Pfaffian formulae for one dimensional coalescing and annihilating systems

The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the n-point density function for coalescing particles is derived.


Introduction and summary of main results
The study of single species reaction diffusion systems A + A → A (coalescence) and A + A → 0 (annihilation) originated in non-equilibrium statistical mechanics (see [13]), but has now a large mathematical literature (see, for example, [1], [6], [7], [4]). In one dimension the systems exhibit strongly non-mean field behaviour due to correlation effects. In this paper we give several examples showing that this correlation structure can be encoded algebraically in a Pfaffian structure. Note that the embedding of annihilating random walks as domain boundaries for a Glauber model makes Pfaffian formulae quite reasonable due to the free fermion structure of the Glauber model (see Felderhof [8]).
In [12] we showed, for n ≥ 1, t 0 , L > 0 and for a variety of initial conditions, the bounds 0 < c 1 (n, L, t 0 ) ≤ ρ (n) t (x 1 , . . . , x n ) t − n 2 − n(n−1) for t ≥ t 0 and |x i | ≤ Lt 1/2 , where the constant c 1 will depend also on the initial condition. The non-linear factor in the power of t illustrates the non mean-field behaviour due to correlations.
In this paper we show that, under the maximal entrance law, the true asymptotic holds in (1) as t → ∞, and identify the limiting constant as the Pfaffian of a certain matrix. The maximal entrance law corresponds intuitively to starting with every point occupied, and can be constructed as the limit of initial Poisson distributions with increasing intensities. This initial condition is natural since, as explained in section 2.3, started from a large class of other initial conditions the distributions at time t become close, as t → ∞, to those of the maximal entrance law.
The presence of the Pfaffian in this asymptotic is a reflection that under the maximal entrance law the particle positions, at a fixed time, form a Pfaffian point process (see the start of section 3 for a definition).
The annihilating versions of Theorems 1 and 2, that is for (instantly) annihilating particles, can be deduced from the thinning relation that connects coalescing and annihilating systems (see section 2.1).
Many probabilities for the fixed t distributions are given by formulae using Pfaffians, and there are many places to start when proving these formulae. We choose to start by considering the following basic fact for product moments for annihilating systems, from which we will deduce all the other Pfaffian fromulae.
Theorem 3 Consider the product moments for annihilating Brownian motions, defined by m (n) t (x 1 , . . . , x n ) = E A (x 1 ,...,xn) i∈It for bounded measurable g, where (x 1 , . . . , x n ) lists the initial positions of the annihilating Brownian motions on R, and (X i t : i ∈ I t ) list the positions of any particles that remain at time t (and an empty product is taken to have value 1). Then for x 1 < x 2 < . . . < x 2n , the even moments m where the right hand side is the Pfaffian of the 2n × 2n anti-symmetric matrix with entries m (2) t (x i , x j ) above the diagonal.
Note these Pfaffians are in variables that determine the initial conditions, allowing us to use p.d.e. methods to characterize these moments. Indeed, the product moments satisfy a closed system of heat equations (with suitable boundary conditions), and we will verify Theorem 3 by simply checking that the Pfaffian uniquely satisfies this system. Markov time-reversal duality (see section 2.2) then immediately implies that certain empty interval formulae P [the intervals (a 1 , a 2 ), (a 3 , a 4 ),...,(a 2m−1 , a 2m ) are empty at time t] for coalescing systems are given by a Pfaffian, where the Pfaffian is now in the variables a 1 < a 2 < . . . < a 2m that determine the end points of the target intervals. This quickly leads to the identification of the Pfaffian point process kernel K(x, y).
We concentrate on Brownian particles but, as we note later, we expect many of our Pfaffian formulae to hold for a large variety of spatial motion processes, and the Pfaffian structure seems to arise from two basic underlying mechanisms: linearly ordered particle motion and instantaneous reactions.

Relation between annihilating Brownian motions and the real Gini-
bre ensemble of random matrices.
The Pfaffian point process defined in Theorem 2 has been originally discovered in the context of random matrices. 1 Namely, consider real Ginibre ensemble [10] defined by the following probability measure on the space of real N × N matrices: where λ N ×N is Lebesgue measure on R N ×N . Even though the real Ginibre ensemble is a classical matrix model, the eigenvalue correlation functions have been computed only recently, see [5], [14], [9], [15] and [16].
