DGFF 4 – Properties of the Green’s function

I’m at UBC this month for the PIMS probability summer school. One of the long courses is being given by Marek Biskup about the Discrete Gaussian Free Field (notes and outline here) so this seems like a good moment to revive the sequence of posts about the DGFF. Here’s DGFF1, DGFF2, DGFF3 from November.

The first draft of this post was about the maximum of the DGFF in a large box V_N, and also about the Green’s function G^{V_N}(x,y), which specifies the covariance structure of the DGFF. This first draft also became too long, so I’m splitting it into two somewhat shorter ones. As we’ll see, some understanding and standard estimates of the Green’s function is enough to say quite a bit about the maximum. In this first post, we’ll explore some ‘low-hanging fruit’ concerning the Green’s function, as defined through a simple random walk, which are useful, but rarely explained in the DGFF literature.

Symmetry of Green’s function

We start with one of these low-hanging fruit. If G^{V_N} is to be a covariance matrix, it has to be symmetric. In the first post, showing that the definition of the DGFF as a random field with given Hamiltonian is equivalent to \mathcal{N}(0,G^{V_N}) certainly can be viewed as a proof of symmetry. However, it would be satisfying if there was a direct argument in the language of the definition of the Green’s function.

To make this self-contained, recall the random walk definition of G^{V_N}(x,y). Let (S_m)_{m\ge 0} be simple random walk on V_N, and \mathbb{P}_x,\,\mathbb{E}_x denote starting the random walk at x\in V_N. As usual, let \tau_y,\,\tau_A denote the hitting time of a vertex y or a set A respectively. Then

G^{V_N}(x,y):= \mathbb{E}_x \left[ \sum_{m=0}^{\tau_{\partial V_N}}1_{(S_m=y) }\right].

That is, G^{V_N}(x,y) is the expected number of visits to y by a random walk from x, before it exits V_N.

Let’s drop the superscript for now, as everything should hold for a more general subset of the lattice. I don’t think it’s immediately obvious at the level of Markov chains why G(x,y)=G(y,x). In particular, it’s not the case that

\mathbb{P}_x(\tau_y < \tau_{D^c}) = \mathbb{P}_y(\tau_x <\tau_{D^c}),

and it feels that we can’t map between paths x \to \partial D and y\to \partial D in a way that preserves the number of visits to y and x, respectively. However, we can argue that for any m

\mathbb{P}_x(S_m=y, \tau_{D^c}>m) = \mathbb{P}_y(S_m=x, \tau_{D^c}>m),

by looking at the suitable paths of (S_m). That is, if we have a path x=S_0,S_1,\ldots,S_m=y that stays within D, then the probability of seeing this path starting from x and its reverse direction starting from y are equal. Why? Because

\mathbb{P}_x(S_0=x,S_1=v_1,\ldots,S_{m-1}=v_{m-1},S_m=y) = \prod_{\ell=0}^{m-1} \frac{1}{\mathrm{deg}(v_\ell)},

and

\mathbb{P}_y(S_0=y,S_1=v_{m-1},\ldots,S_{m-1}=v_1, S_m=x) = \prod_{\ell=0}^{m-1} \frac{1}{\mathrm{deg}(v_{m-\ell})} = \prod_{\ell=1}^m \frac{1}{\mathrm{deg}(v_\ell)}.

Since D\subset \mathbb{Z}^d and x,y are in the interior of D, we must have \mathrm{deg}(x)=\mathrm{deg}(y), and so these two expressions are equal. Summing over all such two-way paths, and then all m gives the result.

Fixing one argument

We now focus on G^D(\cdot,y), where the second argument is fixed. This is the solution to the Poisson equation

\Delta G^D(\cdot,y) = -\delta_y(\cdot),\quad G^D(x,y)=0,\; \forall x\in \partial D.

To see this, can use a standard hitting probability argument (as here) with the Markov property. This is harmonic in D\backslash \{y\}, and since we know

G^D(y,y)= \frac{1}{\mathbb{P}_y(\text{RW hits }\partial D\text{ before returning to }y)},

this uniquely specifies G^D(\cdot,y). Anyway, since harmonic functions achieve their maxima at the boundary, we have G(y,y)\ge G(x,y) for all x\in D. We can also see this from the SRW definition as

G(x,y)=G(y,x) = \mathbb{P}_y (\tau_x < \tau_{\partial D} ) G(x,x) \le G(x,x).

