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arXiv:1001.3401v2 [math.PR] 17 Mar 2010
THE APPROACH TO CRITICALITY IN SANDPILES
ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
Abstract. A popular theory of self-organized criticality relates the critical behavior of
driven dissipative systems to that of systems with conservation. In particular, this theory
predicts that the stationary density of the abelian sandpile model should be equal to the
threshold density of the corresponding fixed-energy sandpile. This “density conjecture” has
been proved for the underlying graph Z. We show (by simulation or by proof) that the
density conjecture is false when the underlying graph is any of Z2, the complete graph Kn,
the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative
sandpiles continue to evolve even after a constant fraction of the sand has been lost at the
sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the
critical behavior of the abelian sandpile model at stationarity.
1. Introduction
In a widely cited series of papers [DVZ98, VDMZ98, DMVZ00, VDMZ00, MDPS+01],
Dickman, Mu˜ noz, Vespignani and Zapperi (DMVZ) developed a theory of self-organized
criticality as a relationship between driven dissipative systems and systems with conservation.
This theory predicts a specific relationship between the classical abelian sandpile model of
Bak, Tang, and Wiesenfeld [BTW87], a driven system in which particles added at random
dissipate across the boundary, and the corresponding “fixed-energy sandpile,” a closed system
in which the total number of particles is conserved.
In this introduction, we briefly define these two models and explain the conjectured rela-
tionship between them in the DMVZ paradigm of self-organized criticality. In particular, we
focus on the prediction that the stationary density of the driven dissipative model equals the
threshold density of the fixed-energy sandpile model. In section 2, we present data from large-
scale simulations which strongly indicate that this conjecture is false on the two-dimensional
square lattice Z2. In the subsequent sections we expand on the results announced in [FLW10]
by examining the conjecture on some simpler families of graphs in which we can provably
refute it.
The difference between the stationary and threshold densities on most of these graphs is
fairly small — typically on the order of 0.01% to 0.2% — which explains why many previous
simulations did not uncover them. The exception is the early experiments by Grassberger
Date: March 9, 2010.
2000 Mathematics Subject Classification. 82C27, 82B27, 60K35.
Key words and phrases. Abelian sandpile model, absorbing state phase transition, fixed-energy sandpile,
parallel chip-firing, self-organized criticality, Tutte polynomial.
1
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2ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
and Manna [GM90], who clearly identified this discrepancy, at least in dimensions 4 and
higher. Later studies focused on dimension 2 and missed the discrepancy.
In some more recent papers such as [BM08], the DMVZ paradigm is explicitly restricted to
stochastic models. In other recent papers [VD05, dCVdSD09] it is claimed to apply both to
stochastic and deterministic sandpiles, although these papers focus on stochastic sandpiles,
for the reason that deterministic sandpiles are said to belong to a different universality class.
Despite our contrary findings, we believe that the central idea of the DMVZ paradigm is a
good one: even in the deterministic case, the dynamics of a driven dissipative system should
in some way reflect the dynamics of the corresponding conservative system. Our results point
to a somewhat different relationship than that posited in the DMVZ series of papers: the
driven dissipative model exhibits a second-order phase transition at the threshold density of
the conservative model. We explain this transition in section 3.
Bak, Tang, and Wiesenfeld [BTW87] introduced the abelian sandpile as a model of self-
organized criticality; for mathematical background, see [Red05]. The model begins with a
collection of particles on the vertices of a finite graph. A vertex having at least as many
particles as its degree topples by sending one particle along each incident edge. A subset of
the vertices are distinguished as sinks: they absorb particles but never topple. A single time
step consists of adding one particle at a random site, and then performing topplings until
each non-sink vertex has fewer particles than its degree. The order of topplings does not
affect the outcome [Dha90]. The set of topplings that occur before the system stabilizes is
called an avalanche.
Avalanches can be decomposed into a sequence of “waves” so that each site topples at
most once during each wave. Over time, sandpiles evolve toward a stationary state in which
the waves exhibit power-law statistics [KLGP00] (though the full avalanches seem to exhibit
multifractal behavior [DMST98, KMS05]). Power law statistics are a hallmark of criticality,
and since the stationary state is reached apparently without tuning of a parameter, the
model is said to be self-organized critical.
To explain how the sandpile model self-organizes to reach the critical state, Dickman et
al. [DVZ98, DMVZ00] introduced an argument which soon became widely accepted: see,
for example, [Sor04, Ch. 15.4.5] and [MQ05, FdBR05, RS09]. Despite the apparent lack of
a free parameter, they argued, the dynamics implicitly involve the tuning of a parameter
to a value where a phase transition takes place. The phase transition is between an active
state, where topplings take place, and a quiescent “absorbing” state where topplings have
died out. The parameter is the density, the average number of particles per site. When
the system is quiescent, addition of new particles increases the density. When the system
is active, particles are lost to the sinks via toppling, decreasing the density. The dynamical
rule “add a particle when all activity has died out” ensures that these two density changing
mechanisms balance one another out, driving the system to the threshold of instability.
To explore this idea, DMVZ introduced the fixed-energy sandpile model (FES), which
involves an explicit free parameter ζ, the density of particles. On a graph with N vertices, the
system starts with ζN particles at vertices chosen independently and uniformly at random.
Unlike the driven dissipative sandpile described above, there are no sinks and no addition
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THE APPROACH TO CRITICALITY IN SANDPILES3
of particles, so the total number of particles is conserved. Subsequently the system evolves
through toppling of unstable sites. Usually the parallel toppling order is chosen: at each time
step, all unstable sites topple simultaneously. In the mathematical literature, this system
goes by the name of parallel chip-firing [BG92, BLS91]. Toppling may persist forever, or it
may stop after some finite time. In the latter case, we say that the system stabilizes; in the
terminology of DMVZ, it reaches an “absorbing state.”
A common choice of underlying graph is the n × n square grid with periodic boundary
conditions. It is believed, and supported by simulations [BCFV03], that there is a threshold
density ζc, such that for ζ < ζc, the system stabilizes with probability tending to 1 as n → ∞;
and for ζ > ζc, with probability tending to 1 the system does not stabilize.
For the driven dissipative sandpile on the n × n square grid, as n → ∞ the stationary
measure has an infinite-volume limit [AJ04], which is a measure on sandpiles on Z2. It
turns out that one gets the same limiting measure whether the grid has periodic or open
boundary conditions, and whether there is one sink vertex or the whole boundary serves
as a sink [AJ04] (see also [Pem91] for the corresponding result on random spanning trees).
The statistical properties of this limiting measure have been much studied [Pri94a, JPR06].
Of particular interest is the stationary density ζsof Z2, defined as the expected number of
particles at a fixed site. Grassberger conjectured that ζsis exactly 17/8, and it is now known
that ζs= 17/8 ± 10−12[JPR06].
DMVZ believed that the combination of driving and dissipation in the classical abelian
sandpile model should push it toward the critical density ζcof the fixed-energy sandpile.
This leads to a specific testable prediction, which we call the Density Conjecture.
Conjecture 1.1 (Density Conjecture, [VDMZ00]). On the square grid, ζc= 17/8.
One can also formulate a density conjecture for more general graphs, where it takes the
form ζc= ζs. We give precise definitions of the densities ζcand ζsin section 2.
Vespignani et al. [VDMZ00] write of the fixed-energy sandpile on the square grid, “the
system turns out to be critical only for a particular value of the energy density equal to that
of the stationary, slowly driven sandpile.” They add that the threshold density ζcof the
fixed-energy sandpile is “the only possible stationary value for the energy density” of the
driven dissipative model. In simulations they find ζc= 2.1250(5), adding in a footnote “It
is likely that, in fact, 17/8 is the exact result.”
Mu˜ noz et al. [MDPS+01] have also expressed this view, asserting that “FES are found to
be critical only for a particular value ζ = ζc(which as we will show turns out to be identical
to the stationary energy density of its driven dissipative counterpart).”