It turns out that the pfaffian point process corresponding to one-dimensional distributions of annihilating Brownian motions is equivalent to the pfaffian point process describing the law of real eigenvalues of Ginibre in the limit N → ∞. Namely, comparing the statement of Theorem 2 with Corollary 9 of [5] we arrive at the following conclusion: Corollary 4 The one-dimensional law of particle positions for the system of annihilating Brownian motions on R at time t > 0 under the maximal entrance law is a Pfaffian point process with the kernel where K Ginibre rr is the N → ∞ limit of the kernel of the Pfaffian point process characterizing the law of real eigenvalues in the real Ginibre ensemble.
In other words, the one-dimensional law of annihilating Brownian motions under the maximal entrance law initial conditions is equivalent to the N = ∞ limiting law of real eigenvalues of a real matrix with independent normal entries. Corollary 4 suggests that real eigenvalues of real matrix-valued Brownian motion might behave like a system of one-dimensional annihilating Brownian motions. In fact, numerical evidence we accumulated up to date compels us to make the following conjecture.
Conjecture. Under the maximal entrance law, all finite-dimensional distributions of particle positions for a system of annihilating Brownian motions on R coincide with N → ∞ limit of multi-time correlation functions of real eigenvalues of gl R (N )-valued Brownian motion.
Here gl R (N ) denotes the linear space of all N × N matrices with real entries.
2 Brief review of some facts for one-dimensional coalescing and annihilating Brownian motions We consider, at first, initial conditions that have only finitely many particles. This paper describes only the one dimensional time distributions, that is at a fixed t > 0, of any remaining particles. We list the positions of the particles at time t as (X i t : i ∈ I t ). The exact details of the labeling system I t will not be important for us, and indeed our results all relate only to the empirical measure N t defined by For the case of annihilating particles, if the initial number of particles is even then it remains so for all time. To a list (x i ) of an even number 2n of disjoint positions we may associate the open set wherex 1 < . . . <x 2n are the ordered positions. Some of the formulae for annihilating particles are then most easily stated in terms of the set valued process

Notation.
We write P C (x 1 ,...,xn) to indicate that we are considering (instantly) coalescing Brownian motions started from initial positions x 1 , . . . , x n . When the particles are annihilating we change the superscript from C to A. When the initial positions are random we change the subscript to Ξ, where Ξ is the law of (X i 0 : i ∈ I 0 ).

The thinning relation
The formulae about coalescing systems in the paper will always come with an analogue for annihilating systems. The close link between the two systems has often been observed. For this paper the formulae can usually be derived from the following thinning relation. For a list of positions (x 1 , . . . , x n ) we let Θ(x 1 , . . . , x n ) be the random subset formed by thinning at rate 1/2, that is by removing each position independently with probability 1/2. We may also thin a random set of positions, for example Θ(X i t : i ∈ I t ), with the understanding that the randomness in the thinning is independent of the randomness in the set of positions. We write Θ(Ξ) for the law of the thinned set of positions that initially have law Ξ. Then the thinning relation between coalescing and annihilating Brownian motions is the following equality in distribution: Such a thinning relation is discussed in Arratia [1] for the scaled limit of reacting random walks, and is related to results in many later papers. There is a simple colouring proof (see ben Avraham and Brunet [3]) that bears repetition here. After the paths of a coalescing system have been realized, independently add random colours as follows. Initially colour each particle red or blue independently with probability 1/2. At coalescences the colours evolve according to the rules R + R → R, B + B → R and R + B → B. Then the resulting system of blue particles evolves as an annihilating system. Moreover the colour of a particle at time t depends on whether there were a odd or even number of ancestors at time zero that were coloured blue. Since distinct particles have disjoint sets of ancestors, the colour of all particles at any time t > 0 remains independently red or blue with equal probability. The thinning relation follows. This argument makes it clear that the result holds much more widely, since the exact nature of the motion process is not relevant, nor is the mechanism of reaction (for example it holds for delayed reactions, when the reactions are controlled by the intersection local time).

