Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration

Two classes of interacting particle systems on $\mathbb{Z}$ are shown to be Pfaffian point processes at fixed times, and for all deterministic initial conditions. The first comprises coalescing and branching random walks, the second annihilating random walks with pairwise immigration. Various limiting Pfaffian point processes on $\mathbb{R}$ are found by diffusive rescaling, including the point set process for the Brownian net.


Introduction and statement of key results
Interacting particle systems of reaction-diffusion type have been an object of mathematical investigation for a long time, see e. g. [9]. It turns out that many such systems can be studied using methods of integrable probability. This paper continues the study in [6], where it was shown that systems of instantaneously annihilating or coalescing random walks on Z are Pfaffian point processes, at any fixed time and for all deterministic initial conditions. The purpose in this paper is to describe two additional mechanisms that preserve this Pfaffian property. The Pfaffian property should be useful to investigate statistics, such as asymptotics of correlation functions (as in Theorem 1 of [18]) or for studying gap probabilities (as in [17], [13] or [12] for examples from random matrix ensembles).
The algebraic structure of the generators of various one dimensional interacting particle systems (without necessarily preservation of particle numbers) has been investigated, and is in many examples intimately linked to Hecke algebras, see the reviews [1] and [10]. A future aim is to better understand how to deduce from these algebraic properties the concrete statistical properties of our models, such as the correlation functions, and in particular what algebraic properties lie behind the emergence of the Pfaffian property.
The connection between coalescing and annihilating systems is well known, both at the analytic level and also via direct couplings of the two processes, see [16] for review. Such pathwise couplings are fundamental to the study of the Brownian Web and Net and the dual Brownian Web and Net (and their discrete analogues -see [15] and [14]). Moreover all the models in this paper can be found 'within' the Net, and so there will be many pathwise couplings available. Our results can be seen as analytic consequences of a strong coupling between our two models, namely coalescing plus branching random walks and annihilating random walks plus pairwise immigration, with finite systems of annihilating random walks. We do not exploit this coupling, but we concentrate on investigating 'fixed time' laws, aimed at revealing an algebraic structure behind the processes. 1

(BCRW) Branching coalescing random walks.
In addition to instantaneously coalescing random walks, we allow nearest neighbour binary branching: any given particle may instantaneously produce an independent copy at a nearest neighbour. The dynamics of this continuous-time model on Z are informally described as follows. Between interactions particles perform independent nearest neighbour random walks with jumps x → x − 1 at rate q, and x → x + 1 at rate p.
If a particle jumps onto an occupied site then the two particles instantaneously coalesce. Independently a particle branches x → {x, x − 1} at rate ℓ, and x → {x, x + 1} at rate r.
Branching events respect coalescence: if a particle branches onto an occupied site then the existing and new particles instantaneously coalesce.
The generator is given in (10), and this characterises the law of a process with values in {0, 1} Z which we denote as the BCRW model. We write (η t : t ≥ 0) for canonical variables and P η for the law corresponding to an initial condition η ∈ {0, 1} Z . As in [6], we fix t ≥ 0 and consider η t as a point process on Z, and our aim is to establish that this is a Pfaffian point process for a suitable Pfaffian kernel K(x, y). A simple point process on Z is called Pfaffian if there is a 2 × 2 complex matrix-valued kernel K on Z × Z, (x, y) → (K αβ (x, y)) 1≤αβ≤2 , such that the n-th intensity function of the process ρ (n) is given by 1≤i,j≤n , n = 1, 2, . . . (see [2] for more details).
Recall the notation from [6]: for η ∈ {0, 1} Z and y ≤ z we write so that η[y, y) = 0. For f : Z → R we write difference operators as 1 A colouring construction of annihilating particles starting from a coalescing system can be used to relate the Pfaffian point processes describing the fixed time laws of coalescing and annihilating random walks or Brownian motions, see [18]. At the moment we do not see a similar link between branching-coalescing random walks and annihilating random walks with pairwise immigration, but a further investigation might uncover such a link.
If the difference operator D ± is applied to the i-th argument of a function of several variables we use the notation D ± i .