Our goal in the present paper is to demonstrate that the density conjecture is more prob-
lematic than it first appears.Table 1 presents data from large-scale simulations which
strongly suggest that ζcis close to but not exactly equal to 17/8 (see also Table 3). We
further consider several other families of graphs, including some for which we can determine
the exact values ζcand ζsanalytically. We find that they are close, but not equal.
All now known information on the threshold density ζcand stationary density ζsis sum-
marized in Table 2. The only graph on which the two densities are known to be equal is Z
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4ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
n #trials
64
128
256
512
1024
2048
4096
8192
16384
estimate of ζc(Z2
2.1249561 ± 0.0000004
2.1251851 ± 0.0000004
2.1252572 ± 0.0000004
2.1252786 ± 0.0000004
2.1252853 ± 0.0000004
2.1252876 ± 0.0000004
2.1252877 ± 0.0000004
2.1252880 ± 0.0000004
2.1252877 ± 0.0000004
n)
228
226
224
222
220
218
216
214
212
64 128 256 512 1024204840968192 16384
n
ζc(Z2
n)
2.125288
2.125
Table 1. Summary of our fixed-energy sandpile simulations on n×n tori Z2
giving our empirical estimate of the threshold density ζc(Z2
deviation in each of our estimates of ζc(Z2
approximated by ζc(Z2
shown in the graph. (The error bars are too small to be visible, so the data
are shown as points.) We conclude that the asymptotic threshold density
ζc(Z2) is 2.125288 to six decimal places. In contrast, the stationary density
ζs(Z2) is 2.125000000000 to twelve decimal places.
n,
n). The standard
n) is 4 × 10−7. The data are well
n) = 2.1252881 ± 3 × 10−7− (0.390 ± 0.001)n−1.7, as
graph
Z
Z2
bracelet
flower graph
ladder graph
complete graph
3-regular tree
4-regular tree
5-regular tree
ζs
1
ζc
1
17/8 = 2.125
5/2 = 2.5
5/3 = 1.666667...
√3
12= 1.605662...
n/2 + O(√n)
3/2
2
5/2
2.125288...
2.496608...
1.668898...
1.6082...
n − O(√nlogn)
1.50000...
2.00041...
2.51167...
7
4−
Table 2. Stationary and threshold densities for different graphs. Exact values
are in bold.
[MQ05, FdBR05, FdBMR09]. On all other graphs we examined, with the possible exception
of the 3-regular tree, it appears that ζc?= ζs.
Taken together, these results show that the conclusions of [MDPS+01] that “FES are
shown to exhibit an absorbing state transition with critical properties coinciding with those
of the corresponding sandpile model” deserve to be re-evaluated. One hope of the DMVZ
paradigm was that critical features of the driven dissipative model, such as the exponents
governing the distribution of avalanche sizes and decay of correlations, might be more easily
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THE APPROACH TO CRITICALITY IN SANDPILES5
studied in FES by examining the scaling behavior of these observables as ζ ↑ ζc. However,
the failure of the density conjecture suggests that the two models may not share the same
critical features.
As Grassberger and Manna observed [GM90], the value of the FES threshold density
depends on the choice of initial condition. One might consider a more general version of FES,
namely adding (ζ−ζ0)N particles at random to a “background” configuration τ of density ζ0
already present on the grid. For example, taking τ to be the deterministic configuration on
Zdof 2d−2 particles everywhere, by [FLP10, Prop 1.4] we obtain a threshold density of ζc=
2d − 2. Many interesting questions present themselves: for instance, for which background
does ζctake the smallest value, and for which backgrounds do we obtain ζc= ζs? It would
also be interesting to replicate the phase transition for driven sandpiles (see section 3) for
different background configurations.
2. Sandpiles on the square grid Z2
In this section we give precise definitions of the stationary and threshold densities, and
present the results of large-scale simulations on Z2. The definitions in this section apply
to general graphs, but we defer the discussion of results about other graphs to subsequent
sections.
2.1. The driven dissipative sandpile and the stationary density ζs. LetˆG = (V,E)
be a finite graph, which may have loops and multiple edges. Let S ⊂ V be a nonempty set of
vertices, which we will call sinks. The presence of sinks distinguishes the driven dissipative
sandpile from its fixed-energy counterpart. To highlight this distinction, throughout the
paper, graphs denoted with a “hat” as inˆG have sinks, and those without a hat as in G do
not.
For vertices v,w ∈ V , write av,w= aw,vfor the number of edges connecting v and w, and
dv=
?
w∈V
av,w.
for the number of edges incident to v. A sandpile (or “configuration”) η onˆG is a map
η : V → Z≥0.
We interpret η(v) as the number of sand particles at the vertex v; we will sometimes call
this number the height of v in η.
A vertex v / ∈ S is called unstable if η(v) ≥ dv. An unstable vertex can topple by sending
one particle along each edge incident to v. Thus, toppling v results in a new sandpile η′
given by
η′= η + ∆v
where
?
av,v− dv,
∆v(w) =
av,w,v ?= w
v = w.
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6ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
Figure 1. The square grid Z2.
Sinks by definition are always stable, and never topple. If all vertices are stable, we say
that η is stable.
Note that toppling a vertex may cause some of its neighbors to become unstable. The
stabilization η◦of η is a sandpile resulting from toppling unstable vertices in sequence, until
all vertices are stable. By the abelian property [Dha90], the stabilization is unique: it does
not depend on the toppling sequence. Moreover, the number of times a given vertex topples
does not depend on the toppling sequence.
The most commonly studied example is the n × n square grid graph, with the boundary
sites serving as sinks (Figure 1).
The driven dissipative sandpile model is a continuous time Markov chain (ηt)t≥0whose
state space is the set of stable sandpiles onˆG. Let V′= V \ S be the set of vertices that
are not sinks. At each site v ∈ V′, particles are added at rate 1. When a particle is added,
topplings occur instantaneously to stabilize the sandpile. Writing σt(v) for the total number
of particles added at v before time t, we have by the abelian property
ηt= (σt)◦.
Note that for fixed t, the random variables σt(v) for v ∈ V′are independent and have the
Poisson distribution with mean t.
The model just described is most commonly known as the abelian sandpile model (ASM),
but we prefer the term “driven dissipative” to distinguish it from the fixed-energy sandpile
described below, which is also a form of ASM. “Driven” refers to the addition of particles,
and “dissipative” to the loss of particles absorbed by the sinks.
Dhar [Dha90] developed the burning algorithm to characterize the recurrent sandpile
states, that is, those sandpiles η for which, regardless of the initial state,
Pr(ηt= η for some t) = 1.
Lemma 2.1 (Burning Algorithm [Dha90]). A sandpile η is recurrent if and only if every
non-sink vertex topples exactly once during the stabilization of η +?
s∈S∆s, where the sum
is over sink vertices S.
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THE APPROACH TO CRITICALITY IN SANDPILES7
The recurrent states form an abelian group under the operation of addition followed by
stabilization. In particular, the stationary distribution of the Markov chain ηtis uniform on
the set of recurrent states.
The combination of driving and dissipation organizes the system into a critical state. To
measure the density of particles in this state, we define the stationary density ζs(ˆG) as
?
#V′
ζs(ˆG) = Eµ
1
?
v∈V′
η(v)
?
where V′= V \S, and µ is the uniform measure on recurrent sandpiles onˆG. The stationary
density has another expression in terms of the Tutte polynomial of the graph obtained fromˆG
by collapsing the set S of sinks to a single vertex; see section 4.
Most of the graphs we will study arise naturally as finite subsets of a locally finite graph
Γ, i.e., Γ is a countably infinite graph in which every vertex has finite degree. (We also study
the complete graph and the flower graph, which do not arise in this way.) LetˆGnfor n ≥ 1
be a nested family of finite induced subgraphs with?ˆGn= Γ. As sinks inˆGnwe take the
Sn=ˆGn−ˆGn−1.
In cases where the free and wired limits are different, such as on regular trees, we will choose
a sequenceˆGncorresponding to the wired limit. We denote by µnthe uniform measure on
recurrent configurations onˆGn.