Duality formulae
We use two duality formulae. For a 1 < a 2 < . . . < a 2m let I k = (a k , a k+1 ) for k = 1, . . . , 2m − 1. Then for disjoint (x i ) The annihilating analogue of this is, writing |A| for the cardinality of a set A, There are various ways to see these formulae, but for coalescing systems a key construction is the Brownian web and its coupling with the dual Brownian web, first considered by Arratia and explored in Toth and Werner [20] (and subsequent papers). We need only part of the Brownian web as follows. For a fixed t > 0, there is a system of coalescing Brownian motions starting from every rational x and running over the time interval [0, t], and a coupled system of backwards coalescing Brownian paths starting at time t at all x ∈ Q and running back to time zero. In fact, the Brownian web has particles starting at all space-time points (s, x) but we will not need this, and it is enough to establish (6) first for rational (x i ) and (a i ). The key property is that, almost surely, none of the forward paths cross any of the backwards paths. (A discrete version of this coupling, that is using simple coalescing simple random walks, is easy to construct -see the appendix in [18] -and illustrates this non-crossing property). From this non-crossing property one sees that the event that N t ((a, b)) = 0 for the forward coalescing system is almost surely equal to the event that the open interval formed by pair of backwards particles starting at a and b does not contain any of the initial forwards particles. The coalescing duality (6) follows immediately, once one notes that S t may be replaced by its closure and that annihilating the backwards particles when they meet will not affect this closure.
The annihilating duality (7) follows from (6) and the thinning relation. Note that thinning a set of n ≥ 1 elements produces a random subset whose size has a binomial B(n, 1/2) distribution, and also that E[(−1) B(n,1/2) ] = 0. Then thinning and (6) show that (where on the right hand side E A (a i ) is the expectation over the annihilating particle system and over the independent thinning). One may then argue by induction on the number n of the initial particles (x 1 , . . . , x n ). When n = 1 the above identity reduces to (7) for a single particle. For general n the identity is a mixture of copies of (7) for initial conditions that are subsets of (x i ). But all but one of the copies will involve n − 1 or less particles allowing an inductive proof. Note also that (6) also follows from (7) -a weighted sum of (7) according to the distribution of Θ(x i ) yields (6).
Remark. Other coalescing duality formulae, such as those in Xiong and Zhou [21], also follow from the Brownian web and its dual, but their proof shows that one may also establish them using the Markov generator duality, as explained in section 4.4 of Ethier and Kurtz, and thus bypass the Brownian web. In particular this generator technique may be extended to show analogous dualities for more general spatial motions, where the web construction does not (as yet) exist. Formally the generator proof shows that the dualities (6) and (7) will hold for instantly reacting continuous Markovian motions, where the motion on the right hand side must be the image of the motion on the left hand side under reflection x → −x. Furthermore the maximal entrance laws constructed in the next section should follow once some moment control is established, which will require some non-degeneracy of the spatial motion to ensure enough reactions take place.

Maximal entrance laws
One may start coalescing systems from infinitely many particles at time zero. A natural state space for the empirical measure is the set M LF P (R) of locally finite point measures on R, which is a closed subset of the space of locally finite measures under the topology of vague convergence of measures. The reactions ensure that the point masses only have mass one, and so we consider the (measurable) subset M 0 of those measures of the form is locally finite in R and has disjoint elements.
(To obtain a process with continuous paths, which does not concern us in this paper, one can quotient M LF P by the minimal relation that ensures µ + 2δ There is a Feller Markov transition kernel p t (µ, dν) on M 0 . Moreover, there is a maximal entrance law, intuitively starting with one particle at every site (as in the Brownian web). This can be characterized by passing to the limit in (6) as (x i ) increase to become dense in the real line. This entrance law, which we denote by P C ∞ , has one dimensional distributions satisfying where τ is the time for complete extinction of the annihilating system. This characterizes the one dimensional laws on M 0 , and these laws are an entrance law for the Markov transition kernel described above. By the scaling property of Brownian motions we have is independent of T > 0. Using (8) this translates into a scaling for the entrance law The law of (T −1 X i Many suitably spread out and non-degenerate initial conditions are attracted to the maximal entrance law as t → ∞. For a large class of initial conditions (x i ), the law of Indeed, using the extension of (6) to countable (x i ), this follows (see the appendix) from The third equality comes from Brownian scaling and the final equality is (8). The convergence holds for deterministic (x i ) for which (T −1 x i ) become dense in any finite interval [a, b] as T → ∞. A large class of random initial conditions will clearly also work, for example non-zero stationary and spatially ergodic.