Theorem 1 Let (η t : t ≥ 0) be the BCRW model with parameter values satisfying For any initial condition η ∈ {0, 1} Z , and at any fixed time t ≥ 0, the variable η t is a Pfaffian point process with kernel K given, for y < z, by and K 12 (y, y) = 1 − 1 φ K t (y, y + 1), with all other entries determined by anti-symmetry, where , for t ≥ 0 and y < z, and φ = 1 + ℓ q = 1 + r p . The same result holds when p = r = 0 and q, l > 0 by taking φ = 1 + ℓ q , or when q = l = 0 and p, r > 0 by taking φ = 1 + r p .
Remarks. 1.1. For certain random initial conditions, including the natural case when the sites (η 0 (x) : x ∈ Z) are independent, the process does remain a Pfaffian point process (see the remarks after the proof of Lemma 6). The invariant measure, which is product Bernouilli, can be considered as a Pfaffian point process (see the remark at the end of section 2.) 1.2. The restriction pl = qr seems to be necessary for the Pfaffian point process property, and we don't fully understand its origin. Luckily, in the large scale diffusive limits explained below this restriction plays no role. Indeed the limit continuum systems depend on three parameters. This is consistent with the three parameters that can be used in the Brownian net, for example the two drift parameters and a common diffusion parameter for the left and right paths, which together determine the branching rate, see [15]. For the four parameter lattice models, there exist Markov duality functions even when the the restriction pl = qr is not true, and it would be of interest to examine whether some other algebraic structure holds for the correlation functions.

1.3.
In section 4 we investigate continuum limits to our Pfaffian point kernels, under space time diffusive scaling, yielding Pfaffian point processes on R. For the branching parameter to have an effect in the continuum kernel, it needs to be scaled to grow suitably fast. This however is well understood in the construction of the Brownian net (see Sun and Swart [15]), and we of course need the same scaling of the branching as in the discrete time branching random walk approximations to the net. The Brownian net is a continuum collection of space time paths found by scaling discrete time branching coalescing random walks started at all space-time lattice points. The point set process (ξ A t : t ≥ 0) within the net, is defined by looking at all points at time t that are on paths that start at time zero in a set A ⊂ R. It is known to be a Feller process taking values in the compact sets of R with a suitable Hausdorff metric (see Theorem 1.11 in [15])). The approximating discrete time branching coalescing random walks used in the construction of the net are not Pfaffian point processes (the discrete time difference equations analogous to the Markov duality below are not solved by Pfaffians). However the difference between the discrete time models and continuous time models does not affect the continuum limits. We leave the verification of these technical details to a forthcoming paper, but we state here the resulting Pfaffian property for the Brownian net point set process, which answers the first open problem in section 8.3 of the survey paper on the net [14]. The Brownian net can have a parameter b ≥ 0, controlling the branching rate, where the embedded left-right paths have drifts ±b. The standard Brownian net corresponds to b = 1, and the Brownian web corresponds to the case b = 0.

Proposition 2
The transition density p t (A, dB) for the point set process (ξ A t ), within the Brownian net with parameter b ≥ 0, is equal to the distribution of the closed support of a Pfaffian point process on R with the kernel K A t of the form for y < z, where D i is a partial derivative in the ith variable, and where (K A t (y, z) : y, z ∈ R 2 : y ≤ z) is the unique bounded solution to the PDE The examples in section 4 will illustrate how continuum kernels arise of this form.

(ARWPI) Annihilating random walks with pairwise immigration.
As for the previous model, particles jump left or right at rates q and p. If a particle jumps onto an occupied site then the two particles instantaneously annihilate. In addition, independently there is immigration of a pair of particles on sites {x, x − 1} at rate m.
Immigration respects annihilation: if a particle immigrates onto an occupied site then the existing and new particles instantaneously annihilate. The generator is given in (13) and characterises a Markov process that we denote as the ARWPI model.

Theorem 3
For any initial condition η ∈ {0, 1} Z for the ARPI model, and at any fixed time t ≥ 0, the variable η t is a Pfaffian point process with kernel K given, for y < z, by and K 12 (y, y) = − 1 2 D + 2 K t (y, y), with other entries determined by anti-symmetry, where , for t ≥ 0 and y < z.