We are interested in the stationary density
set of boundary vertices
ζs(Γ) := lim
n→∞ζs(ˆGn).
When Γ = Zd, it is known that the infinite-volume limit of measures µ = limn→∞µnexists
and is translation-invariant [AJ04]. In this case it follows that the limit defining ζs(Γ) exists
and equals
ζs= Eµ[η(0)].
For other families of graphs we consider, we will show that the limit defining ζs(Γ) exists.
Much is known about the limiting measure µ in the case Γ = Z2. The following expressions
have been obtained for ζsand the single site height probabilities. The symbol
expressions that are rigorous up to a conjecture [JPR06] that a certain integral, numerically
evaluated as 0.5 ± 10−12, is exactly 1/2.
ζs(Z2)[JPR06],
?= denotes
?= 17/8
µ{η(x) = 0} =
µ{η(x) = 1}
µ{η(x) = 2}
µ{η(x) = 3}
2
π2−
4−
8+1
4
π3
[MD91],
?=1
1
2π−
π−12
1
2π+
2
π2+12
π3
[Pri94a, JPR06],
?=3
π3
[JPR06], and
?=3
8−
1
π2+
4
π3
[JPR06].
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8ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
2.2. The fixed-energy sandpile model and the threshold density ζc. Next we describe
the fixed-energy sandpile model, in which the driving and dissipation are absent, and the
total number of particles is conserved. As before, let G be a finite graph, possibly with loops
and multiple edges. Unlike the driven dissipative model, we do not single out any vertices
as sinks. The fixed-energy sandpile evolves in discrete time: at each time step, all unstable
vertices topple once in parallel. Thus the configuration ηj+1at time j + 1 is given by
ηj+1= ηj+
?
v∈Uj
∆v
where
Uj= {v ∈ V : ηj(v) ≥ dv}
is the set of vertices that are unstable at time j. We say that η0 stabilizes if toppling
eventually stops, i.e. Uj= ∅ for all sufficiently large j.
If η0stabilizes, then there is some site that never topples [Tar88] (see also [FdBMR09,
Theorem 2.8, item 4] and [FLP10, Lemma 2.2]). Otherwise, for each site x, let j(x) be
the last time x topples. Choose a site x minimizing j(x). Then each neighbor y of x has
j(y) ≥ j(x), so y topples at least once at or after time j(x). Thus x receives at least dx
additional particles and must topple again after time j(x), a contradiction. This criterion is
very useful in simulations: as soon as every site has toppled at least once, we know that the
system will not stabilize.
Let (σλ(v))λ≥0be a collection of independent Poisson point processes of intensity 1, indexed
by the vertices of G. So each σλ(v) has the Poisson distribution with mean λ. We define the
threshold density of G as
ζc(G) = EΛc,
where
Λc= sup{λ : σλstabilizes}.
We expect that Λcis tightly concentrated around its mean when G is large. Indeed, if Γ is an
infinite vertex-transitive graph, then the event that σλstabilizes on Γ is translation-invariant.
By the ergodicity of the Poisson product measure, this event has probability 0 or 1. Since
this probability is monotone in λ, there is a (deterministic) threshold density ζc(Γ), such
that
Pr[σλstabilizes on Γ] =
?
1,
0,
λ < ζc(Γ)
λ > ζc(Γ).
We expect the threshold densities on natural families of finite graphs to satisfy a law of large
numbers such as the following.
Conjecture 2.2. With probability 1,
Λc(Z2
n) → ζc(Z2)as n → ∞.
Previous simulations (n = 160 [DVZ98]; n = 1280 [VDMZ98]) to estimate the threshold
density ζc(Z2) found a value of 2.125, in agreement with the stationary density ζs(Z2)
By performing larger-scale simulations, however, we find that ζcexceeds ζs.
?= 17/8.
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THE APPROACH TO CRITICALITY IN SANDPILES9
grid size
(n2)
#samplesζc(Z2
n)
distribution of height h of sand
Pr[h = 0] Pr[h = 1]
0.0735550.173966
0.0735050.173866
0.073488 0.173835
0.073481 0.173826
0.073479 0.173822
0.073478 0.173821
0.073477 0.173821
0.0734770.173821
0.0734780.173821
0.0736360.173900
#topplings
÷n3
0.197110
0.197808
0.198789
0.200162
0.201745
0.203378
0.205323
0.206475
0.208079
Pr[h = 2]
0.306447
0.306567
0.306609
0.306626
0.306633
0.306635
0.306637
0.306638
0.306638
0.306291
Pr[h = 3]
0.446032
0.446062
0.446068
0.446067
0.446066
0.446065
0.446064
0.446064
0.446064
0.446172
642
1282
2562
5122
10242
20482
40962
81922
163842
Z2(stationary)
Table 3. Fixed-energy sandpile simulations on n × n tori Z2
column gives our empirical estimate of the threshold density ζc(Z2
four columns give the empirical distribution of the height of a fixed vertex in
the stabilization (σλ)◦, for λ just below Λc. Each estimate of the expectation
ζc(Z2
The total number of topplings needed to stabilize σλappears to scale as n3.
268435456
67108864
16777216
4194304
1048576
262144
65536
16384
2.124956
2.125185
2.125257
2.125279
2.125285
2.125288
2.125288
2.125288
2.125288
2.125000
4096
n. The third
n). The next
n) and of the marginals Pr[h = i] has standard deviation less than 4·10−7.
Table 3 summarizes the results of our simulations, which indicate that ζc(Z2) equals
2.125288 to six decimal places. In each random trial, we add particles one at a time at
uniformly random sites of the n × n torus. After each addition, we perform topplings until
either all sites are stable, or every site has toppled at least once since the last addition. For
deterministic sandpiles on a connected graph, if every site topples at least once, the system
will never stabilize [FdBMR09, FLP10]. We record m/n2as an empirical estimate of the
threshold density, where m is the maximum number of particles for which the configuration
stabilizes. We then average these empirical estimates were over many independent trials.
The one-site marginals we report are obtained from the stable configuration just before the
(m + 1)stparticle was added, and the number of topplings reported is the total number of
topplings required to stabilize the first m particles.
We used a random number generator based on the Advanced Encryption Standard (AES-
256), which has been found to exhibit good randomness properties. Our simulations were
conducted on a High Performance Computing (HPC) cluster of computers.
3. Sandpiles on the bracelet
Next we examine a family of graphs for which we can determine ζcand ζsexactly and
prove that they are not equal. Despite this inequality, we show that an interesting connection
remains between the driven dissipative and conservative dynamics: the threshold density of
the conservative model is the point at which the driven dissipative model begins to lose a
macroscopic amount of sand to the sink.
The bracelet graph Bn(Figure 2) is a multigraph with vertex set Zn(the n-cycle) with the
usual edge set {(i,i + 1 mod n) : 0 ≤ i < n} doubled. Thus all vertices have degree 4. The
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10ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
graphˆBnis the same, except that vertex 0 is distinguished as a sink from which particles
disappear from the system. We denote by B∞the infinite path Z with doubled edges.
For λ > 0, let σλbe the configuration with Poisson(λ) particles independently on each
site ofˆBn. Let ηλ= (σλ)◦be the stabilization of σλ, and let
ρn(λ) =
1
n − 1
n−1
?
x=1
ηλ(x)
be the final density. The following theorem gives the threshold and stationary densities of
the infinite bracelet graph B∞, and identifies the n → ∞ limit of the final density ρn(λ) as
a function of the initial density λ.
Theorem 3.1. For the bracelet graph,
(1) The threshold density ζc(B∞) is the unique positive root of ζ =5
ζc= 2.496608).
(2) The stationary density ζs(B∞) is 5/2.
(3) ρn(λ) → ρ(λ) in probability as n → ∞, where
?