Changing the domain

Now we want to consider nested domains D\subset E, and compare G^D(\cdot,\cdot) and G^E(\cdot,\cdot) on DxD. The idea is that for SRW started from x\in D, we have \tau_{\partial D}\le \tau_{\partial E}, since one boundary is contained within the other. From this, we get

G^D(x,y)\le G^E(x,y),\quad \forall x,y\in D,

and we will use the particular case y=x.

For example, if x\in V_N, the box with width N, then the box with width 2N centred on x contains the whole of V_N. So, if we set \bar {V}_{2N}:= [-N,N]^d, then with reference to the diagram, we have

G^{V_N}(x,x)\le G^{\bar{V}_{2N}}(0,0),\quad x\in V_N.

As we’ll see when we study the maximum of the DGFF on V_N, uniform control over the pointwise variance will be a useful tool.

Maximising the Green’s function

The idea of bounding G^{V_N}(x,x) by G^{\bar V_{2N}}(0,0) for any x\in V_N is clever and useful. But a more direct approach would be to find the value of x that maximises G^{V_N}(x,x). We would conjecture that when V_N has a central vertex, then this is the maximiser.

We can prove this directly from the definition of the Green’s function in terms of random walk occupation times. Let’s assume we are working with \bar{V}_N for even N, so that 0 is the central vertex. Again, since

G^D(x,x)=\frac{1}{\mathbb{P}_x(\text{RW hits }\partial D\text{ before returning to }x)}, (*)

it would suffice to show that this probability is minimised when x=0. This feels right, since 0 is furthest from the boundary. Other points are closer to the boundary in some directions but further in others, so we can’t condition on the maximum distance from its start point achieved by an excursion of SRW (we’re vertex-transitive, so these look the same from all starting points), as even allowing for the four possible rotations, for an excursion of diameter slightly larger than N, starting at the centre is maximally bad.

However, intuitively it does feel as if being closer to the boundary makes you more likely to escape earlier. In fact, with a bit more care, we can couple the SRW started from 0 and the SRW started from r=(r^x,r^y)\ne 0 such that the latter always exits first. For convenience we’ll assume also that r^x,r^y are both even.

I couldn’t find any reference to this, so I don’t know whether it’s well-known or not. The following argument involves projecting into each axis, and doing separate couplings for transitions in the x-direction and transitions in the y-direction. We assume WLOG that x is in the upper-right quadrant as shown. Then, let 0=S_0,S_1,S_2,\ldots be SRW started from 0, and we will construct r=R_0,R_1,R_2,\ldots on the same probability space as (S_m)_{m\ge 0} as follows. For every m, we set the increment R_{m+1}-R_m to be \pm(S_{m+1}-S_m). It remains to specify the sign, which will be determined by the direction of the S-increment, and a pair of stopping times. The marginal is therefore again an SRW, started from r. Temporarily, we use the unusual notation S_m= (S^x_m,S^y_m) for the coordinates of S_m.

So, if S_{m+1}-S_m=(1,0), (-1,0), ie S moves left or right, then we set

R_{m+1}-R_m = \begin{cases} -(S_{m+1}-S_m) &\quad \text{if }m<T^x\\ S_{m+1}-S_m&\quad \text{if }m>T^x.\end{cases} (*)

where T^x:= \min\{m\,:\, R^x_m=S^x_m\}. That is, R^x moves in the opposing direction to S^x until the first time when they are equal (hence the parity requirement), and then they move together. WLOG assume that r^x>0. Then suppose S^x_m=\pm N and such m is minimal. Then by construction, if m\ge T^x, then R^x_m=\pm N also. If m<T^x, then we must have S^x_m=-N, and so since R^x‘s trajectory is a mirror image of S^x‘s, in fact R^x_m = N+r^x>N, so R^x hit +N first. In both cases, we see that R^x hits \pm N at the same time or before S^x.

