sum runs over v = (v1, . . . , vr−1) ∈ N r−1 (or N r ) such that vi ∈ {y i+1 , . . . , xi ∧ z i }, for all i. The notation can be depicted visually as time n time n + 1/2 time n + 1 level k y z y Here is a description in words. For both k = 2r and k = 2r − 1, the kth level has r particles, so after the (k − 1)th level evolves as S k−1 (without dependence on what happens on the kth level), there is a (r − 1)-fold double product corresponding to the left and right jumps of the r − 1 particles away from the wall. For the particle closest to the wall, the evolution is as R when k is odd, and when k is even the evolution fits into the previous (r − 1)-fold double product. In order to show (8), there need to be explicit expressions and identities for these laws. The next lemma provides this.
For (x, y, a) ∈ N 3 such that a ≤ y and y ≤ x For (x, y, a) ∈ N 3 such that y ≤ a and x ≤ y For y ∈ N, y ∈ N * such that y ≤ a a v=y q v∨y−2v →v R (y ∧ v, y ) = 1 1 − q q −a R(y, y ).
Proof. Note that (12) and (13) are precisely statements from Lemma 8.3 of [4]. Before showing the remaining identities are true, it is necessary to have formulas for these laws. The following two statements are from Lemma 8.2 of [3]: For a, x, y ∈ N such that a ≤ y ≤ x This next formula follows from direct computation. For b, x, y ∈ N such that b ≥ y, Furthermore, a v=y q v∨y−2v →v R (y ∧ v, y ) = q y ∨y−2y →y R (y ∧ y , y ) + a v=y +1 q v∨y−2v →v R (y ∧ v, y ) = q y ∨y−2y 1 1 + q q y (q −y∧y + q y∧y +1 ) + a v=y +1 Note that in each case, the summation over v is of the form v q −v , and in all cases simplifies to 1 1−q q −a R(y, y ). y), (u, z, y))Q k ((u, z, y), (x, z , y )).
Assume for now that k = 2r. Then L k Q k is equal to where the sum runs over (u, v) ∈ N r × N r−1 such that ur ∈ {0, . . . , z r }, vi ∈ {y i+1 , . . . , xi ∧ z i }, ui ∈ {vi ∨ yi+1, . . . , z i }, for i ∈ {1, . . . , r − 1}. Thus L k Q k equals v∈N r−1 Now evaluate the sum over u and v. For each fixed v the sum over u is equal to Now, identities (11) and (12) of Lemma 4.2 imply that the sum over u equals i.e. q xr ∨z r +yr −2z r + r−1 i=1 y i +v i −z i (1 − q).

Thus
L2rQ2r((z, y), (x, z , y )) = Identity (13) of Lemma 4.2 gives that r i=2 which is quickly seen to be equal to S2rL2r. This finishes the proof when k = 2r. Similarly when k = 2r − 1, where the sum runs over (u, v

Asymptotics
The interacting particle system from [2] is a determinantal point process. In general, a determinantal point processes on a discrete space S is uniquely characterized by an object called a correlation kernel, which is a function on S × S. In [2], the asymptotics were calculated for Plancherel representations of Sp(∞). Here, we find the asymptotics for the pure alpha representations. By Theorem 1.2 of [2], the correlation kernel at integer times T is given by where (s, k), (t, m) ∈ N × Z+ and recall that φ(x) is the function from Proposition 3.1. If φ(x) is replaced with e (x−1) and T is allowed to take any nonnegative value, then K becomes the correlation kernel corresponding to the Plancherel characters.

Symmetric Pearcey
Define the symmetric Pearcey kernel K on R+ × R as follows (see Theorem 1.5 of [2]). Let By substituting x → cx and u → cu, Proof. Since the proof is almost identical to the proof of Theorem 5.8 from [1] and Theorem 1.5 from [2], some of the details will be omitted. If x = N 1/2 (x + 1) and u = N 1/2 (u + 1), then by an (unnumbered) equation on page 41 of [2], and if s ∼ N 1/4 ν then x To analyze the other terms in the integrand, define Thus the expression in x becomes Note that after taking the determinant, the conjugating factors (−2) r 2 −r 1 (−1) s 1 −s 2 2 an 2 −an 1 have no affect.
Note that L only depends on r1, r2 through their difference r1 − r2.
Theorem 5.2. Let T depend on N in such a way that T /N → t. Let r1, . . . , r l depend on N in such a way that ri/N → l and their differences ri − rj are fixed finite constants. Here, t, l > 0. Fix s1, s2, . . . , s l to be finite constants. Let , α = 2q 1 − q Then setting kj = 2rj + aj − 1/2, Proof. The proof of Theorem 4.1 of [5] carries over here. The only difference is the parameters in the Jacobi polynomials, but these have no effect in the asymptotics. A yellow arrow means that the particle has been pushed by a particle below it. A green arrow means that the particle has jumped by itself. A red line means that the particle has been blocked by a particle below.
In the table, keep in mind that ξ k (k+1)/2 (n + 1/2) actually correspond to left jumps, but occur at the same time as the right jumps.