Part 3 of this theorem shows that the final density undergoes a second-order phase tran-
sition at ζc: the derivative of ρ(λ) is discontinuous at λ = ζc(Figure 3). Thus in spite of
the fact that ζs?= ζc, there remains a connection between the conservative dynamics used
to define ζcand the driven-dissipative dynamics used to define ζs. For λ < ζc, very little
dissipation takes place, so the final density equals the initial density λ; for λ > ζca sub-
stantial amount of dissipation takes place, many particles are lost to the sink, and the final
density is strictly less than the initial density. The sandpile continues to evolve as λ increases
beyond ζc; in particular its density keeps changing.
We believe that this phenomenon is widespread. As evidence, in section 5 we introduce
the “flower graph,” which looks very different from the bracelet, and prove (in Theorem 5.5)
that a similar phase transition takes place there.
For the proof of Theorem 3.1, we compare the dynamics of pairs of particles on the bracelet
graph to single particles on Z. At each vertex x of the bracelet, we group the particles starting
2−1
2e−2ζ(numerically,
ρ(λ) = minλ,5 − e−2λ
2
?
=
?
λ,
5−e−2λ
2
λ ≤ ζc
λ > ζc.,
Figure 2. The bracelet graph B20.
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THE APPROACH TO CRITICALITY IN SANDPILES11
0123
1
2
2.4 2.6 2.5
2.48
2.49
2.5
ρρ
λλ
ζc
ζc
ζc
ζs= 5/2
5−e−2λ
2
ζs= 5/2
5−e−2λ
2
Figure 3. Density ρ(λ) of the final stable configuration as a function of initial
density λ, for the driven sandpile on the bracelet graphˆBn as n → ∞. A
second-order phase transition occurs at λ = ζc. Beyond this transition, the
density of the driven sandpile continues to increase, approaching the stationary
density ζsfrom below.
at x into pairs, with one “passive” particle left over if σλ(x) is odd. Since all edges in the
bracelet are doubled, we can ensure that in each toppling the two particles comprising a pair
always move to the same neighbor, and that the passive particles never move. The toppling
dynamics of the pairs are equivalent to the usual abelian sandpile dynamics on Z.
We recall the relevant facts about one-dimensional sandpile dynamics:
• In any recurrent configuration on a finite interval of Z, every site has height 1, except
for at most one site of height 0. Therefore, ζs= 1 [Red05].
• On Z, an initial configuration distributed according to a nontrivial product measure
with mean λ stabilizes almost surely (every site topples only finitely many times) if
λ < 1, while it almost surely does not stabilize (every site topples infinitely often) if
λ ≥ 1 [FdBMR09]. Thus, ζc= 1.
Proof of Theorem 3.1 parts 1 and 2. Given λ > 0, let λ∗be the pair density E⌊σλ(x)/2⌋,
and let
podd(λ) = e−λ?
be the probability that a Poisson(λ) random variable is odd. Then λ and λ∗are related by
m≥0
λ2m+1
(2m + 1)!=1 2(1 − e−2λ).
λ = 2λ∗+ podd(λ).(1)
The configuration σλstabilizes on B∞if and only if the pair configuration σ∗
Thus ζc(B∞)∗= ζc(Z). Setting λ = ζc(B∞) in (1), using the fact that ζc(Z) = 1, and that λ∗
λstabilizes on Z.
Page 12
12ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
is an increasing function of λ > 0, we conclude that ζc(B∞) is the unique positive root of
ζ = 2 + podd(ζ),
or ζ =5
For part 2, by the burning algorithm, a configuration σ onˆBnis recurrent if and only
if it has at most one site with fewer than two particles. Thus, in the uniform measure on
recurrent configurations onˆBn,
Pr(σ(x) = 2) = Pr(σ(x) = 3) =1
2−
We conclude that ζs(ˆBn) = Eσ(x) =5
2−1
2e−2ζ. This proves part 1.
1
2n,
Pr(σ(x) = 0) = Pr(σ(x) = 1) =
1
2n.
2−2
n→5
2as n → ∞.
?
To prove part 3 of Theorem 3.1, we use the following lemma, whose proof is deferred to
the end of this section. LetˆZnbe the n-cycle with vertex 0 distinguished as a sink. Let σ′
be a sandpile onˆZndistributed according to a product measure (not necessarily Poisson)
of mean λ. Let η′
density after stabilization.
λ
λbe the stabilization of σ′
λ, and let ρ′
n(λ) =
1
n−1
?n−1
x=1η′
λ(x) be the final
Lemma 3.2. OnˆZn, we have ρ′
n(λ) → min(λ,1) in probability.
Proof of Theorem 3.1, part 3. Let ηλ be the stabilization of σλ onˆBn, and let η∗
stabilization of σ∗
λbe the
λ= ⌊σλ/2⌋ onˆZn. Then
ηλ(x) = 2η∗
λ(x) + ωλ(x)(2)
where ωλ(x) = σλ(x) − 2σ∗
λ(x) is 1 or 0 accordingly as σλ(x) is odd or even. Let
ρ∗
n(λ) =
1
n − 1
λonˆZn. Then
n−1
?
x=1
η∗
λ(x)
be the final density after stabilization of σ∗
ρn(λ) = 2ρ∗
n(λ) +
1
n − 1
n−1
?
x=1
ωλ(x).
By the weak law of large numbers,
λ < ζc, then λ∗< 1, so by Lemma 3.2, ρ∗
1
n−1
?n−1
x=1ωλ(x) → podd(λ) in probability as n → ∞. If
n(λ) → λ∗in probability, and hence
ρn(λ) → 2λ∗+ podd(λ) = λ
n(λ) → 1 in probability, hence
ρn(λ) → 2 + podd(λ) =5 − e−2λ
2
in probability. This proves part 3.
in probability. If λ ≥ ζc, then λ∗≥ 1, so by Lemma 3.2, ρ∗
?
Page 13
THE APPROACH TO CRITICALITY IN SANDPILES13
Proof of Lemma 3.2. We may viewˆZnas the path in Z from an= −⌊n/2⌋ to bn= ⌈n/2⌉,
with both endpoints serving as sinks. For x ∈ˆZn, let un(x) be the number of times that
x topples during stabilization of the configuration σ′
of times x topples during stabilization of σ′
volumes” [FdBMR09] shows that un(x) ↑ u∞(x) as n → ∞.
We consider first λ < 1. In this case u∞(x) is finite almost surely (a.s.). The total number
of particles lost to the sinks onˆZnis un(an+1)+un(bn−1), so the final density is given by
1
n − 1
x=an+1
By the law of large numbers,
n−1
finite, we have
n−1
Next we consider λ ≥ 1. In this case we have un(x) ↑ u∞(x) = ∞, a.s. Let p(n,x) =
Pr(un(x) = 0) be the probability that x ∈ˆZndoes not topple. By the abelian property,
adding sinks can not increase the number of topplings, so
λonˆZn. Let u∞(x) be the number
λon Z. The procedure of “toppling in nested
ρ′
n(λ) =
?
bn−1
?
?σλ(x) → λ in probability as n → ∞. Since u∞(x) is a.s.
σλ(x) − un(an+ 1) − un(bn− 1)
?
.
1
un(an+1)+un(bn−1)
→ 0 in probability, so ρ′
n(λ) → λ in probability.
p(n,x) ≤ p(m,0)
where m = min(x − an,bn− x). Let
Yn=
bn−1
?
x=an+1
1{un(x)=0}
be the number of sites inˆZnthat do not topple. Since un(0) ↑ ∞ a.s., we have p(n,0) ↓ 0,
hence
EYn
nn
x=an+1
as n → ∞. Since Yn≥ 0 it follows that Yn/n → 0 in probability.
In an interval where every site toppled, there can be at most one empty site. We have
Yn+ 1 such intervals. Therefore, the number of empty sites is at most 2Yn+ 1. Hence
n − 2Yn− 2
n − 1
The left side tends to 1 in probability, which completes the proof.
=1
bn−1
?
p(n,x) ≤2
n
n/2
?
m=1
p(m,0) → 0
≤ ρ′
n(λ) ≤ 1.
?