In other words, when S^x_m has non-negative x coordinate, the lazy random walk R^x follows the same trajectory as S^x, and when it has negative x coordinate, the R^x mirrors S^x. At some time, it may happen that S^x_m= R^x_m=0 (recall the parity condition on r). Call this time T^x. We then adjust the description of the coupling so that (*) is the mechanism for m<T^x, and then for m\ge T^x, we take S^x_m=R^x_m.

Similarly, if S_{m+1}-S_m =(0,1), (0,-1), ie S moves up or down, then we set

R_{m+1}-R_m = \begin{cases} -(S_{m+1}-S_m)&\quad \text{ if }m<T^y\\  S_{m+1}-S_m&\quad \text{if }m\le T^y,\end{cases}

with corresponding definition of the stopping time T^y.

This completes the coupling, and by considering T^x\wedge T^y, we have shown what that the exit time for the walk started from zero dominates the exit time for walk started from r. Recall that so far we are in the case where the box has even width and r=(r^x,r^y) has even coordinates.

This exit time comparison isn’t exactly what we need to compare G^N(0,0) and G^N(x,x). It’s worth remarking at this stage that if all we cared about was the Green’s function on the integer line [-N,N], we would have an easier argument, as by the harmonic property of G(\cdot,y)

G^{[-N,N]}(0,r)=\frac{N-r}{N}G^{[-N,N]}(0,0),

G^{[-N,N]}(r,0) = \frac{N}{N+r}G^{[-N,N]}(r,r),

and so G(0,0)>G(r,r) follows by symmetry. To lift from 1D to 2D directly, we need a bit more than this. It’s possible that S returns in both x- and y- coordinates more often than R, but never at the same time. Fortunately, the coupling we defined slightly earlier does give us a bit more control.

Let \tau^x(S), \tau^x(R) be the first times that S^x, R^x hit \pm N. Under this coupling, for any m\ge 0

\mathbb{P}(S^x_m=0, m<T^x) = \mathbb{P}(R^x_m=r^x, m<T^x)

since these events are literally equal. Since we showed that \tau^x(R)\le \tau^x(S) almost surely, we can further deduce

\mathbb{P}(S^x_m=0,m<T^x\wedge \tau^x(S)) \ge \mathbb{P}(S^x_m=0,m<T^x\wedge \tau^x(R))

=\mathbb{P}(R^x_m=r^x, m <T^x \wedge \tau^x(R)).

To address the corresponding events for which m\ge T^x, we apply the strong Markov property at T^x, to obtain SRW Z_m started from r/2, and let \tau_{-N},\tau_{+N} be the hitting times of -N,+N respectively and \tau_{\pm N}=\tau_{-N}\wedge \tau_{+N}. It will now suffice to prove that

\mathbb{P}(Z_m=0, m< \tau_{\pm N}) \ge \mathbb{P}(Z_m=r,m<\tau_{\pm N}), (**)

as then we can apply the law of total probability and sum over values of T^x and m\ge 0.

To prove this result, we consider the following bijection between trajectories of length m from r/2 to {0,r}. We decompose the trajectories into excursions away from r/2, and then a final meander from r/2 to {0,r} that stays on the same side of r/2. We construct the new trajectory by preserving all the initial excursions, but reversing all the steps of the final meander. So if the original trajectory ended up at 0, the image ends up at r. Trivially, the initial excursions in the image only hit \pm N if the excursions in the original trajectory did this too. But it’s also easy to see, by a similar argument to the coupling at the start of this section, that if the original trajectory ends at r and does not hit \pm N, then so does the image. However, the converse is not true. So we conclude (**), and thus

\mathbb{P}(S_m^x=0) \ge \mathbb{P}(R_m^x=0)

for all m by combining everything we have seen so far. And so we can now lift to a statement about S_m itself, that is considering both coordinates separately.

 

The remaining cases for r require a little more care over the definition of T^x, though the same projection argument works, for fundamentally the same reason. (Note that in the above argument, if S^x_m=-N and m<T^x, then in fact R^x_m\ge N+2, and so it’s not hard to convince yourself that a sensible adjustment to the stopping time will allow a corresponding result with R^x_m\ge N+1 in the odd r^x case.) The case for N odd is harder, since in one dimension there are two median sites, and it’s clear by symmetry that we can’t couple them such that RW from one always exits at least as early as RW from the other. However, the distributions of exit times started from these two sites are the same (by symmetry), and so although we can’t find a coupling, we can use similar stopping times to obtain a result in probability.