For annihilating systems a Markov transition kernel can also be constructed, using (7) and it's extension to countable (x i ) as a means of characterization. We can define an entrance law P A ∞ for the annihilating system by taking the thinned copy of the entrance law for the coalescing system. This satisfies the formula which again determines one dimensional laws on M 0 that form an entrance law for the annihilating system. The domain of attraction of this entrance law is more delicate. The example in section 3 of Bramson and Griffeath [6] suggests that different approximations to a maximal entrance law may yield different laws at times t > 0 (their example uses varying intensities of nearby pairs at time zero). For initial conditions that fill the lattice λ −1 Z, or that are Poisson with intensity λ, the one-dimensional time distributions converge as λ → ∞ to those of the entrance measure, or for a fixed λ the large time distribution rescales to those of the entrance law, by the argument above.
Since we found it difficult to find a full account in the literature, we give, in the appendix, a brief sketch of the proofs of the results in this subsection.

Review of Pfaffians
We give a short summary, targeted at beginners like us, of the facts we shall use about Pfaffians (mostly proved in [19] section 2), and of the definition of a Pfaffian point process. We write Pf (a ij : 1 ≤ i < j ≤ 2n) (or just Pf (a ij : i < j)) for the Pfaffian of the real anti-symmetric matrix whose elements are a ij for i < j.
The determinant of an anti-symmetric matrix of odd order is zero. Suppose A is an anti-symmetric 2n × 2n matrix. Then det(A) is the square of a polynomial of degree n in the matrix elements, called the Pfaffian of A and written as Pf (A). One can define the Pfaffian as a suitable sum over permutations of products of matrix elements. Indeed, where Σ 2n is the set of permutations σ of {1, 2, . . . , 2n} given by σ(2k − 1) = i k , σ(2k) = j k for k = 1, . . . , n for which the choices (i k ), (j k ) satisfy i k < j k for all k and i 1 < i 2 < . . . < i n . A convenient way to calculate the sign of such a permutation is via crossings.
Then the sign of σ ∈ Σ 2n equals (−1) M where M is the number of crossings. To visualize these crossings easily one can embed the integers 1, . . . , 2n into the x-axis of the plane and join i k to j k for each k with a loop in the upper half plane.
It is worth recording the smallest cases: The explicit 4 × 4 case was used to guess many of the Pfaffian formulae in this paper.
Pfaffians have many similar properties to determinants. It follows from the definition that Pf (λ i λ j a ij ) = Pf (a ij ) k λ k . For any 2n × 2n matrix B the product B T AB is anti-symmetric and Pf (B T AB) = det(B)Pf (A). The Pfaffian can be decomposed along a row, or column, of the matrix. For example if A is a 2n × 2n anti-symmetric matrix it satisfies the recursion, for any i ∈ {1, 2, . . . , 2n}, where submatrix formed by removing the ith and jth rows and columns. We will also use a decomposition formula for the Pfaffian of a sum of two 2n × 2n anti-symmetric matrices A and B, namely where: the sum is over all subsets J ⊆ {1, 2, . . . , 2n} with an even number of terms; J c = {1, 2, . . . , 2n} \ J; s(J) = j∈J j (and s(∅) = 0); and where A| J means the submatrix of A formed by the rows and columns indexed by elements of J (and the Pfaffian of the empty matrix is taken to have value 1).

Suppose a measurable kernel
is anti-symmetric, in the sense K ij (x, y) = −K ji (y, x) for all i, j ∈ {1, 2} and x, y ∈ R. Suppose it also acts as a kernel for a a bounded operator on L 2 (R) ⊕ L 2 (R). A point process (X i : i ∈ I) with n-point density functions ρ (n) (x 1 , . . . , x n ) is called (see Soshnikov [17]) a Pfaffian point process with kernel K if ρ (n) (x 1 , . . . , x n ) is given by the Pfaffian of the 2n×2n anti-symmetric matrix formed by the n 2 two-by-two matrix entries (K(x i , x j ) : i, j = 1, . . . , n). The kernel is not uniquely determined.