4. Sandpiles on the complete graph
Let Knbe the complete graph on n vertices: every pair of distinct vertices is connected by
an edge. InˆKn, one vertex is distinguished as the sink. The maximal stable configuration
onˆKnhas density n−2, while the minimal recurrent configurations have exactly one vertex of
each height 0,1,...,n−2, hence densityn−2
and threshold densities are quite far apart: ζs is close to the minimal recurrent density,
while ζcis close to the maximal stable density.
2. The following result shows that the stationary
Page 14
14ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
Theorem 4.1.
ζs(ˆKn) =n
2+ O(√n)
ζc(Kn) ≥ n − O(
?
nlogn).
The proof uses an expression for the stationary density ζsin terms of the Tutte polynomial,
due to C. Merino L´ opez [ML97]. Our application will be to the complete graph, but we state
Merino L´ opez’ theorem in full generality. Let G = (V,E) be a connected undirected graph
with n vertices and m edges. Let v be any vertex of G, and writeˆG for the graph G with v
distinguished as a sink. Let d be the degree of v.
Recall that the Tutte polynomial TG(x,y) is defined by
TG(x,y) =
?
A⊆E
(x − 1)c(A)−c(E)(y − 1)c(A)+|A|−|V |
where c(A) denotes the number of connected components of the spanning subgraph (V,A).
Theorem 4.2 ([ML97]). The Tutte polynomial TG(x,y) evaluated at x = 1 is given by
TG(1,y) = yd−m?
σ
y|σ|
where the sum is over all recurrent sandpile configurations σ onˆG, and |σ| denotes the
number of particles in σ.
Differentiating and evaluating at y = 1, we obtain
d
dyTG(1,y)
????
y=1
=
?
σ
(d − m + |σ|).(3)
Referring to the definition of the Tutte polynomial, we see that TG(1,1) is the number
of spanning trees of G, and that the left side of (3) is the number of spanning unicyclic
subgraphs of G. (In evaluating TGat x = y = 1, we interpret 00as 1.) The number of
recurrent configurations equals the number of spanning trees of G, so the stationary density
ζsmay be expressed as
1
nTG(1,1)
ζs(ˆG) =
?
σ
|σ|.
Combining these expressions yields the following:
Corollary 4.3.
ζs(ˆG) =1
n
?
m − d +u(G)
κ(G)
?
where κ(G) is the number of spanning trees of G, and u(G) is the number of spanning
unicyclic subgraphs of G.
Page 15
THE APPROACH TO CRITICALITY IN SANDPILES15
Note that m − d is the minimum number of particles in a recurrent configuration, so the
ratio u(G)/κ(G) can be interpreted as the average number of excess particles in a recurrent
configuration.
Everything so far applies to general connected graphs G. The following is specific for the
complete graph.
Theorem 4.4 (Wright [Wri77]). The number of spanning unicyclic subgraphs of Knis
??π
Proof of Theorem 4.1. ForˆKnwe have
m − d =n(n − 1)
2
From Corollary 4.3, Theorem 4.4, and Cayley’s formula κ(Kn) = nn−2, we obtain
?(n − 2)(n − 1)
=n
2+
u(Kn) =
8+ o(1)
?
nn−1
2.
− (n − 1) =(n − 2)(n − 1)
2
.
ζs(ˆKn) =1
n2
+u(Kn)
κ(Kn)
?√n.
?
??π
8+ o(1)
On the other hand, if we let
λ = n − 2
?
nlogn
and start with σ(v) ∼Poisson(λ) particles at each vertex v of Kn, then for all v
Pr[σ(v) ≥ n] <
So
Pr[σ(v) ≥ n for some v] <1
in other words, with high probability no topplings occur at all. Thus
?
which completes the proof.
1
n2.
n;
PrΛc(Kn) ≥ n − 2
?
nlogn
?
> 1 −1
n
?
One might guess that the large gap between ζsand ζcis related to the small diameter
ofˆKn: since the sink is adjacent to every vertex, its effect is felt with each and every toppling.
This intuition is misleading, however, as shown by the lollipop graphˆLnconsisting of Kn
connected to a path of length n, with the sink at the far end of the path. Since Lnhas the
same number of spanning trees and unicyclic subgraphs as Kn, we have by Corollary 4.3
ζs(ˆLn) =
1
2n
m
????
n(n − 1)
2
+ n−1 +u(Ln)
κ(Ln)
=n
4+ O(√n).
Page 16
16ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
On the other hand, by first stabilizing the vertices on the path, close to half of which end
up in the sink without reaching the Kn, it is easy to see that with high probability
Λc(Ln) ≥2n
3
− O(
?
nlogn).
5. Sandpiles on the flower graph
An interesting feature of parallel chip-firing is that further phase transitions appear above
the threshold density ζc. On a finite graph G = (V,E), since the time evolution is deter-
ministic, the system will eventually reach a periodic orbit: for some positive integer m, we
have ηt+m= ηtfor all sufficiently large t. The activity density, ρa, measures the proportion
of vertices that topple in an average time step:
ρa(λ) = Eλlim
t→∞
1
t
t−1
?
s=0
1
#V
?
x∈V
1{ηs(x)≥dx}.
The expectation Eλrefers to the initial state η0, which we take to be distributed according
to the Poisson product measure with mean λ. Note that the limit in the definition of ρacan
also be expressed as a finite average, due to the eventual periodicity of the dynamics.
Bagnoli et al. [BCFV03] observed that ρatends to increase with λ in a sequence of flat
steps punctuated by sudden jumps. This “devil’s staircase” phenomenon is so far explained
only on the complete graph [Lev08]: The number of flat stairs increases with n, and in the
n → ∞ limit there is a stair at each rational number height ρa= p/q.
On the cycle Zn[Dal06] there are just two jumps: at λ = 1, the activity density jumps from
0 to 1/2, and at λ = 2, from 1/2 to 1. For the n × n torus, simulations [BCFV03] indicate
a devil’s staircase, which is still not completely understood despite much effort [CDVV06].
In this section we study the “flower” graph, which was designed with parallel chip-firing
in mind: the idea is that a graph with only short cycles should give rise to short period
orbits under the parallel chip-firing dynamics. We find that there are four activity density
jumps (Theorem 5.4). In addition, we determine the stationary and threshold densities of
the flower graph, and find a second-order phase transition at ζc(Theorem 5.5).
The flower graph Fnconsists of a central site together with n ≥ 1 petals (Figure 4). Each
petal consists of two sites connected by an edge, each connected to the central site by an
edge. Thus the central site has degree 2n, and all other sites have degree 2. The number of
sites is 2n + 1. The graphˆFnis the same, except one petal serves as sink.
Recall that we defined the density of a configuration as the total number of particles,
divided by the total number of sites. Since the flower graph is not regular, the central site
has a different expected number of particles than the petal sites.
Proposition 5.1. For parallel chip-firing on the flower graph Fn, every configuration has
eventual period at most 3.
The proof uses the following two lemmas.
Lemma 5.2 ([Pri94b] Lemma 2.5). If the eventual period is not 1, then after some finite
time, every site x has height at most 2dx− 1.
Page 17
THE APPROACH TO CRITICALITY IN SANDPILES17
Figure 4. The flower graph F20.
We also use an observation from [BCFV03]; it was stated and proved there for Z2, but the
same proof works for general graphs.
Lemma 5.3 ([BCFV03]). Let two height configurations η and ξ be “mirror images” of each
other, that is, η(x) = 2dx− 1 − ξ(x) for all x. Then after performing for each a parallel
chip-firing time step, the two configurations are again mirror images of each other.
Proof of Proposition 5.1. Suppose at time t the model has settled into periodic orbit, and
the period is not 1. Then by Lemma 5.2, at time t every petal site has height at most 3.
Say that a petal is in state ij if it has i particles at one site and j particles at the other site.