Remarks. 1.4. The Glauber spin chain on Z is an assignment of ±1 spin values to each site which independently flip according to rates determined by nearest neighbour spins [8]. Sites favour aligned spin and at zero temperature a site surrounded by spins of the same sign does not flip, and the domain wall between regions of constant spin form a system of annihilating random walks on the dual lattice. At positive temperature, a spin may spontaneously flip regardless of its neighbours, and this corresponds to the creation of a pair of neighbouring domain walls. Since the Glauber model can be solved at all temperatures by mapping to a system of free fermions (Felderhof [5]), it is reasonable that the extra immigration of pairs does not destroy the Pfaffian property of solutions. A model with Poisson immigration of single particles is perhaps of more interest, but we do not see a simple algebraic structure behind this model.

1.5.
We give in section 3 a spatially inhomogeneous version of Theorem 3, where the parameters p x , q x , m x may be site dependent, in particular allowing immigration of particles at different rates at different places. Continuum limits can also be found for the ARWPI model under diffusive rescaling, where the immigration rate m must be scaled suitably so that they have a non-trivial effect in the limit. It is a pleasant fact that in many cases the Pfaffian kernel can be found completely explicitly. For example in example (d) in section 4, which we call the Brownian firework, pairs of particles are immigrated only at the origin at an infinite rate. This has a steady state X (c) ∞ where the immigration and the annihilation balance each other: The corresponding intensity is given by ρ when y 1 , . . . , y n < 0 < z 1 , . . . , z m and hence that X are independent point processes. The infinite strength firework of particles at the origin leads to the two half spaces being independent.
1.6. One simple consequence of the Pfaffian structure is an estimate showing exponential convergence to equilibrium (which is a product Bernouilli distribution). Indeed, writing ρ (N ) t (y 1 , . . . , y N ) for the N point intensity function of the process as time t, we claim that there exist C N for all N ≥ 1 so that uniformly over for all y 1 , . . . , y N and over all initial conditions. (Recall that m is the immigration rate of pairs). This follows for a deterministic initial condition η once we show that since the entries in the kernel K are differences of the bounded function K t , so that the Pfaffian formula for ρ (N ) t is given by a finite linear combination of finite products of K t (y i , y j ). For a general initial condition, one can first condition on the initial condition and then use the fact that the above estimates are uniform in η.
(To show (8) we can solve for K t (y, z) explicitly. Indeed, fixing a deterministic initial condition η, the kernel K t (y, z) has a representation in terms of a pair of independent continuous time random walkers (Y t , Z t ) with generator qD Then the equation solved by K t (y, z) (see Lemma 7) shows that and the estimate (8) follows easily from subtracting these two probabilistic representations. Solving explicitly we have and K 12 (y, y) = (1 − θ)/2. A little manipulation shows that this is a kernel for a product Bernouilli(θ) distribution, wherê Indeed by conjugating E T KE with an elementary matrix E for row and column operations (which leaves the corresponding point process unaltered) K(y, z) for y < z can be changed successively to while leaving K 12 (y, y) = (1 − θ)/2 unchanged. Then the Pfaffian for ρ (N ) has only a single non-zero entry on the top row, and expanding along this row one finds We remark that no exponential convergence statement such as (7) holds for the BCRW model, since long empty gaps in the initial condition are only filled at linear speed. However, for many initial conditions there is weak convergence to a Bernoulli invariant measure -see the remarks in section 2.
We start with a terse summary of the main steps: the proof in [6] for coalescing systems without branching uses the empty interval duality function; this function remains a duality function for the branching model, but the empty interval probabilities are no longer given by a Pfaffian; however the duality function can be adjusted by a suitable phase factor in a way that again yields Pfaffians. The use of empty interval probabilities to study branching systems is not new, see for example Krebs et al. [11] where the equations for a single empty interval are used to study various finite systems.