In the next post, we’ll see how to apply this uniform bound on G^{V_N}(x,x) to control the maximum of the DGFF on V_N. In particular, we address how the positive correlations of DGFF influence the behaviour of the maximum by comparison with independent Gaussians at each site.

DGFF 2 – Boundary conditions and Gibbs-Markov property

In the previous post, we defined the Discrete Gaussian Free Field, and offered some motivation via the discrete random walk bridge. In particular, when the increments of the random walk are chosen to be Gaussian, many natural calculations are straightforward, since Gaussian processes are well-behaved under conditioning and under linear transformations.

Non-zero boundary conditions

In the definition of the DGFF given last time, we demanded that h\equiv 0 on \partial D. But the model is perfectly well-defined under more general boundary conditions.

It’s helpful to recall again the situation with random walk and Brownian bridge. If we want a Brownian motion which passes through (0,0) and (1,s), we could repeat one construction for Brownian bridge, by taking a standard Brownian motion and conditioning (modulo probability zero technicalities) on passing through level s at time 1. But alternatively, we could set

B^{\mathrm{drift-br}}(t) = B(t)+ t(s-B(1)),\quad t\in[0,1],

or equivalently

B^{\mathrm{drift-br}}(t)=B^{\mathrm{br}}(t)+ st, \quad t\in[0,1].

That is, a Brownian bridge with drift can be obtain from a centered Brownian bridge by a linear transformation, and so certainly remains a Gaussian process. And exactly the same holds for a discrete Gaussian bridge: if we want non-zero values at the endpoints, we can obtain this distribution by taking the standard centred bridge and applying a linear transformation.

We can see how this works directly at the level of density functions. If we take 0=Z_0,Z_1,\ldots,Z_{N-1},Z_N=0 a centred Gaussian bridge, then the density of Z=\mathbf{z}\in \mathbb{R}^{N+1} is proportional to

\mathbf{1}\{z_0=z_N=0\}\exp\left( -\frac12 \sum_{i=1}^N (z_i-z_{i-1})^2 \right). (3)

So rewriting z_i= y_i- ki (where we might want k=s/N to fit the previous example), the sum within the exponent rearranges as

-\frac12 \sum_{i=1}^N (y_i-y_{i-1} - k)^2 = -\frac12 \sum_{i=1}^N (y_i-y_{i-1})^2 - 2k(y_N-y_0)+ Nk^2.

So when the values at the endpoints z_0,z_n,y_0,y_N are fixed, this middle term is a constant, as is the final term, and thus the density of the linearly transformed bridge has exactly the same form as the original one.

In two or more dimensions, the analogue of adding a linear function is to add a harmonic function. First, some notation. Let \varphi be any function on \partial D. Then there is a unique harmonic extension of \varphi, for which \nabla \varphi=0 everywhere on D, the interior of the domain. Recall that \nabla is the discrete graph Laplacian defined up to a constant by

(\nabla \varphi) _x = \sum\limits_{x\sim y} \varphi_x - \varphi_y.

If we want h^D instead to have boundary values \varphi, it’s enough to replace h^D with h^D+\varphi. Then, in the density for the DGFF ( (1) in the previous post), the term in the exponential becomes (ignoring the \frac{1}{4d} )

-\sum\limits_{x\sim y} \left[ (h^D_x-h^D_y)^2 + (\varphi_x-\varphi_y)^2 +2(h^D_x - h^D_y)(\varphi_x-\varphi_y)\right].

For each x\in D, on taking this sum over its neighbours y\in \bar D, the final term vanishes (since \varphi is harmonic), while the second term is just a constant. So the density of the transformed field, which we’ll call h^{D,\varphi} is proportional to (after removing the constant arising from the second term above)

\mathbf{1}\left\{h^{D,\varphi}_x = \varphi_x,\, x\in\partial D\right\} \exp\left( -\frac{1}{4d} \sum\limits_{x\sim y} \left( h^{D,\varphi}_x - h^{D,\varphi}_y \right)^2 \right).