A priori there are 10 possible petal states, listed below, where each one has two possible
successor states, depending on whether or not the central site is stable. If a petal is in state
ij, then by S(ij) we denote the state that it is in after one time step in which the central site
does not topple, and likewise by U(ij) after one time step in which the central site topples.
state S(state) U(state)
0000
0101
0201
03 11
1111
1202
1312
22 11
2312
33 22
11
12
12
22
22
13
23
22
23
33
From this we see that a petal will be in state 00 only if the central site is always stable,
and consequently each site is always stable, in which case the period is 1. Similarly, petal
state 33 only occurs if the central site is unstable each step, in which case each site must
be unstable each step, and the period is again 1. State 03 is not a successor of any state of
Page 18
18ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
these states, so it will not be a periodic petal state either. Thus the set of allowed periodic
petal states is {01,02,11,12,13,22,23}.
If the central site is stable every other time step, then the possible petal states are 12 →
02 → 12, 22 → 11 → 22, and 13 → 12 → 13, each of which has period 2. Then the period
of the entire configuration is 2.
Thus if the period is larger than 2, the central site must be stable for at least two con-
secutive time steps, or else unstable for at least two consecutive time steps. We will label a
time step S if the central site is stable in that time step, otherwise we label it U. So, if the
period is larger than 2 we will see SS or UU in the time evolution. In the latter case, we can
study the mirror image, which will have the same period, and for which we will see SS.
Eventually the central site must be unstable again, since otherwise the period would be 1.
Therefore, we can examine three time steps labeled SSU. Examining the evolution of the
central site together with the petals, we see
SSU
01, 02 01 01 12
12, 23 02 01 12
11, 22 11 11 22
2312 02 12
Whenever we have SSU, during the second and third time steps each petal contributes at
most two particles to the central site, while the central site topples, so the central site must
again be stable. Thus SSUU cannot occur, and we see SSUS.
There are two cases for what the central site does next. Let us first consider SSUSU.
SSUSU
01, 02 01 01 12 02 12
12, 23 02 01 12 02 12
11, 22 11 11 22 11 22
2312 02 12 02 12
During the last two time steps, each petal contributes exactly 2 particles to the central site,
and the central site topples once. Thus after two time steps not only the petals, but also
the central site is in the same state. Therefore, the period becomes 2.
Next we consider SSUSS. At this stage each petal is in state 01 or 11, so if there were yet
another S, the sandpile would be periodic with period 1. So we see SSUSSU, and because
SSUU is forbidden, we conclude that we see SSUSSUS.
SSUSSUS
01, 02 01 01 12 02 01 12
12, 23 02 01 12 02 01 12
11, 22 11 11 22 11 11 22
2312 02 12 02 01 12
At the time of the third S, each petal is in state 12 or 22. Between the third S and the fifth
S, each petal contributes exactly two particles to the central site and returns to the same
Page 19
THE APPROACH TO CRITICALITY IN SANDPILES19
state, while the central site topples once. Thus the configuration is periodic with period
3.
?
We conclude from the above case analysis that the activity ρais always one of 0, 1/3,
1/2, 2/3, or 1. Table 4 summarizes the behavior of the periodic sandpile states for different
values of ρa.
periodic sandpile states
1/31/2
SUS
01 12 02 01 12 02 23 12 13 ≥ 22
11 22 11 11 22 11 22 11 22
0013 12
activity ρa
central site
petals
0
S
2/3
U
1
USUSU
Table 4. Behavior of the central site and petals as a function of the activity ρa.
The following theorem shows that parallel chip-firing on the flower graph exhibits four
distinct phase transitions where the activity ρajumps in value: For each α ∈ {0,1
there is a nonvanishing interval of initial densities λ where ρa= α asymptotically almost
surely.
3,1
2,2
3,1},
Theorem 5.4. Let ζcbe the unique root of5
root of
probability tending to 1 as n → ∞, the activity density ρaof parallel chip-firing on the flower
graph Fnis given by
Proof. In a given petal, let X denote the difference modulo 3 of the number of particles on
the two sites of the petal. Observe that X is unaffected by toppling. Let Z denote the
number of petals for which X = 0, and R denote the total number of particles, in a given
initial configuration. Using Table 4, we can relate Z, R, and the activity ρa.
When ρa = 0, we have less than 2n particles at the central site, at most two particles
for the Z petals of type X = 0, and exactly one particle for the other n − Z petals, so
0 ≤ R < 2n + 2Z + (n − Z) = 3n + Z.
When ρa= 1/3, by considering the U time step, we have R ≥ 2n+2Z +(n−Z) = 3n+Z.
By considering the preceding S time step, we have R < 2n + 2Z + 2(n − Z) = 4n.
When ρa= 1/2, by considering the U time step, we have R ≥ 4n, and by considering the
S time step, we get R < 2n + 4Z + 4(n − Z) = 6n.
When ρa= 2/3, by considering the second U step, we have R ≥ 2n+4Z +4(n−Z) = 6n.
By considering the S time step, we have R < 2n + 4Z + 5(n − Z) = 7n − Z.
3+1
3e−3ζ= ζ, and let ζ′
cbe the unique positive
c= 3.3333182....) With
10
3−1
3e−3ζ= ζ. (Numerically, ζc= 1.6688976... and ζ′
ρa=
0,
1/3,
1/2,
2/3,
1
if 0 ≤ λ < ζc
if ζc< λ < 2
if 2 < λ < 3
if 3 < λ < ζ′
if ζ′
c
c< λ.
Page 20
20 ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
When ρa= 1, we have R ≥ 2n + 4Z + 5(n − Z) = 7n − Z.
Since for given n and Z, these intervals on the values of R are disjoint, we see that the
converse statements hold as well: the values of R and Z determine the activity ρa. We
summarize these bounds:
Everything so far holds deterministically; next we use probability to estimate R and Z.
By the weak law of large numbers, R/n → 2λ and Z/n → Pr(X = 0) in probability. Thus,
to complete the proof it suffices to show
Pr(X = 0) =1
ρa=
0
1/3
1/2
2/3
1
if and only if 0 ≤ R < 3n + Z
if and only if 3n + Z ≤ R < 4n
if and only if 4n ≤ R < 6n
if and only if 6n ≤ R < 7n − Z
if and only if 7n − Z ≤ R.
3(1 + 2e−3λ).(4)
We can think of building the initial configuration σλby starting with the empty config-
uration and adding particles in continuous time. Then the value of X for a single petal
as particles are added is a continuous time Markov chain on the state space {0,±1} with
transitions 0 → ±1 at rate 2, and ±1 → 0 and ±1 → ±1 each at rate 1. Starting in state 0,
after running this chain for time λ we obtain
?
where P =
1 1/2
, and I is the 2 × 2 identity matrix. The eigenvalues of P − I are 0
and −3
obtain (4).
Pr(X = 0)
Pr(X ?= 0)
?
= exp{2λ(P − I)}
?
1
0
?
?
0 1/2
?
2, with corresponding eigenvectors v1= [1
2] and v2= [1
−1]. Since [1
0] =1
3v1+2
3v2, we
?
The following theorem describes a phase transition in the driven sandpile dynamics on the
flower graph analogous to Theorem 3.1 for the bracelet graph. We remark on one interesting
difference between the two transitions: for λ > ζc, the final density ρ(λ) is increasing in λ
for the bracelet, and decreasing in λ for the flower graph.
For λ > 0, let σλbe the configuration with Poisson(λ) particles independently on each
site ofˆFn. Let ηλ= (σλ)◦be the stabilization of σλ, and let
ρn(λ) =
1
n − 1
n−1
?
x=1
ηλ(x)
be the final density.
Theorem 5.5. For the flower graph with n petals, in the limit n → ∞ we have
(1) The threshold density ζcis the unique positive root of ζ =5
(2) The stationary density ζsis 5/3.
3+1
3e−3ζ.
Page 21
THE APPROACH TO CRITICALITY IN SANDPILES21
0123
1
2
1.61.71.8
1.65
1.66
1.67
ρρ
λλ
ζc
ζc
ζc
ζs= 5/3ζs= 5/3
5+e−λ
3
5+e−λ
3
Figure 5. Density ρ(λ) of the final stable configuration as a function of initial
density λ on the flower graphˆFnfor large n. A second-order phase transition
occurs at λ = ζc. Beyond this transition, the density of the driven sandpile
decreases with λ.