The generator for BCRW is given, for suitable F : where η x,y (resp. η b x,y ) is the new configuration resulting from a jump (respectively a branch) from x onto y, that is For n ≥ 1 and y = (y 1 , . . . , y 2n ) with y 1 ≤ · · · ≤ y 2n we define the function Σ y (η) as the inidcator that the intervals [y 1 , y 2 ), . . . , [y 2n−1 , y 2n ) are all empty; explicitly We define a one-particle operator, acting on f : Z → R, by Lemma 4 For y 1 < · · · < y 2n the action of the generator L on Σ y (η) is given by where the subscript y i indicates the variable upon which the operator acts.
Proof of Lemma 4. A direct check shows that the terms of L coming from left and right jumping contribute to LΣ y (η) (see [6] for the details of this calculation). It remains to compute the terms arising from branching. Consider the term from left branching. The modified branching configuration η b x,x−1 differs from η only at the site x − 1, so for each x there can be a change in at most one of the indicators in Σ y , so we may write Fix y < z and consider the generator contribution for a single empty interval indicator, namely ℓ The terms indexed by x ≤ y and x ≥ z+1 are zero, as the modified configuration is unchanged in the interval [y, z). The terms indexed by y ≤ x ≤ z − 1 are also zero since there must be a particle at x to branch to the left from, in which case both empty interval indicators are zero. The remaining summand, when x = z, is given by A similar calculation reveals that the term of the generator arising from right branching satisfies r Collecting up contributions gives the claimed action.
The expression for LΣ y (η) in Lemma 4 has different operators acting on even co-ordinates y 2i and odd co-ordinates y 2i−1 . The proof of the Pfaffian property in Lemma 6 below is facilitated if each co-ordinate has the same operator acting on it, and the aim is to introduce a suitable phase factor precisely to have this effect. The phase factor is defined by φ (y 2i −y 2i−1 ) for y = (y 1 , . . . , y 2n ) ∈ R 2n and n ≥ 1 and the following lemma shows that the value φ = 1 + ℓ q = 1 + r p is the correct choice.
Lemma 5 Suppose the rates p, q, r, l satisfy (1) and φ is chosen as in the statement of Theorem 1. Then for y 1 < · · · < y 2n Proof. This is a direct calculation. For a function of one variable we find that a change of f : Z → R tof (y) = c y f (y), for c = 0, produces the change L p,qf (y) = c y L pc −1 ,qc f (y) + (pc −1 + qc − p − q)c y f (y).
We apply this in the even coordinates with c = φ and in the odd co-ordinates with c = φ −1 .
The value of φ is chosen so that the corresponding difference operators now agree on both sets of coordinates. Different potential terms are created at odd or even co-ordinates, but these can be summed and then redistributed equally between all co-ordinates, which yields the constant c 0 . The equivalent expressions for c 0 follow from the definition of φ.
Lemma 6 For all η ∈ {0, 1} Z , for all n ≥ 1, y 1 ≤ · · · ≤ y 2n and t ≥ 0 Proof of Lemma 6. We follow closely the arguments for the pure coalescing case, as in Lemma 7 of [6], pointing out the changes needed here but refer the reader to [6] details. We recall the notation for the discrete cells V 2n and their faces (∂V , defined as follows: V 2n = {y ∈ Z 2n : y 1 < y 2 < . . . < y 2n }, = {y ∈ Z 2n : y 1 < y 2 < . . . < y i = y i+1 < . . . < y 2n }. We also use the notation y i,i+1 for the vector y with coordinates y i and y i+1 removed; thus when n ≥ 2, for y ∈ ∂V To establish the identity in the Lemma, one checks that both sides are solutions to the system of ODEs, in this case taking u (0) ≡ 1. This infinite system can be shown by induction on n to have unique solutions, within the class of functions that suitable exponential growth at infinity. As in [6], the fact that (t, y) → E η [Φ(y)Σ y (η t )] is a solution follows from Lemma 5 and the extra fact that the phase factor satisfies Φ(y) = Φ(y i,i+1 ) on the boundary ∂V The fact that the Pfaffian is also the solution to this system follows as in the non-branching case in [6]. The starting point is that the function K t (y, z) solves the equation with the boundary condition K t (y, y) = 1 for all t > 0. The Pfaffian is made up terms that are products of copies of K using all the variables y 1 , . . . , y 2n , each of which can be checked to be a solution of the the full system. The combination of terms in the Pfaffian is exactly what is needed t satify the boundary conditions of the system. The details of these arguments are explained in Lemma 7 of [6] with the only change being that we need to verify the extra phase term does not affect the initial condition being satisfied. However we may rewrite the entries in the Pfaffian at time zero using The Pfaffian Pf(K (2n) (0, y)) is therefore the limit of Pfaffians of a matrix A with entries in quotient form A ij = a i /a j for i < j. For such matrices Pf(A) = a 2 a 4 . . . a 2n /a 1 a 3 . . . a 2n−1 (see the appendix of [6] for example) and hence, taking a < min{y i }, as required.