So h^{D,\varphi}:= h^D + \varphi satisfies the conditions for the DGFF on D with non-zero boundary conditions \varphi.

Harmonic functions and RW – a quick review

Like the covariances in DGFF, harmonic functions on D are related to simple random walk on D stopped on \partial D. (I’m not claiming a direct connection right now.) We can define the harmonic extension \varphi to an interior point x by taking \mathbb{P}_x to be the law of SRW x=Z_0,Z_1,Z_2,\ldots started from x, and then setting

\varphi(x):= \mathbb{E}\left[ \varphi_{\tau_{\partial d}} \right],

where \tau_{\partial D} is the first time that the random walk hits the boundary.

Inverse temperature – a quick remark

In the original definition of the density of the DGFF, there is the option to add a constant \beta>0 within the exponential term so the density is proportional to

\exp\left(-\beta \sum\limits_{x\sim y} (h_x-h_y)^2 \right).

With zero boundary conditions, the effect of this is straightforward, as varying \beta just rescales the values taken by the field. But with non-zero boundary conditions, the effect is instead to vary the magnitude of the fluctuations of the values of the field around the (unique) harmonic function on the domain with those BCs. In particular, when \beta\rightarrow \infty, the field is ‘reluctant to be far from harmonic’, and so h^D \Rightarrow \varphi.

This parameter \beta is called inverse temperature. So low temperature corresponds to high \beta, and high stability, which fits some physical intuition.

A Markov property

For a discrete (Gaussian) random walk, the Markov property says that conditional on a given value at a given time, the trajectory of the process before this time is independent of the trajectory afterwards. The discrete Gaussian bridge is similar. Suppose we have as before 0=Z_0,Z_1,\ldots, Z_N=0 a centred Gaussian bridge, and condition that Z_k=y, for k\in\{1,\ldots,N-1\}, and y\in\mathbb{R}. With this conditioning, the density (3) splits as a product

\mathbf{1}\{z_0=z_N=0, z_k=y\}\exp\left(-\frac12 \sum\limits_{i=1}^N (z_i-z_{i-1})^2 \right) =

\mathbf{1}\{z_0=0,z_k=y\} \exp\left(-\frac12 \sum\limits_{i=1}^k (z_i-z_{i-1})^2 \right) \cdot \mathbf{1}\{z_k=y,z_N=0\} \exp\left(-\frac12 \sum\limits_{i=k+1}^N (z_i-z_{i-1})^2 \right).

Therefore, with this conditioning, the discrete Gaussian bridge splits into a pair of independent discrete Gaussian bridges with drift. (The same would hold if the original process had drift too.)

The situation for the DGFF is similar, though rather than focusing on the condition, it makes sense to start by focusing on the sub-domain of interest. Let A\subset D, and take B=\bar D\backslash A. So in particular \partial A\subset B.

img_20161106_194123472_compressedThen we have that conditional on h^D\big|_{\partial A}, the restricted fields h^D\big|_{B\backslash \partial A} and h^D\big|_A are independent. Furthermore, h^D\big|_A has the distribution of the DGFF on A, with boundary condition given by h^D\big|_{\partial A}. As in the discrete bridge, this follows just by splitting the density. Every gradient term corresponds to an edge in the underlying graph that lies either entirely inside \bar A or entirely inside B. This holds for a general class of Gibbs models where the Hamiltonian depends only on the sum of some function of the heights (taken to be constant in this ‘free’ model) and the sum of some function of their nearest-neighbour gradients.

One additional and useful interpretation is that if we only care about the field on the restricted region A, the dependence of h^D\big|_A on h^D\big|_{D\backslash A} comes only through h^D\big|_{\partial A}. But more than that, it comes only through the (random) harmonic function which extends the (random) values taken on the boundary of A to the whole of A. So, if h^A is an independent DGFF on A with zero boundary conditions, we can construct the DGFF h^D from its value on D\backslash A via

h^D_x \stackrel{d}= h^A_x + \varphi^{h^D\big|_{\partial A}},

where \varphi^{h^D\big|_{\partial A}} is the unique harmonic extension of the (random) values taken by h^D on \partial A to \bar A.