A very convenient tool for manipulating Pfaffians is the Berezin integral. We provide arguments that avoid this tool in this paper, and so do not describe the rules for manipulating these integrals. However they were used repeatedly while exploring these results, and in the next section we show how the Berezin integral can considerably shorten the argument. A very readable account of Berezin integrals can be found in Itzykson and Drouffe [11]. The key property linking the Berezin integral to Pfaffians is (compare with the normalizing determinant for multi-dimensional Gaussian integrals)

Proof of Theorem 3, the product moment Pfaffians
We start with the product moment, defined for bounded measurable g : R → R and disjoint (x i ) by where the product over an empty set, occurring when all the particles have been annihilated, is defined to have value 1. Note that m (1) t (x) is given by the Brownian semigroup applied to g. We also set m (0) ≡ 1. To show that m (2n) is given by a Pfaffian, we shall give a p.d.e. derivation similar in spirit to that showing the Karlin McGregor formula for the transition density for non-intersecting Brownian motions is given by a determinant.
t (x) solves the heat equation, and we must examine the boundary conditions. For n ≥ 2 and when g is bounded and continuous, the functions m (n) are continuous on [0, ∞) × V n and extend to a continuous function in C((0, ∞) × V n ). There are lots of pieces to the boundary of V n , but the most important are the faces F i,n defined by x i = x i+1 and where the remaining x k are disjoint. On F i,n the continuous extension agrees with the lower order moment m (n−2) (x (i,i+1) ), where x (i,j) ∈ R n−2 is the (n − 2)tuple formed by removing x i and x j from (x 1 , . . . , x n ). This can be seen by showing that near the boundary the hitting time between particles starting at x i and x i+1 is likely to occur before any other collision and before time t. On other parts of the boundary the extension agrees with other lower moments.
The system of heat equations for (m (n) : n = 1, 2, . . .) i+1) ) for x ∈ F i,n and i = 1, . . . , n − 1, m forms a closed system, in that each equation has boundary conditions formed by equations of lower order. Note that, typically, the initial condition does not match the boundary conditions. Taking g bounded and smooth, the system has unique solutions in It is enough to specify boundary conditions only on each face F i,n -the Feynman-Kac formula makes it clear that the other parts of the boundary of V n do not affect the value of m (n) .
To establish the Pfaffian (2) stated in Theorem 3, it is enough, by an approximation argument, to treat the case where g is smooth. We shall prove (2) by showing the Pfaffian Pf m (2) t (x i , x j ) : 1 ≤ i < j ≤ 2n solves the system of heat equations above. Note that (2) holds when t = 0 since The Pfaffian is a finite sum of product terms (see (12)) of the form where σ is a permutation given by σ(2k − 1) = i k , σ(2k) = j k for k = 1, . . . , n. Since m  t (x, y) extends continuously to (0, ∞) × {(x, y) : x ≤ y}, the Pfaffian extends continuously to (0, ∞) × V 2n . By the uniqueness for the system of heat equations, it remains to check that the Pfaffian satisfies the required boundary conditions on each face F i,2n which will complete the proof of Theorem 3.
We show the argument for the face F 1,2n where x 1 = x 2 (other faces are similar). We may argue inductively, and suppose that m (k) is given by the Pfaffian for k = 0, 2, . . . , 2n − 2. Our quickest proof is using the representation (15) in terms of Berezin integrals. This gives Pf m The sum M = 2n k=3 m t (x 1 , x k )ψ k is independent of ψ 1 and ψ 2 and the dψ 2 dψ 1 integral becomes (using the rules for Berezin integrals) This simplification leaves only dψ 2n . . . dψ 3 e − 1 2 2n i,j=3 ψ i m (2) t (x i ,x j )ψ j which is the Berezin integral for m (2n−2) (x 3 , . . . , x 2n ).
An argument that avoids Berezin integrals is as follows. Using the recursive relation for Pfaffians (13) we see that the Pfaffian in (2) equals (1,k) ).
Since m (2) t (x 1 , x 2 ) extends to the function 1 on x 1 = x 2 , it remains only to check that vanishes when x 1 = x 2 and t > 0. But this follows from expressing m (2n−2) using (12). Indeed, fix j, k ≥ 3. Then for an expression of the form arising from the kth term in (16) arising from the jth term in (16). These terms agree on x 1 = x 2 and a careful check of the signs of the permutations, and the factors (−1) j and (−1) k in (16), shows they will cancel. One way to do this check is to compare the sign of by counting crossings. The loop joining 2 to j in σ must be replaced by a loop joining 2 to k in σ ′ . This may affect crossings with any of the loops emanating from sites between j and k, and will do so unless a pair of them are joined to each other. There are |k −j|−1 sites between j and k so it will change the parity of the number of crossings exactly when |k − j| is even.

