Annealed bounds for the return probability of the simple random
walk on critical percolation clusters
Florian Sobieczky
Mathematisches Institut Jena, Universität Jena
Abstract: Critical Bernoulli percolation on a unimodular transitive graph and on the
2-dim. euclidean lattice has almost surely finite connected components. Estimating the
expected return probability of the simple random walk is difficult, due to the heavy tails
of the cluster-size distribution[1]. Annealed upper and lower bounds are presented and
compared to the conventional technique: instead of only estimating the spectral gap, the
whole spectrum of the graph Laplacian is involved. This leads to an improvement of the
upper bound [2]. In the case of regular trees, this is good enough to distinguish the decay
of the expected return probability on finite clusters from that on the incipient infinite
cluster, as it is predicted by the Alexander-Orbach conjecture[3,4]. The proof involves
using the property of cartesian products of finite graphs with cycles of a specific length to
be Hamiltonian[5].
(This project was funded by the Austrian Science Foundation FWF P18703.)
1. W.Kirsch, P.Müller: ‘Spectral properties of the Laplacian on bond-percolative graphs’:
Math. Z. 252, 899-916, 2006
2. F.Sobieczky: ‘Bounds for the return probability of the delayed random walk on finite
percolation clusters in the critical case’, arXiv:0812.0117v2
3. M. T. Barlow, T. Kumagai: ‘Random walk on the incipient infinite cluster on trees’,
Illinois Journ. Math., 50, 33-65, 2006
4. G. Kozma, A. Nachmias: ‘The Alexander-Orbach conjecture holds in high dimensions’, arXiv:0806.1442v1
5. V. Batagelj, T. Pisanski: ‘Hamiltonian cycles in the cartesian product of a tree and
a cycle’, Discr. Math,. 38, 311-312, 1982
Keywords: Interlacing, random walk, critical percolation, return-probability, IDS
Introduction of the Model, Problem-Definition
Simple <−> Delayed Random Walk
Transitive Graph G
(e.g. <Z2, N.N.>)
δ=4
Percolation
Random Walk
on random Co
Unimodularity & spectral Properties
X
t
Expected Return−
Probability: P(o) = E[P [X = X ]]
t
t
0
Presentation of the results
Simple <−> Delayed Random Walk
Transitive Graph G
(e.g. <Z2, N.N.>)
δ=4
Percolation
Unimodularity & spectral Properties
Random Walk
on random Co
X
t
Expected Return−
Probability: P(o)
= E[P [Xt = X 0]]
t
For Bernoulli−Perc. at Criticality:
Expected Return−Probability
−1
−η
Co < c’ t
c t−ξ < P(o)
−
Ε
t
Integrated Density of States
ζ
ρ
b E < N(E) − N(0) < b’ E
with
ξ, ε, ζ, ρ,c, c’, b, b’ > 0 known.
Consequences for the IIC
Incipient Infinite Cluster
c’’
o
E [P[X = o]] <
IIC
t
(c’’>0)
Illustration of methods
Simple <−> Delayed Random Walk
Transitive Graph G
(e.g. <Z2, N.N.>)
δ=4
Percolation
Random Walk
Unimodularity & spectral Properties
on random Co
X
t
Expected Return−
Probability: P(o)
= E[P [Xt = X 0]]
t
For Bernoulli−Perc. at Criticality:
Expected Return−Probability
−1
−η
Co < c’ t
c t−ξ < P(o)
−
Ε
t
Integrated Density of States
ζ
Consequences for the IIC
Incipient Infinite Cluster
c’’
o
E [P[X = o]] <
IIC
t
(c’’>0)
ρ
b E < N(E) − N(0) < b’ E
ξ, ε, ζ, ρ,c, c’, b, b’ > 0 known.
with
Method B: The Spectrum at the edge
Method A: Bulk−Spectrum
^−1^
^
^ ) 1=β >β ...
^)
P = D A { βi } = σ( P
σ( P
1 2
Edge−Removal:
1
Co
P
σ( P )
β < β
N’
PN
j
Edge removal and insertion: S
R = Rank(S) << |Co | + N’ =: N
Interlacing:
βj < β j − R
1
j
=
δ=3
P(o)
P(o)=
P (o)
t
t
2t
Invariant Percolation on transitive Graphs
Transitive Graph G
(e.g. <Z2, N.N.>)
Percolation
Random Walk
on random Co
X
t
Expected Return−
Probability: P(o)
= E[P [Xt = X t ]]
t
G = hV, Ei,
Γ ≤ Aut(G)
o ∈ V (“Root”),
(Ω, F, µ) :
simple, transitive, infinite
f.a. v, w ∈ V t.e. γ ∈ Γ : v = γ(g)
δ = deg(G)
Ω = 2E = {η : E → {0, 1}},
µ : F → [0, 1],
ω ∈ Ω : H(ω) = hV, EH (ω)i,
Γ-invariant
F=
N
e∈E
F
EH (ω) = ω −1 ({1}) ⊂ E
Co(ω) = connected component of o in H(ω)
Ho (ω) := H(ω)|Co(ω) =: hCn(ω), Eo (ω)i
µ-a.s. finite
Nearest Neighbour Random Walks on Ho (ω)
Simple <−> Delayed Random Walk
δ=4
Unimodularity & spectral Properties
Xnω : Ω̃ω → Co(ω) (discrete time n ∈ N)
Choose among two initial distributions:
νo (Atom at o), and ν̄ (UNIF(Co ))
Choose among two transition kernels:
1/degω (k) k ∼ l in Ho(ω),
1. Pkl =
0
otherwise.