Remark. 2.1. The last lemma is the point at which to observe that for certain random initial conditions, the Pfaffian property is still true. Indeed suppose that η 0 is random but that E [Φ(y)Σ y (η 0 )] is still given by a 2n × 2n Pfaffian with entries K 0 (y i , y j ) for i < j, for some K 0 of exponential growth. The statement of the lemma then still holds, and so does Theorem 1, which is deduced from the lemma without any changes. A simple example is when η 0 (x) are independent Bernoulli(θ x ) variables. Then the condition above is true with A similar observation holds for the AWRPI model discussed in the next section.
Proof of Theorem 1. As in [6], the desired particle intensities E η [η t (x 1 ) . . . η t (x n )] may be recovered from the empty interval probabilities. From Lemma 6 we have t, y)).
Note the ij'th entry of the matrix D(y)K (2n) (t, y)D(y) is still a function only of the variables y i , y j . We now follow the argument in [6], where the intensities are derived from the empty interval probabilities via discrete derivatives. This leads to the process η t being a Pfaffian point process with a kernelK(y, z) where for y < z φ y+z K t (y, z) −D + z (φ y−z K t (y, z)) −D + y (φ z−y K t (y, z)) D + y D + z (φ −y−z K t (y, z)) , andK 12 (y, y) = −D + z φ y−z K t (y, z) | z=y = 1 − φ −1 K t (y, y + 1).
It remains to massage this kernelK into the form K stated in the Theorem, which uses only row and column operations that can be represented by conjugation with suitable matrices, that is we may replaceK(y, z) by A(y)K(y, z)A T (z) for any 2-by-2 matrix A(y) (depending only on the variable y) that has determinant 1.
Expanding out the discrete derivatives inK using the discrete product rule, and then conjugating the final matrix with a diagonal matrix A(y) = φ −y 0 0 φ y leads to an equivalent kernel, which we still denoteK, with entrieŝ Subtracting the first row and column from the second row and column, and then further conjugating with a diagonal matrix A(y) = φ −1/2 0 0 φ 1/2 gives the equivalent kernel K K(y, z) = φ −1 K t (y, z) −K t (y, z + 1) −K t (y + 1, z) K t (y + 1, z + 1) , withK 12 (y, y) = 1 − φ −1 K t (y, y + 1). Finally, the desired kernel K is obtained by again subtracting the first row and column from the second.

Remarks. 2.2.
Letting t → ∞ the process, for any non-zero initial condition, converges to an invariant Bernoulli distribution. It is fun to see this via the Pfaffian kernels by solving for K t (y, z) explicitly. Consider first the maximal initial condition η 0 (x) = 1 for all x. Then K 0 (y, z) = 0 and the kernel K t (y, z) has a representation in terms of a pair of independent continuous time random walkers (Y t , Z t ) with generator qφD Solving explicitly we find K ∞ (y, z) = φ −(z−y) , and the Pfaffian kernel of the form (2) corresponding to K ∞ (y, z) is and K 12 (y, y) = 1 − φ −2 . A little manipulation (using row and column operations as in (9) for the ARWPI model kernel) shows that this is a kernel for a product Bernoulli(1 − φ −2 ) distribution. Convergence of K t (y, z) implies that all entries in the Pfaffian kernel converge, which in turn implies that the process η t converges as t → ∞ to the product Bernoulli(1−φ −2 ) in distribution in the product topology.