(3) ρn(λ) → ρ(λ) in probability, where
ρ(λ) = min
?
λ,
5
3+13e−λ
?
=
?
λ,
5
3+1
λ ≤ ζc
λ > ζc.
3e−3λ,
Proof. Part 1 follows from Theorem 5.4.
For Part 2, we use the burning algorithm. In all recurrent configurations onˆFn, the central
site has either 2n − 1 or 2n − 2 particles. All other sites have at most one particle, and in
each petal (except the sink) there is at least one particle. For each petal that is not the sink,
there are two possible configurations with 1 particle, and one with 2 particles. Each of these
occurs with equal probability in the stationary state, so the expected number of particles in
the petals is (n − 1)(2
ζs= limn→∞
2n−1
For part 3, for the driven dissipative sandpile onˆFn, we first stabilize all the petals,
then topple the center site if it is unstable, then stabilize all the petals, and so on. For
each toppling of the center site, the sandpile loses O(1) particles to the sink. If the center
topples at least once, then each petal will be in one of the states 11, 01, or 10, after which
the number of particles at the center site is R − n − Z + O(1). Recall from the proof of
Theorem 5.4 that R/n → 2λ and Z/n →1
R−n+1−Z
2n
→ λ −2
amount of sand, and ρn(λ) → λ in probability.
If λ > ζc, then the number of particles that remain after stabilization is 2n+n+Z +O(1).
In this case, we have ρn(λ) =3n+Z
3· 1 +1
+ o(1) = 5/3.
3· 2) =
4n
3+ O(1) as n → ∞. Therefore, the total density is
2n+4n/3
3(1 + 2e−3λ) in probability. Thus if λ ≤ ζc, then
3e−3λ≤ 1 in probability, so the sandpile does not lose a macroscopic
3+1
2n+1+ o(1) →5
3+1
3e−3λin probability.
?
Page 22
22ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
6. Sandpiles on the Cayley tree
Dhar and Majumdar [DM90] studied the abelian sandpile model on the Cayley tree (also
called the Bethe lattice) with branching factor q, which has degree q + 1. Implicit in their
formulation is that they used wired boundary conditions, i.e., where all the vertices of the
tree at a certain large distance from a central vertex are glued together and become the sink.
(The other common boundary condition is free boundary conditions, where all the vertices
at a certain distance from the central vertex become leaves, and one of them becomes the
sink. The issue of boundary conditions becomes important for trees, because in any finite
subgraph, a constant fraction of vertices are on the boundary. This is in contrast to Z2, where
free and wired boundary conditions lead to the same infinite volume limit. See [LP10].)
The finite regular wired tree Tq,n is the ball of radius n in the infinite (q + 1)-regular
tree, with all leaves collapsed to a single vertex s. InˆTq,nthe vertex s serves as the sink.
Maes, Redig and Saada [MRS02] show that the stationary measure on recurrent sandpiles
onˆTq,nhas an infinite-volume limit, which is a measure on sandpiles on the infinite tree.
Denoting this measure by Prq, if h denotes the number of particles at a single site far from
the boundary, then we have [DM90]
Prq[h = i] =
1
(q2− 1)qq
i?
m=0
?q + 1
m
?
(q − 1)q+1−m.
From this formula we see that the stationary density is
ζs= Eq[h] =q + 1
2
.
For 3-regular, 4-regular, and 5-regular trees, these values are summarized below:
Figure 6. The Cayley trees (Bethe lattices) of degree d = 3,4,5.
Page 23
THE APPROACH TO CRITICALITY IN SANDPILES 23
tree
degree
3
4
5
Eq[h]
distribution of height h of sand
Prq[h = 1]Prq[h = 2]
4/127/12
2/91/3
27/160153/640
q
2
3
4
Prq[h = 0]
1/12
2/27
81/1280
Prq[h = 3]Prq[h = 4]
3/2
2
5/2
10/27
21/80341/1280
Large-scale simulations on Tq,nare rather impractical because the vast majority of vertices
are near the boundary. Consequently, each simulation run produces only a small amount of
usable data from vertices near the center.
To experimentally measure ζcfor the Cayley trees, we generated large random regular
graphs Gq,n, and used these as finite approximations of the infinite Cayley tree. We used the
following procedure to generate random connected bipartite multigraphs of degree q + 1 on
n vertices (n even). Let M0be the set of edges (i,i+ 1) for i = 1,3,5,...,n− 1. Then take
the union of M0with q additional i.i.d. perfect matchings M1,...,Mqbetween odd and even
vertices. Each Mjis chosen uniformly among all odd-even perfect matchings whose union
with M0is an n-cycle.
Most vertices of Gnwill not be contained in any cycle smaller than logqn+O(1) (see e.g.,
[Bol86]), so these graphs are locally tree-like. For this reason, we believe that as n → ∞ the
threshold density Λc(Gn) will be concentrated at the threshold density of the infinite tree.
Since the choice of multigraph affects the estimate of ζc, we generated a new independent
random multigraph for each trial. The results for random regular graphs of degree 3, 4 and
5 are summarized in Tables 5, 6, and 7. We find that for the 5-regular tree, the threshold
density is about 2.511 rather than 2.5, for the 4-regular tree the threshold density is very
close to but decidedly larger than 2, while for the 3-regular tree the threshold density is
extremely close to 1.5, with a discrepancy that we were unable to measure. However, for the
3-regular tree there is a measurable discrepancy (about 2 × 10−6) in the probability that a
site has no particles.
Page 24
24ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
n#samples
E[h]
distribution of height h of sand
Pr[h = 0]Pr[h = 1]
0.08333260.332903
0.08333210.333031
0.08333140.333121
0.0833311 0.333185
0.0833311 0.333230
0.08333070.333262
0.08333070.333285
0.08333070.333300
0.08333080.333311
0.08333080.333319
0.08333070.333325
0.08333330.333333
#topplings
÷nlog1/2n
1.263145
1.258046
1.253092
1.247642
1.242359
1.237317
1.232398
1.227548
1.222371
1.214431
1.212751
Pr[h = 2]
0.583764
0.583637
0.583548
0.583484
0.583439
0.583407
0.583385
0.583369
0.583358
0.583350
0.583344
0.583333
1048576
2097152
4194304
8388608
16777216
33554432
67108864
134217728
268435456
536870912
1073741824
∞ (stationary)
Table 5. Data for the fixed-energy sandpile on a pseudorandom 3-regular
graph on n nodes. Each estimate of E[h] has standard deviation less than
7 · 10−8, and each estimate of the marginals Pr[h = i] has standard deviation
less than 3 · 10−7. The data for E[h] appears to fit 3/2 + const/√n very well,
and extrapolating to n → ∞ it appears that E[h] → 1.500000 to six decimal
places. However, apparently Pr[h = 0] → 0.083331 < 1/12.
2097152
1048576
524288
262144
131072
65536
32768
16384
8192
4096
2048
1.5004315
1.5003054
1.5002161
1.5001528
1.5001081
1.5000765
1.5000540
1.5000382
1.5000269
1.5000191
1.5000136
1.5
n #samples
E[h]
distribution of height h of sand
Pr[h = 0]Pr[h = 1]
0.073884 0.221887
0.0738810.221978
0.0738800.222037
0.0738780.222075
0.0738770.222100
0.0738770.222114
0.0738770.222123
0.0738760.222130
0.073876 0.222134
0.0738760.222136
0.0738760.222138
0.074074 0.222222
#topplings
÷nlog1/2n
0.623322
0.618848
0.620894
0.631324
0.649328
0.670838
0.691040
0.699706
0.695065
0.684507
0.673061
Pr[h = 2]
0.333466
0.333547
0.333599
0.333631
0.333651
0.333664
0.333673
0.333678
0.333681
0.333683
0.333684
0.333333
Pr[h = 3]
0.370763
0.370593
0.370484
0.370416
0.370372
0.370345
0.370328
0.370316
0.370310
0.370305
0.370303
0.370370
1048576
2097152
4194304
8388608
16777216
33554432
67108864
134217728
268435456
536870912
1073741824
∞ (stationary)
Table 6. Data for the fixed-energy sandpile on a pseudorandom 4-regular
graph on n nodes. Each estimate of E[h] and of the marginals Pr[h = i] has
standard deviation less than 3 · 10−7.