Remark 1.
For odd moments there is also a Pfaffian representation, namely, when where we adopt the convention that m t (x k ). This Pfaffian involves a linear combination of terms of the form sgn(σ)m (1) t (x j 1 )m (2) (x i 2 , x j 2 ) . . . m (2) (x in , x jn ) which again shows that it solves the heat equation when [0, ∞) × V 2n−1 . The recursive Pfaffian relation gives Expanding the Pfaffian along its first row using (13) we obtain for x = (x 1 , . . . , where we again write superscripts x (i,j,...) to mean that we remove the indicated coordinates. The terms with k = 1 and k = 2 cancel on the face F 2n−1,1 where x 1 = x 2 . Moreover on this face, for k ≥ 3, m (1,2) ) along the first row, and this shows the boundary conditions are correct on F 2n−1,1 . Other faces are similar.

But this is the decomposition of
Remark 2. Since our proof relies only on uniqueness for the underlying system of heat equations, the extension of these product moment Pfaffians to more general spatial motions looks quite straightforward, for example to Markovian spatial motions that are continuous and suitably non-degenerate. The Pfaffians in the next section would then also follow for these more general motions, just by algebraic manipulation, once maximal entrance laws characterized by (8) and (11) are established.

Proof of Theorem 2, the Pfaffian point process kernel
Fixing a 1 < . . . < a 2m and choosing g(x) = (−1) i χ(x≤a i ) in (2) we see that both sides of the duality (7) are Pfaffians in the variables (x i ). Choosing g = 0, recalling that an empty product takes the value 1, we see that P A (x i ) [τ < t] is a Pfaffian. The entrance law dualities (8) and (11) show that are Pfaffians in the variables (a i ). The entries in this last Pfaffian are explicit since are all equal to (by Brownian hitting probabilities) We switch dummy variables for the rest of this section, taking x 1 < x 2 < . . . < x 2n and I k = (x k , x k+1 ) so that we start from To prove Theorem 2, we shall identify the Pfaffian point process kernel by differentiating the empty interval Pfaffian (20) above. By scaling we may take t = 1. Differentiate the identity (20) for t = 1 in the variables x 1 , x 3 , . . . , x 2n−1 . The left hand side becomes, formally, Letting x 2l ↓ x 2l−1 for l = 1, . . . , n we reach the n-point density ρ In the appendix 4.3 we give more details verifying the formal differentiation above is valid, by using distributional derivatives.
On the right hand side of (20) we will also differentiate in the variables x 1 , x 3 , . . . , x 2n−1 . Note that each product term in the permutation expansion (12) of the Pfaffian contains exactly one element that involves the variable x 1 . So differentiating in x 1 leads to a similar permutation expansion, but where all the terms that involve x 1 have been differentiated. Repeating this argument, differentiating in x 1 , x 3 , . . . , x 2n−1 yields the Pfaffian where each term in the matrix has been differentiated in the variables x 1 , x 3 , . . . , x 2n−1 that is where the 2 × 2 block formed by the rows 2j − 1, 2j and columns 2k − 1, 2k is given by when j ≤ k. (Note that F ′′ is an odd function and so no sgn is needed in the 2j −1, 2k −1 entry.) Letting x 2l ↓ x 2l−1 for l = 1, . . . , n we obtain the kernel K stated in Theorem 2. The decay in F, F ′ , F ′′ implies that K acts as a suitable bounded operator. The scaling relation (9) implies that the kernel of the distribution time t is t −1/2 K(xt −1/2 , yt −1/2 ).
Remark 4. Starting from the Pfaffian (20) certain other probabilities can, by algebraic manipulation, also be expressed as Pfaffians. We give three examples, leaving details of the derivations to the appendix. In each case F is the 2n × 2n anti-symmetric matrix with elements F ij = P C ∞ [N t ((x j , x k )) = 0] = F (t −1/2 (x j − x i )) as in (19).