For general non-zero initial conditions the same is true. Rather than analyse the kernel, we use a simple coupling argument for attractive nearest neighbour systems. All non zero solutions can be coupled between the maximal solution and a solution started from a single point. It therefore is enough to prove convergence for the process η 0 t started from a single occupied site, say the origin. But this process can be coupled with the process η Z t started from all occupied sites. Indeed by a graphical construction (or equivalently solving a system of equations using the same Poisson drivers for jump and branch events) shows that where l t , r t mark the leftmost and rightmost occupied site in η 0 t . The behaviour of the pair {l t , r t } is however easy to understand: provided p+l > q and q+r > p we can ensure l t → −∞ and r t → ∞. Under these conditions the process looks like η Z t in a growing interval, and we already know η Z t converges to Bernoulli equilibrium.

2.3.
It is natural to look for a spatially inhomogeneous version of the BCRW model, where p x , q x , l x , r x are allowed to be site dependent. This was explored in the thesis [7] and the Pfaffian property can survive, but under a somewhat stronger condition on the parameters that we do not yet fully understand.

Proof of Theorem 3.
The result for the annihilating model with immigration follows by very similar lines, and we remark only on the changes caused by the new immigration term. The result holds for systems with spatially inhomogeneous rates. There is no additional complexity in the proof, so we continue in this general framework.
The generator for (spatially inhomogeneous) ARWPI is given, for suitable F : {0, 1} Z → R by where η x,y is the new configuration resulting from a jump from x onto y, that is η x,y (z) = η(z) for z ∈ {x, y}, η x,y (x) = 0, η x,y (y) = η(x) + η(y) mod (2), and where η i x−1,x is the new configuration resulting from a pair immigration onto {x − 1, x} defined by Note that any immigrating particle instantly annihilates with any existing particle on the target site. We suppose m x , p x , q x are uniformly bounded, so that this generator uniquely determines a Markov process.
The following spin product function is a Markov duality function for this generator (as used for the pure annihilating model in [6]). For n ≥ 1 and y = (y 1 , . . . , y 2n ) with y 1 ≤ · · · ≤ y 2n we define the spin product We define the one-particle operator L, acting on f : Z → R, by Lemma 7 For y 1 < · · · < y 2n the action of the generator L on Σ y (η) is given by Proof of Lemma 7. As in [6] the terms of L coming from particle motion contribute It remains to compute the immigration term. Note that the modified immigration configuration η i x−1,x differs from η on at most two sites, x − 1 and x. Since the y i are strictly ordered, the intervals [y 2i−1 , y 2i ) are separated by at least one site, whereby a pair of adjacent sites −1, x can intersect at most one of the intervals. In particular any change due to immigration affects at most one interval [y 2i−1 , y 2i ), leading to the formula Fix y < z and consider the generator contribution for a single spin product (−1) η[y,z) , namely The terms indexed by x ≤ y − 1 and x ≥ z + 1 are zero, as the modified configuration is unchanged in the interval [y, z). The terms y + 1 ≤ x ≤ z − 1 are also zero, since the immigration of two particles does not change the parity of η[y, z). The remaining terms give identical non-zero contributions: for x = y or x = z All together the immigration term is given by Collecting the jump and immigration terms gives the desired expression.
Proof of Theorem 3. Following the argument from [6], we first claim that for all η ∈ {0, 1} Z , for all n ≥ 1, y 1 ≤ · · · ≤ y 2n and t ≥ 0 where K (2n) (t, y) is the anti-symmetric 2n × 2n matrix with entries K t (y i , y j ) for i < j, can then be recovered from product spin expectations via discrete derivatives and yield the stated kernel K(y, z).

Some continuum Pfaffian point process limits.