1/δ
k ∼ l in Ho(ω),

2. Pbkl = 1 − degω (w)/δ
k=l

0
otherwise.
(1. Simple Random Walk; 2. Delayed Random Walk)
Continuous time random walk
Transitive Graph G
(e.g. <Z2, N.N.>)
Percolation
Random Walk
on random Co
X
t
Expected Return−
Probability: P(o)
= E[P [Xt = X t ]]
t
Wait an exponentially (Parameter=1) distributed time
Perform one random step according to P or Pb
:k
The Sum of exponentially distr. Variables is Poisson-distr.:
∞
P
P e−ttk k
k
Poo
P[#(Jumps) up to t = k]Poo =
Po[Xt = o] =
k!
k=0
k
−t(1−P )
]
Annealed Return-Probability: Eµ Po[Xt = o] = E[ e
o,o
Generator of the semigroup exp(−t(1 − P )) is P − 1.
b=D
b − A,
b Pb = D
b −1 A
b (DLR),
(Note: L = D − A, P = D−1A (SRW), L
b −1A
b = I − δ −1 (A + δI − D) = δ −1(D − A) = δ −1 L)
so: I − Pb = I − D
Unimodularity: The Mass-Transport-Principle
Simple <−> Delayed Random Walk
δ=4
Unimodularity & spectral Properties
MTP: For diagonally invariant functions f : V × V → R :
X
f (v, w) =
v∈V
Then: Eµ [Po [Xt = o]]
X
f (w, v).
v∈V
=
Eµ [P− [Xt = X0]].
Proof:
X
v∈V
Eµ Po [Xt = o]
χ{v∈Co }
|Co |
=
X
v∈V
Eµ Pv [Xt = v]
χ{o∈Cv }
|Cv |
which equals the right-hand-side, since v ∈ Co ⇔ o ∈ Cv .
Note:
Pt(o) = E[ |C1o| Tr exp(−t(1 − Pb))].
=
X
v∈V
Eµ Pv [Xt = v]
χ{v∈Co }
|Co |
,
Annealed Return Probability on finite Clusters
bt be the continuous time delayed ranTheorem 1: Let X
dom walk on the µ-a.s. finite critical percolation cluster of a
unimodular transitive graph. If a, ν > 0, such that P[|Co| ≥
m] ≥ a m−ν , and b > 12 such that Eµ[|Co|2b−1] < ∞, then the
annealed return probability Pt(o) = EµPo[Xt = o] satisfies
−1
e−δ
Eµ [|Co|2b−1]
a 1+ν ≤ Pt(o) − Eµ |Co|
≤ c(δ)
.
⌈t⌉
tb
For G = hZ2, N.N i, and α > 0 s.t. E[|Co|α ] < ∞,
−1
1
e−4 −3/2
≤ cµ t− 2 (1+α).
⌈t⌉
≤ Pt(o) − Eµ |Co|
2
Moreover, if G is the homogeneous tree of degree three, then
ν = 3/4 − ǫ, for any ǫ > 0.
M. T. Barlow, T. Kumagai: ‘Random walk on the incipient infinite cluster on trees’,
Illinois Journ. Math., 50, (2006), pp. 33-65
Integrated Density of states on finite Clusters
Theorem 2: For the integrated density of states E 7→ NN (E)
of the Neumann Laplacian corresponding to ciritical 2D-Bernoulli
bond percolation there is a constant α > 0, such that for
ν :=√21 (1 + α), and for all positive E < 0.0002 and cµ =
(8 + 3π)Eµ[|Co|α ]
e ν 1
E 3/2
1
2 (1+α) .
E
≤
N
(E)
−
N
(0)
≤
c
N
N
µ
1000 (log 1/E)3/2
ν
N(E)
−1
E[|Co| ]
E
Theorem 3: On the infinite incipient cluster
Consequences for the IIC
Incipient Infinite Cluster
c’’
o
E [P[X = o]] <
IIC
t
(some
c’’ > 0)
E[Po[Xt = o]|o ↔ ∂Br (o)] → EIIC [Po[Xt = o]], as r → ∞.
[1] H. Kesten: ‘Subdiffusive behaviour of random walk on a random cluster’, Annales de l’Institut Henri Poincaré , Probabilités et Statistiques 22,(1986), 425-487
[2] G. Kozma, A. Nachmias: ‘The Alexander-Orbach conjecture holds in high dimensions’, arXiv:0806.1442v1 (2008)
With Hro := {ω ∈ Ω | o ↔ ∂Br (o)},
1
o
EICC [P− [Xt = Xo ]|H2r
]
′′
c
≤
and [2] (P[Hro ] ∼ r −1 ), t.e. c′′ > 0, s.t.
E[Po [Xt = o]|]
≤
o
c′′ EICC [P− [Xt = Xo ]|Hr/2
].
Interlacing for the eigenvalues in the middle of σ(Pb)
Method A: Bulk−Spectrum
^−1^
^
^ ) 1=β >β ...
P = D A { βi } = σ( P
1 2
Co
P
N’
PN
Edge removal and insertion: S
R = Rank(S) << |Co | + N’ =: N
Interlacing:
βj < β j − R
Cartesian Products with cycles of length δ
Method B: The Spectrum at the edge
Edge−Removal:
1
βj < β j
1
=
δ=3
P(o)
P(o)=
P (o)
t
t
2t
^)
σ( P
σ( P )