• Let I = I 2n be the 2n×2n anti-symmetric matrix with entries 1 above the diagonal.
The annihilating analogue of this is • Let O = O 2n be the 2n × 2n anti-symmetric matrix formed by n copies of the 2 × 2 matrix 0 1 −1 0 down the diagonal and zeros elsewhere. Then Again there is an annihilating analogue.
• LetÔ =Ô 2n be the 2n × 2n anti-symmetric matrix with entrieŝ  3.4 Proof of Theorem 1, the asymptotics for ρ (2n) t We work throughout under the entrance measure P C ∞ . By thinning the corresponding density for annihilating systems differs only by a multiplicative factor 2 −n . The n-point density function ρ is discussed in [12]. Furthermore there we established the following upper bound: for all L > 0 there exists C L < ∞ so that As t → ∞ the entries in the Pfaffian for ρ (n) t are of the form F, F ′ , F ′′ evaluated at points t −1/2 (x j − x i ) close to zero. One may approximate these by using the Taylor expansion for F (z) at small values of z. However, considerable cancellation occurs in the many terms of the Pfaffian and it is not immediately clear how to read off the leading asymptotic decay in t. Indeed the following argument shows at F needs to be expanded to a large number of terms to obtain the correct answer.
We shall analyze first a modified density functionρ (2n) t (x) for x ∈ V 2n , which is a density for the measure (where we recall that I k = (x k , x k+1 )). We claim that where φ(z) = z exp(−z 2 /4). This follows formally, as in section 3.3, by differentiating (20) in all variables x 1 , x 2 , . . . , x 2n , and using that, (which follows from differentiating each term in the permutation expansion (12) of the Pfaffian). We give more details in the appendix 4.3.
The advantage of the representation (27) is that it is a Pfaffian all of whose entries are of the form f (x i − x j ) for a single function f , and this allows us to apply the following lemma, proved at the end of this section, that gives an expansion for a Pfaffian whose entries are close to the zero of an odd function.
Each term in this sum is of a smaller order in t by (26).
Examination of the proof shows that we need not let the values of x 1 , . . . , x 2n be fixed, and that in fact we may take the supremum over any positions (x i (t)) provided that sup i |x i (t)|t −1/2 → 0 as t → ∞.
Proof of Lemma 5. Let Φ be the 2n × 2n anti-symmetric matrix with entries given by Φ ij = φ(y j − y i ). The aim is to show, for small y, that where J and R are as in the lemma (with n fixed and suppressed) and V is the 2n × 2n Vandermond matrix given by V ij = y i−1 j . Since det(V ) = 1≤i<j≤2n (y j − y i ), the conclusion then holds from Pf (V T (J + R)V ) = det(V )Pf (J + R) For small |y| we expand by analyticity (writing φ k (0) for the kth derivative of φ at zero) where we have rearranged using l = n − k in the penultimate equality. Note that It remains to re-express the remaining terms in (29) as the desired remainder.
Recall the symmetric polynomials σ 2n k (y) defined for y ∈ R 2n by Note that σ 2n k is a polynomial of order k. Since σ 2n 0 ≡ 1 we may choose λ = y i to see that Multiplying by y p i we see that where τ 2n,p+2n k (y) is a polynomial of order p + 2n − k + 1. Using this substitution in the remaining terms of (29), that is where k or l is at least 2n + 1, we find (formally) that Note the lowest order of the polynomial entries in the terms for R pq is of order 1. In the appendix 4.4 we check that this rearrangement of (29) is valid when |y| is suitably small and that the required error bound |R pq (y)| ≤ C(n, φ)|y| holds.
Thanks. We would like to thank our colleague Dmitriy Rumynin for advice on the use of symmetric polynomials.

Details for section 2.3
We give a few details on (one approach to) the results surveyed in section 2.3. For coalescing systems one can use monotonicity, adding initial particles one by one, to construct the infinite system. This is not available for annihilating systems, so we sketch a weak convergence argument that applies to both systems.
One can control moments by bounds on the n-point density function. Indeed ρ on V n , depends on the initial condition, but satisfies the bound ρ (n) t (x) ≤ C n t −n/2 uniformly over all possible finite initial conditions (x i ). This follows by duality for n = 1 and by anti-correlation for n > 1 (see [12]). It follows that E C (x i ) [N p t (a, b)] is bounded, for each t, p > 0, a, b ∈ R, uniformly over finite initial conditions (x i ).