The entries for the Pfaffian kernels K(x, y) in both the branching model and the immigration model, are determined by a scalar function K t (x, y) that solves a certain discrete heat equation. Under diffusive space time scaling, and with suitable scaling of the parameters, we can obtain natural limiting Pfaffian point processes X (c) on R, with associated continuum kernels K (c) (where the superscript (c) stands for continuum). We record here certain examples, simply to add to the rather small current list of explicit Pfaffian point process kernels. Two points are perhaps of greatest interest: 1. Unlike the continuum examples from [6], alongside the diffusive scaling of space-time, the reaction parameters controlling branching and immigration must be simultaneously scaled, so that they have a non-trivial effect on the continuum limit. Indeed branching but instantly coalescing Brownian motions do not have a simple meaning, and nor does immigration of instantly annihilating pair of Brownian motions onto the same point. However, since both discrete process are Pfaffian whose entire statistics are controlled by a kernel whose entries solve a discrete PDE, the correct scaling for the parameters is easily revealed by examining the convergence for the differential equations.
2. The discrete equations behind coalescing models with branching and annihilating models with pairwise immigration are both discrete heat equations with a constant potential. This can be used to show there is an equality in law for the fixed time particle positions between these two models, if parameter values and initial values are chosen carefully. This connection exists at the discrete level (see [7]) but is most transparent for the limiting continuum models, and we detail this in the remarks after example (b).
In each of the four examples below we define where η t is one of the models studied earlier, with an initial condition and ǫ dependent The unique bounded solution K (c) t (y, z) has a probabilistic representation in terms of a two dimensional Brownian motion (Y t , Z t ), run at rate 2α (that is scaled to have variance 2αt at time t) and started at (y, z), namely Solving for K (c) t (y, z) explicitly allows one to read off the one point density (where the error function is defined by erf(x) = (2/ √ π) x 0 exp(−t 2 )dt.) The kernel also has a limit as t → ∞, in particular K It is no longer enough in the continuum to just examine convergence of K (c) t , but an examination of the exact formula shows that both K (c) and its first two derivatives converge, uniformly over y, z, as t → ∞, which implies that the continuum point processes X (c) t converge as t → ∞ to a point process X and K (c) ∞,12 (y, y) = 1 2 β α . One can identify the limit, this time a disguised form for the kernel for a Poisson process. Indeed the same row and column operations as in the discrete case allow the Pfaffian of the above kernel to be easily computed explicitly and the n-point intensity is given by Thus the distribution of the (continuum) point process in the large time limit is a Poisson process rate 1 2 β α . The four entries in the kernel converge exponentially to the t = ∞ limit. As in the discrete ARWPI model, this can be used to show the exponentially fast convergence ρ (n) t (y 1 , . . . , y n ) → ρ (n) ∞ (y 1 , . . . , y n ) as t → ∞, for any fixed n and uniformly over y i . Explicit formulae can be found for a variety of other initial conditions (see [7]). For example for an initial Bernouilli(ǫθ) condition, where θ is fixed, only the initial condition in (16) changes to K (ǫ) 0 (y, z) = (1 − 2ǫθ) ǫ −1 (z−y) , and the initial condition for the limiting PDE (18) changes to K (c) 0 (y, z) = exp(−2θ(z − y)). In the maximal case, η x = 1 for all x, the initial conditions K (ǫ) 0 (y, z) = (−1) ǫ −1 (z−y) are extermely oscilatory, but they converge in distribution to the zero function which is sufficient to imply suitable convergence of the kernels at a fixed times t > 0.
Similar row and column manipulations as in the discrete case show that this kernel encodes a Poisson process of rate 2 β α .
Remarks. 4.1. In many formulations of determinantal point processes, the determinantal kernel D is associated to an integral operator D on L 2 (R), and the diagonal values D(y, y) are linked to those of (D(y, z) : y < z) by the fact that the operator is assumed to be of trace class. One might ask the same for the Pfaffian case, asking for four operators K ij on L 2 (R). For our examples this link is broken: the operators K ij acting on L 2 (R) would have discontinuities along y = z and are not expected to be trace class. The diagonal values K 12 (y, y) are not given, for example, as even the continuous limit of K 12 (y, z). This is also the case for classical Pfaffian kernels, for example for GOE.