This Markov property is crucial to much of the analysis to come. There are several choices of the restricted domain which come up repeatedly. In the next post we’ll look at how much one can deduce by taking A to be the even vertices in D (recalling that every integer lattice \mathbb{Z}^d is bipartite), and then taking A to be a finer sublattice within D. We’ll use this to get some good bounds on the probability that the DGFF is positive on the whole of D. Perhaps later we’ll look at a ring decomposition of \mathbb{Z}^d consisting of annuli spreading out from a fixed origin. Then the distribution of the field at this origin can be considered, via the final idea discussed above, as the limit of an infinite sequence of random harmonic functions given by the values taken by the field at increasingly large radius from the origin. Defining the DGFF on the whole lattice depends on the existence or otherwise of this local limit.

Duality for Interacting Particle Systems

Yesterday I introduced the notion of duality for two stochastic processes. My two goals for this post are to elaborate on the idea of why duality is useful, which we touched on in passing in the previous part, and to discuss duality of interacting particle systems. In the latter case, there are often nice ways to consider the forward and backward processes together that make the relation somewhat more natural.

The starting point is to assume a finite state space. This will be reasonable when we start to consider interacting particle systems, eg on \{0,1\}^{[n]}. As before, call the spaces R and S, and a duality function H(x,y). Since the state-spaces are finite, it is entirely natural to think of this as a matrix, and hence as an operator. Of course, a function defined on a finite state-space can be thought of as a vector, so it is clear what this operator will actually operate on. (I’ve chosen H rather than h for the duality function so it is more clear that it is acting as an operator here.)

We have some choice about which way round to define it, but for now let’s say that given some function f(.) on S

Hf(x):=\sum_{y\in S} H(x,y)f(y).

Note that this is a) exactly the definition of matrix (left-)multiplication; b) We should think of Hf as a function on R – perhaps (Hf)(x) might be more clear? and c) the operator H acts \mathbb{R}^S\rightarrow \mathbb{R}^R. If we want the corresponding operator \mathbb{R}^R\rightarrow\mathbb{R}^S, we simply multiply by H on the right instead.

But note also that the generator of a finite state-space Markov process is also a matrix, indeed a Q-matrix. So if we take our definition of the duality function as

\mathcal{G}_X h(x,y)=\mathcal{G}_Y h(x,y),

which, importantly, holds for all x,y, we can convert this into an algebraic form as

\mathcal{G}_X H = H \mathcal{G}_Y^\dagger.

In the same way that n-step transition probabilities for a discrete-time Markov chain are given by the product of the one-step transition matrix, general time transition probabilities for a continuous-time Markov chain are given by exponents of the Q-matrix. In particular, if X and Y have transition kernels P and Q respectively, then P_t=e^{tG_X}, and after doing some manipulation, we can show that

P_t H=H Q_t^\dagger,

also. This is really useful as in general we would hope that H might be invertible, from which we derive

P_t=HQ_t^\dagger H^{-1}.

So this is a powerful statement about the relationship between the evolutions of the two processes. In particular, it shows a correspondence (given by H) between left eigenvectors of P, and right eigenvectors of Q, and vice versa naturally.

The reason why this is useful rather than merely cute, is that when we re-interpret everything in terms of the original stochastic processes, we get a map between stationary distributions of X, and harmonic functions of Y. Stationary distributions are often hard to describe in any terms other than the left-1-eigenvector, or through some convergence property that is typically hard to work with. Harmonic functions, on the other hard, can be much more tractable. An example of a harmonic function is the survival probability started from a given state. This is useful for specifying the stationary distribution, but perhaps even more so for describing properties of the set of stationary distributions. In particular, uniqueness and existence are carried across this equivalence. So, for example, if the dual does not survive almost surely, then this says the only stationary measure is zero, and so the process is transient or similar.

Jan Swart’s course in Luminy last October dealt with duality, with a focus mainly on interacting particle systems. There are a couple of themes I want to talk about, without going into too much detail.