Fix µ ∈ M 0 and take finite measures µ n so that µ n → µ (recall we are using vague convergence). The moment bounds above imply that the laws of N t on M LF P under P C µn are tight. Take a subsequence n ′ along which they converge to a limit, which we denote Q. The functions ν → F (a i ) (ν) := χ(ν(I 1 ) = ν(I 3 ) = . . . = ν(I 2n−1 ) = 0) are discontinuous on M LF P . However the moment bound E C (x i ) [N t (a, b)] ≤ C(t)(b − a) holds also for the limit law Q and implies that ν({a i }) = 0, Q(dν) almost surely. This shows that Q does not charge the discontinuity set of F (a i ) . Then we may pass to the limit in (6) to deduce that These functionals do not characterize a law on M LF P , but they do characterize a law that is supported on M 0 . To see this note that for ν ∈ M 0 From this one may use (34) to find ν([x 1 , y 1 ]) . . . ν([x n , y n ])Q(dν) which, by the moment bounds, determine Q. To see that Q is supported on M 0 note that This bound holds uniformly over n and hence also for the limit law Q. Then the conclusion follows from the usual covering argument, for instance Thus the law Q is determined and we may define p t (µ, dν) to equal Q(dν).
The remainder of the results in section 2.3 follow using similar tools. For example, for the continuity of µ → p t (µ, dν), that is the Feller property, suppose that µ n → µ in M 0 . The moment bounds, which still hold for infinite initial conditions, imply the tightness of p t (µ n , dν). Passing to the limit in shows that any limit point of p t (µ n , dν) must be p t (µ, dν). The semigroup property, for bounded continuous F : M LF P → R, which is valid for finite measures µ extends to hold for µ ∈ M 0 by approximation, using the Feller property. The same tightness and characterization methods establish the existence of a law characterized by (8), and justify the arguments in (10) that many initial laws are attracted to it. That (8) determines an entrance law can be established by passing to the limit in (35) along µ = k δ λ −1 k as λ → ∞.
The annihilating case follows the same lines, with moments controlled since the npoint density and moments for annihilating systems are bounded by the corresponding coalescing system. The coalescing duality formula (6) is replaced by the annihilating duality formula (7), and to see that this will characterize the law note that for ν ∈ M 0
Pf (I) = 1 for I the 2n × 2n anti-symmetric matrix with entries 1 above the diagonal, and the formula (14) specializes to (using for a 2n×2n anti-symmetric matrix A, that Pf (−A) = (−1) n Pf (A)). We combine this with a simple combinatorial identity (which can be checked by induction on n): suppose that (m j,k : 1 ≤ j < k ≤ n) satisfy the collapsing product m j,k m k,l = m j,l for all j, k, l; then where the final sum is over all subsets J of {1, 2, . . . , n} of even size, and if J = {k 1 , . . . , k 2m } where k 1 < . . . < k 2m then m J = m k 1 ,k 2 m k 3 ,k 4 . . . m k 2m−1 ,k 2m (and with m ∅ = 1). If n is even then the last term of this series is m 1,2 m 3,4 . . . m n−1,n . Note that m j,k = α k−j m j,k also satisfym j,kmk,l =m j,l and applying the above form one obtains a decomposition for n−1 k=1 (1 + αm k,k+1 ). In particular for α = −1 we get Now apply this with m j,k = χ(N t ((a j , a k )) = 0). These satisfy the collapsing products almost surely under the probability P C ∞ . The Pfaffian (20) shows that E C ∞ [m J ] = Pf (F | J ) and so We may apply the same argument for the annihilating case taking m j,k = (−1) Nt((a j ,a k )) , where 1 − m j,k = 2χ(N t ((a j , a k )) is odd), to find (22). where the sum is over all J 1 of the form in (36) (including the empty set). We use another combinatorial identity, also straightforward by induction on n: where the sum is over all J 1 of the form in (36) (including the empty set). Arguing as in the previous example leads to (23).
In the last step we have passed the derivatives onto the Pfaffian, which is smooth since F is smooth, and used F ′′ (x) = (4π) −1/2 φ(x).