The diffusive evaporation-deposition model
and the voter model
Benjamin Graham
University of Warwick
b.graham@warwick.ac.uk
November 12, 2012
Abstract
We consider the Takaysu model with desorption. Particles are deposited on the hypercubic lattice Zd . At each site individual particles
arrive at rate q. The top-most particle at each non-empty site evaporates at rate p. The particles also diffuse in the manner of coalescing
random walks: at rate one, the stack of particles sitting at a site is
moved to a randomly chosen neighbor, joining any particles that were
there already. The model has a critical point qc (p) above which the
density of particles diverges with time.
We look at the connection between this model and the voter model,
both on Zd and in the mean-field setting. In the mean-field setting we
show that points where particles accumulate corresponds to the size
of the influence of the corresponding voter.
1
Introduction
The evaporation-deposition model [1], also known as the Takaysu model with
desorption [4], is a model for monomers undergoing diffusion, coagulation,
growth and evaporation. It is defined on the integer lattice Zd with nearestneighbor edges. Let Mtd (x) ∈ Z+ denote the number of particles at site
x ∈ Zd at time t. As an initial condition we take Mtd0 (x) = 0 for all x ∈ Zd .
The configuration Mtd : Zd → Z+ then evolves for t > t0 in the following
ways.
(i) Deposition: Particles are deposited at each site x at rate q, causing
Mtd (x) to increase by one.
1
(ii) Evaporation: The top-most particle at each non-empty site x (with
Mtd (x) > 0) evaporates at rate p, causing Mtd (x) to decrease by one.
(iii) Coagulation: For each pair of neighboring particles x and y, the stack
of particles sitting at x is moved to y joining any particles that were
there already there; Mtd (x) takes the value zero and Mtd (y) takes the
d
d
(y).
(x) + Mt−
value Mt−
Mtd is stochastically increasing as t0 decreases. Take the limit t0 → −∞ to
give the equilibrium distribution. Given p, the critical point qcd (p) is then
defined
qcd (p) = inf{q : Ep,q [M0d (0)] = ∞}.
If there was no diffusion the critical point would simply be p. The effect
of the diffusion is to increase the proportion of empty sites. This decrease
the rate of evaporation, increasing the net flow of particles into the system:
qcd (p) < p.
We will study the mean-field version of the evaporation deposition model.
The mean-field model is believed to capture many properties of the finite
dimensional model [1]. It is an open problem to show that the critical point
converges to the critical point of the corresponding mean-field model. We
prove that the mean-field model provides an upper bound on the critical
point.
The mean-field evaporation-deposition-diffusion model MtMF is defined by
considering a single site that interacts with its own probability distribution.
Let L(MtMF ) denote a random variable that is independent of MtMF but with
the same distribution. Setting MtMF
= 0 let MtMF evolve as follows.
0
(i) Deposition: MtMF increases by one at rate p.
(ii) Evaporation: If MtMF > 0 then MtMF deceases by one at rate q.
(iii) Coagulation+: At rate 1, MtMF increases by L(MtMF ). This corresponds
to particles arriving at a site.
(iv) Coagulation−: At rate 1, MtMF is set to zero. This corresponds to the
particles leaving a site. MtMF is a renewal process with respect to these
events.
Again take the limit t0 → −∞ to give the system in equilibrium. The critical
point qcMF of the mean-field model is defined by
qcMF (p) = inf{q : Ep,q [M0MF ] = ∞}.
2
The supercritical phase is characterised by a net flow q−pP[M0MF (0) > 0] > 0
of particles into the system:
(
> 1 − q/p if q > qcMF ,
Pp,q [M0MF (0) = 0]
= 1 − q/p if q < qcMF ,
The location of this critical
√ point has been found using generating functions
MF
[4]: qc (p) = p + 2 − 2 p + 1 and
(
exp(−m/m∗ )m−3/2 q < qcMF (p),
P(M0MF > m) ∼
(1.1)
m−1/2
q > qcMF (p).
We are using f (n) ∼ g(n) to mean limn→∞ f (n)/g(n) ∈ (0, ∞).
We will show that the supercritical mean-field model places mass according to a corresponding voter model.
2
Particle histories and the voter model
The distribution of M0d can be expressed in terms of a graphical construction. Let N (Zd ) denote the set of 2d unit vectors in Zd . Let Pp , Pq , P→
denote Poisson processes with rates p, q, 1/(2d), respectively, on the sets
Zd × R, Zd × R, Zd × N (Zd ) × R. The points of the Poisson processes correspond to evaporation (if the site is not already empty), deposition, and
particles movements in the direction of the unit vector.
The space-time path particles takes, assuming that they do not evaporate
along the way, is a function of P→ : let X→ (x, s, t) denote the location at time
t of the particles which were at site x at time s. Define the history H d of the
origin at time 0 in terms of P→ ,
H d = {(x, s) : s < 0, X→ (x, s, 0) = 0)}.
See Figure 1. If we reverse the direction of all the arrows in Figure 1, then
H d becomes equal in distribution to the spread of a single voter’s opinion in
the voter model; H d is almost surely finite.
Similarly, the recursive definition of M0MF implicitly defines a history
structure H MF ; see Figure 2. H MF is a rooted binary tree. The root
node corresponds to time zero. The branch points of the tree correspond
to coagulation+ events: for each of these events there is an extra, independent copy of MtMF to keep track off. The leaves of the tree correspond to the
renewal times of the MtMF processes.