2097152
1048576
524288
262144
131072
65536
32768
16384
8192
4096
2048
2.001109
2.000853
2.000688
2.000584
2.000518
2.000477
2.000451
2.000434
2.000424
2.000417
2.000413
2
Page 25
THE APPROACH TO CRITICALITY IN SANDPILES25
n#samples
E[h]
distribution of height h of sand
Pr[h = 1]Pr[h = 2]
0.1665470.237230
0.166579 0.237256
0.166599 0.237272
0.1666130.237283
0.1666210.237289
0.1666270.237293
0.166630 0.237296
0.1666320.237297
0.1666340.237299
0.1666330.237300
0.1666340.237300
0.1687500.239063
#topplings
÷n
1.666086
1.666244
1.666404
1.666589
1.667322
1.668196
1.669392
1.671613
1.675479
1.677092
1.688093
Pr[h = 0]
0.062271
0.062269
0.062268
0.062267
0.062267
0.062267
0.062267
0.062267
0.062266
0.062267
0.062266
0.063281
Pr[h = 3]
0.264711
0.264727
0.264737
0.264743
0.264748
0.264750
0.264752
0.264755
0.264753
0.264755
0.264755
0.262500
Pr[h = 4]
0.269242
0.269169
0.269123
0.269093
0.269075
0.269063
0.269056
0.269050
0.269048
0.269045
0.269044
0.266406
1048576
2097152
4194304
8388608
16777216
33554432
67108864
134217728
268435456
536870912
1073741824
∞ (stationary)
Table 7. Data for the fixed-energy sandpile on a pseudorandom 5-regular
graph on n nodes. Each estimate of E[h] and of the marginals Pr[h = i] has
standard deviation less than 2 · 10−6.
1048576
524288
262144
131072
65536
65536
32768
16384
8192
4096
2048
2.512106
2.511947
2.511847
2.511781
2.511743
2.511716
2.511700
2.511689
2.511683
2.511680
2.511677
2.5
Page 26
26ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
7. Sandpiles on the ladder graph
The examples in previous sections suggest that the density conjecture can fail for (at least)
two distinct reasons: local toppling invariants, and boundary effects. A toppling invariant
for a graph G is a function f defined on sandpile configurations on G which is unchanged
by performing topplings; that is
f(σ) = f(σ + ∆x)
for any sandpile σ and any column vector ∆xof the Laplacian of G. Examples we have seen
are
f(σ) = σ(x) mod 2
where x is any vertex of the bracelet graph Bn; and
f(σ) = σ(x1) − σ(x2) mod 3
where x1,x2are the two vertices comprising any petal on the flower graph Fn. Both of these
toppling invariants are local in the sense that they depend only on a bounded number of
vertices as n → ∞.
The Cayley tree has no local toppling invariants, but the large number of sinks, comparable
to the total number of vertices, produce a large boundary effect. The density conjecture fails
even more dramatically on the complete graph (Theorem 4.1). One might guess that this is
due to the high degree of interconnectedness, which causes boundary effects from the sink to
persist as n → ∞. A good candidate for a graph G satisfying the density conjecture, then,
should have
• no local toppling invariants,
• most vertices far from the sink.
The best candidate graphs G should be essentially one-dimensional, so that the sink is well
insulated from the bulk of the graph, keeping boundary effects to a minimum. Indeed, the
only graph known to satisfy the density conjecture is the infinite path Z.
J´ arai and Lyons [JL07] study sandpiles on graphs of the form G × Pn, where G is a finite
connected graph and Pnis the path of length n, with the endpoints serving as sinks. The
simplest such graphs that are not paths are obtained when G = P1has two vertices and
one edge. These graphs are a good candidate for ζc= ζs, for the reasons described above.
Nevertheless, we find that while ζcand ζsare very close, they appear to be different.
First we calculate ζs. Jarai and Lyons [JL07, section 5] define recurrent configurations as
Markov chains on the state space
?
X = (3,3),(3,2),(2,3),(3,1),(1,3),(3,2),(2,3)
?
Figure 7. The ladder graph.
Page 27
THE APPROACH TO CRITICALITY IN SANDPILES27
describing the possible transitions from one rung of the ladder to the next. States (i,j) and
(i,j) both represent rungs whose left vertex has i − 1 particles and whose right vertex has
j −1 particles. The distinction between states (3,2) and (3,2) lies only in which transitions
are allowed. The adjacency matrix describing the allowable transitions is given by
Its largest eigenvalue is 2 +√3, and the corresponding left and right eigenvectors are
√3,1 +
√3,1 +
A =
1 1 1 1 1 0 0
1 1 1 1 1 0 0
1 1 1 1 1 0 0
1 0 0 0 0 1 0
1 0 0 0 0 0 1
1 0 0 0 0 1 0
1 0 0 0 0 0 1
.
u = (1 +
√3,1 +
√3,1 +
√3,1,1,1,1)
√3,1 +
v = (3 +
√3,1 +
√3,1,1)T
By the Parry formula [Par64], the stationary probabilities are given by p(i) = uivi/Z, where
Z is a normalizing constant. So
p(3,3) = (1 +
√3)(3 +
√3)2/Z
√3)/Z
√3)/Z
p(2,3) = p(3,2) = (1 +
p(1,3) = p(3,1) = (1 +
p(2,3) = p(3,2) = 1/Z
where
Z = (1 +
√3)(3 +
√3) + 2(1 +
√3)2+ 2(1 +
√3) + 2.
Thus we find that for the ladder graph in stationarity, the number h of particles at a site
satisfies
Pr[h = 0] = −1
Pr[h = 1] =5
√3
4= 0.6830127...
ζs= E[h] =7
2+
√3
3= 0.0773503...
4−7√3
4+
√3
12= 1.60566243...
12= 0.2396370...
Pr[h = 2] =1
4−
In contrast, the threshold density for ladders appears to be about 1.6082. Table 8 summarizes
simulation data on finite 2 × n ladders.
Page 28
28ANNE FEY, LIONEL LEVINE, AND DAVID B. WILSON
n#samples
E[h]
distribution of height h of sand
Pr[h = 0]Pr[h = 1]
0.076950.24043
0.076560.23996
0.076360.23970
0.07626 0.23960
0.076210.23952
0.076180.23950
0.076180.23949
0.076170.23948
0.076160.23948
0.07617 0.23946
0.076150.23949
0.077350.23964
#topplings
÷n5/2
0.094773
0.095366
0.095864
0.096316
0.096545
0.096753
0.097113
0.096944
0.097342
0.097648
0.096158
Pr[h = 2]
0.68262
0.68349
0.68393
0.68414
0.68426
0.68432
0.68434
0.68435
0.68436
0.68437
0.68436
0.68301
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
∞ (stationary)
Table 8. Data for the fixed-energy sandpile on 2 × n ladder graphs. Each
estimate of E[h] and of the marginals Pr[h = i] has a standard deviation
smaller than 10−5. To four decimal places, the threshold density ζc equals
1.6082, which exceeds the stationary density ζs= 7/4−√3/12 = 1.6057. The
total number of topplings appears to scale as n5/2.
4194304
2097152
1048576
524288
262144
131072
65536
32768
16384
8192
4096
1.60567
1.60693
1.60757
1.60788
1.60805
1.60814
1.60816
1.60818
1.60820
1.60820
1.60821
1.60566
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Anne Fey, Delft Institute of Applied Mathematics, Delft University of Technology,
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Lionel Levine, Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, MA 02139, http://math.mit.edu/~levine
David B. Wilson, Microsoft Research, Redmond, WA 98052, http://dbwilson.com
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