The operator formulation is useful, for example when applying the theory of Fredholm determinants or Fredholm Pfaffians, and in classification theorems. However, we state our continuum Pfaffian kernels in the form of the five measurable functions, namely (K ij (y, z) : y, z ∈ R, y < z)) i.j∈{1,2} and (K 12 (y, y) : y ∈ R). These five functions are what appear in the Pfaffian formulae for the intensities ρ (N ) . The kernel (22) can be adjusted, by row and column operations, so that for example the the diagonal values K 12 (y, y) are given as the continuous limit of K 12 (y, z) as z ↓ y, for example to We hope that this form may be more useful for example when manipulating Fredholm Pfaffians (as for example in the manipulations for the gap probabilities for the GOE spectrum).

4.2.
With our convention on kernels just as measurable functions, a Poisson rate γ process can be realised as a Pfaffian point process with kernel γJ where J(y, z) = 0 for y < z and J 12 (y, y) = γ. (The same convention would allow Poisson processes to be determinantal processes with a purely diagonal kernel. ) The kernel in example (b) is connected to Poisson thickening. A locally finite point process X can be γ thickened by adding the points of an independent Poisson process Y of rate γ, producing a new point process X + Y . If the original point process was Pfaffian with kernel K then the thickened process remains Pfaffian with kernel K + γJ. Indeed, since the points of X and the Poisson process never meet, the intensities for the thickened process are given by where |J| is the size of the subset J. But the Pfaffian Pf(K + γJ) can be expanded by the Pfaffian sum formula to give exactly this relation.
A locally finite point process X can be γ thinned by removing each point independently with probability γ, producing a new point process which we denote Θ γ (X). If X is Pfaffian with kernel K then the thinned process Θ γ (X) remains Pfaffian, with the kernel γK.
Since the PDE behind both the continuum branching process and the continuum pairwise immigration model is the same, the heat equation with constant potential, it is not surprising that there is a connection between their fixed time distribution. Using thickening and thinning we can state this: let X 1 = the diffusion limit of BCRW with p = q = α, l = r = 2ǫ √ αβ and η ≡ 1; X 2 = the diffusion limit of ARWPI with p = q = α, m = ǫ 2 β and η ≡ 1; Y = a Poisson point process of rate 1 2 β/α, independent of X 2 .
Then, as point processes on R, Θ 1/2 (X 1 ) The proof is just the verification that the Pfaffian kernels are identical.
There is a nice dynamic coupling argument that connects annihilating Brownian motions with coalescing Brownian motions (see [18]), but we do not know a dynamic coupling that explains the above equality of distributions.
A similar identity also works for carefully chosen Poisson initial conditions, and also for the processes on Z with suitably chosen initial conditions (many details are in [7]).
(c) Branching and coalescing model with a single initial particle.
We take the same parameter choices as in example (a) but choose an initial condition that is a single particle at the origin. The initial conditions for (21) and (22) where ψ t (x) = 2 erfc x − 2 √ αβt √ 2αt .
As in the discrete setting, the fixed time distribution started from a single site is quite easy to understand. The limit behaviour of the leftmost and rightmost particles {l t , r t }, under the parameter scaling we have used, is known to become that of a sticky pair {L t , R t }, that is the solution of dL t = I(L t = R t )dB L t + I(L t = R t )dB − 2 αβdt, L 0 = 0, dR t = I(L t = R t )dB R t + I(L t = R t )dB + 2 αβdt, R 0 = 0, where B R , B L , B are three independent Brownian motions run at rate 2α. Uniqueness in law holds and L t ≤ R t for all t ≥ 0 (see Proposition 2.1 in [15]). Let Y be an independent Poisson process of rate 2 β α . The point process X (c) t can be constructed as the pair of particles L t and R t together with the particles from Y that lie inside (L t , R t ). Indeed then P[X t [y, z) = 0] = P[R t < y] + P[L t ≥ z] + e Weak (and strong) invariance principles for random walks and their local times are a widely The solution is K (c) ∞ (y, z) = 1 + 2β πα ∞ 0 e − β α s arctan y s + |z| − arctan z s + |y| ds.
A further limit can be taken as the immigration rate at the origin β increases to an infinite rate, and this yields the kernel (6) stated in the introduction for the infinite rate Brownian firework.