3
0
Zd
0
time
Figure 1: Two examples of histories for Zd . The black arrows correspond to
points in P→ that shift particles to the origin at time 0. The boundary of
H d is determined by points in P→ that move particles away from the origin;
they are represented on the left hand side by blue arrows. On the left H d
forms a binary tree; on the right it does not.
MtM F L(MtM F ) L(MtM F ) L(MtM F )
time
Figure 2: Left: History H MF for the mean-field process. The horizontal axis
correspond to just enough copies of the mean-field process to sample one copy
of the mean-field process at time 0. The arrows correspond to coagulation+
events; the black dots correspond to coagulation− events. Right: H MF corresponds to a binary tree. The number of leaves is N = 4 and the maximum
number of generations from top to bottom is G = 3. The black vertical lines
on the left hand side have the exponential distribution with mean one. On
the right-hand side the have the exponential distribution with mean 1/2.
4
Remark 2.1. Let N count the number of leaves of H MF . The number of
coagulation± events in the tree is 2N − 1. The number of rooted binary trees
with n leaves is given by the (n − 1)-th Catalan number. Hence
P(N = n) =
Cn−1
∼ n−3/2
2n−1
2
and
P(N > n) ∼ n−1/2 .
Remark 2.2. Let G denote the height of the tree H MF —the maximum number of bifurications between the root and a leaf. H MF is a critical GaltonWatson process so P(G > g) ∼ g −1 [Kolmogorov, 1938].
The following follows almost immediately from the graphical representation.
Proposition 2.3. The total-variation distance between M0MF and M0d (0)
tends to zero as d → ∞.
Proof. There is a natural coupling between H MF and H d . Embed H MF into
Zd by associating with each of the 2N − 1 coagulation events a randomly
chosen unit vector. Call the embedding good if it remains a binary tree,
that is if no loops are formed. The probability that the embedding is good
tends to one as d → ∞. By Remark 2.1, with high probability N 6 d1/3 . If
N 6 d1/3 then with high probability the unit vectors are orthogonal so no
loops are formed.
To calculate M0d (0), and M0MF when the embedding is good, look at the
intersection of Pp and Pq with H d .
3
Comparison with the Voter model
In this section we show that the mean field evaporation-deposition model
has a large number of particles whenever the corresponding voter has large
influence.
The converse result, that the mean field evaporation-deposition model
has a small number of particles when the voter’s influence is small, is also
true. However, it is follows immediatley from the properties of the Poisson
distribution so we omit a formal statement and proof.
Theorem 3.1. If q > qcMF (p) then for all integers m,
lim P[M0MF 6 m | G = g] = 0.
g→∞
Here we use G, rather than say N , to measure the size of a voter. Of
course log2 N 6 G 6 N . In fact G grows with the root of N .
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Remark 3.2. E[G | N = n] ∼
√
n [2].
We will prove a lemma before proving Theorem 3.1.
Lemma 3.3. Let q > qcMF (p) and n ∈ Z+ . For some constant α = α(p, q) >
0, P[M0MF > αn | N = n] > α.
Proof of Lemma 3.3. If N = n, the H MF is composed of 2n − 1 intervals.
The number of depositions per interval is exponential with mean q/2 and so
the total number of depositions has the negative binomial distribution with
mean (2n − 1)q/2. By Hoeffding’s inequality, for some c > 0,
P[M0MF > 2qn | N 6 n] 6 exp(−cn).
(3.4)
By (1.1), Remark 2.1 and (3.4), we can set
a := lim inf P[M0MF > 2qn | N > n] > 0.
n→∞
By Remark 2.1,
lim inf P[M0MF > 2qn | 4a−2 n > N > n] > a/2.
n→∞
Notice that P[M0MF > m | N = n] is increasing in n.
Proof of Theorem 3.1. Consider the black vertical lines in Figure 2; their
lengths are given by the exponential distribution with mean 1/2. Suppose
that at the bottom of one of these lines MtMF = m; let B(m) denote the
distribution of MtMF at the top of the line. B(m) may be less than m because
of the evaporation over the time interval. The number of evaporation-events
is a geometric random variable with mean p/2.
Let F ∈ {2, 4, 6, . . . , 2g } denote the number of leaves in generation g, the
final generation. Define conditional versions of M0MF :
d
M g = M0MF | {G = g},
d
M g,2 = M0MF | {G = g, F = 2}
d
M <g = M0MF | {G < g}
By monotonicity P[M g 6 m] 6 P[M g,2 6 m]. Conditional on {G = g, F =
2}, the history has a backbone that descends g − 1 generations from the root
of the tree, finishing with a copy of M 1 . Attached to the backbone are g − 1
subtrees. The subtrees are independent copies of M <k , k = 1, . . . , g − 1 (cf.
[3]). We can write
d
M g,2 = B(M <(g−1) + B(M <(g−2) + B(· · · + B(M <2 + B(M <1 + M 1 ))))).
6
Recall Remark 3.2: applying Markov’s inequality to G | {N = n} we can
conclude that for C is sufficiently large
inf P[G 6 Cn | N = n] > 1 − α/2.
n
(3.5)
By Lemma 3.3 and the above, for g > Cn,
P[M0MF > αn | N = n, G < g] > α/2,
(3.6)
Consider the event
A = {∃i : m 6 i 6 g/2 and M <(g−i) > (2p + 1)i}.
By Lemma 3.3, (3.5) and (3.6), for some c = c(p, q) > 0,
P[A] > 1 −
g/2 Y
√
1 − c/ i → 1 as g → ∞.
i=m
Under the event A, the number of particles reaching the backbone due to
the subtree of H MF corresponding to i in the definition of Aα is greater than
(2p + 1)i − pi with high probability by Hoeffding’s inequality.
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