A typical interacting particle system will take place on a locally finite graph. At each vertex, there is either a particle, or there isn’t. Particles move between adjacent vertices, and sometimes interact with particles at adjacent vertices. These interactions might involve branching or coalescence. We will discuss shortly the set of possible forms such interaction might take. The state space is \{0,1\}^{V(G)}, with G the underlying graph. Then given a state, there is some set of actions which might happen next, and we consider the possibility that they happen with exponential rates.

At this stage, it seems like the initial configuration is important, as this affects what set of moves can happen immediately, and also thereafter. It is not clear how quickly this dependence fades. One useful idea is not to restrict ourselves to interactions involving the particles currently present in the system, but instead to consider a Poisson process of all possible interactions. Only the moves actually permitted by the current state will happen, but having this extra information allows for coupling between initial configurations.

It’s probably easier to consider a concrete example. The picture below shows the set-up for a branching random walk up an integer lattice. Each particle moves to one of the two state directly above its current state, or it branches and sends particles to both of them.DSC_2589In the diagram, we have glued arrows onto every state at every time, which tells us what to do if there is a particle there at each time. As a coupling, we can now think of the process as a deterministic walk through a random environment. The environment is given by some probability space, which in continuous time might have the appearance of a Poisson process on the set of ‘moves’, and the initial condition of the walk is up to us.

We can generalise this to a broader class of interacting particle systems. If we want all interactions to be between pairs of adjacent states, there are six possible things which could happen:

  • Annihilation: two adjacent particles destroy each other. ( 11 -> 00 )
  • Branching: one particle becomes two particles. ( 01 or 10 -> 11 )
  • Coalescence: two particles merge. ( 11 -> 01 or 10 )
  • Death: A particle is removed. ( 01 or 10 -> 00 )
  • Exclusion: a particle moves. ( 01 -> 10 )
  • Birth: a particle is created. ( 00 -> 01 or 10 )

For now we exclude the possibility of birth. Note that the way we have set this up involving two-site interactions excludes the possibility of a particle trying to move to an already-occupied site.

DSC_2588Let us say that in process X the rates at which each of these events happen are a, b, c, d and e, taking advantage of the helpful choice of naming. There is some flexibility about whether the rates are the same between every pair of vertices of note. For this post we assume that they are. Then it is a result of Lloyd and Sudbury that given some real q\neq 1, the process X’ with corresponding rates given by:

a'=a+2q\gamma, b'=b+\gamma, c'=c-(1+q)\gamma, d'=d+\gamma, e'=e-\gamma,

for \gamma:= \frac{a+c-d+qb}{1-q},

is dual to X, with duality function given by h(Y,Z)=q^{|Y\cap Z|}, for Y and Z possible states.

I want to make two comments:

1) This illustrates one of the differences between the dual and the time-reversal. It is clear that the time-reversal of branching is coalescence and vice versa, and exclusion is invariant under time-reversal. But the time-reversal of death is definitely birth, but there is no birth component in the dual of a process which features death. I don’t have a strong intuition for why this is the case, but see the final paragraph of this post. However, at least it seems plausible that both processes might simultaneously be recurrent, since in the dual, both the branching rate and the death rate have increased by the same amount.

2) This settles one problem of uniqueness of the dual that I mentioned last time, since we can vary q and get a different dual to the same original process. For example, in the voter model, we have b=d=1, and a=c=e=0, as in any update, the opinions of neighbours which were previously different become the same. Anyway, for any q\in[-1,0] there is a choice of dual, where at the extremes q=0 corresponds to coalescing random walk, and q=-1 to annihilating random walk. (Note that the duality function for q=0 is the indicator function that the systems are different.)

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As a final observation without much justification, suppose we add in arrows in the gaps of the branching random walk picture we had earlier, and direct them in the opposite direction. It turns out that this corresponds precisely to the dual of the process. This provides an appealing visual idea of why the dual of branching might be death. It also supports the general idea based on the coupling described earlier that the dual process is in some sense a deterministic walk in the opposite direction through the random environment specified by the original process.

REFERENCES

J.M. Swart – Duality and Intertwining of Markov Chains (mainly using chapters 2.1 and 2.7)

Thanks for Daniel Straulino for direction towards the branching random walk duality example.

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