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Diffusion on fractals: Branching Processes and Random Fractals

Ben Hambly Mathematical Insitute University of Oxford

Diffusion on fractals:Branching Processes andRandom Fractals – p. 1

Contents

Some motivation Random self-similar fractals General branching processes Spectral problems for bounded domains and fractals Sharp spectral asymptotics for random strings, the CRT and the CRG Spectral asymptotics and heat kernel estimates for the critical percolation cluster on the diamond hierarchical lattice

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Percolation and Random Media

In the 1970s De Gennes suggested the ‘ant in the labyrinth’ to investigate the transport properties of percolation cluster models for random media. Early toy models for clusters were structures with exact self-similarity which enabled explicit renormalization group calculations such as the Sierpinski gasket. The first mathematical work was Kesten 1986 showing subdiffusivity

• f random walk on the incipient infinite cluster. Recent developments

have focused on the high dimensional/mean field case. Aim: To examine simple models for random fractals which allow calculations to be done.

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Percolation on Zd

Fix p ∈ [0, 1]. For each edge e in Zd, let µe be independent random variables with P(µe = 1) = p, P(µe = 0) = 1 − p. The edges such that µe = 1 are called open. Let η be the set of open edges. The connected components of the graph (Zd, η) are called open clusters. There exists pc ∈ (0, 1) such that, a.s.,

• if p < pc, all clusters are finite,
• if p > pc, then there exists a unique infinite cluster, C∞.
• if p = pc, no infinite cluster.

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p = 0.5

Diffusion on fractals:Branching Processes andRandom Fractals – p. 5

p = 0.5

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Random walk on C∞

The supercritical case p > pc is reasonably well understood. Fix a percolation configuration ω. Let G = C∞(ω), E be the open bonds in C∞(ω). This defines an (infinite, connected) weighted graph. Let Yt be the continuous time random walk on (C∞(ω), µ(ω)). Its transition density is qω

t (x, y)µy(ω) = P x ω(Yt = y).

There is an invariance principle There are Gaussian heat kernel bounds There is a local CLT

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The critical case

In the critical case there is no infinite cluster with probability 1 (at least for d = 2, d ≥ 19). We must define an ‘Incipient infinite cluster’ (IIC). This critical cluster should have fractal structure. Kesten (1986): random walk on the IIC in d = 2 is subdiffusive. Barlow, Kumagai (2007): random walk on the tree (‘d = ∞’) has sub-Gaussian heat kernel estimates. Barlow, Jarai, Kumagai and Slade (2007): sub-Gaussian estimates for random walk on high dimensional spreadout oriented percolation. Kozma and Nachmias (2008): Alexander-Orbach conjecture holds in high dimensions: ds = 4/3. Jarai has some estimates on resistance in d = 2.

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Random self-similar sets

Two possible simple randomizations of the Sierpinski gasket: The LHS is a random recursive fractal, in that each triangle is randomly subdivided into 3 or 6. The RHS is a homogeneous random fractal, in that at each scale we choose randomly to divide all triangles into 3 or 6.

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The homogeneous random tree

The first stages and the tree for a homogeneous random gasket where at each level 2 or 3 is independently chosen with probability p, 1 − p

F(2) F(2) F(2) F(2) F(3) F(3) F(3)

The fractal dimension is df = p log 3 + (1 − p) log 6 p log 2 + (1 − p) log 3.

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The random recursive tree

The first stages of a random recursive gasket where each 2, 3 is independently chosen with probability p, 1 − p within each triangle. The tree of cell addresses is now a Galton-Watson branching process. However we need a more sophisticated model to compute the dimension.

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General (CMJ) branching processes

To tackle a range of examples like this we use a branching process description. An individual x in a general branching process has

• 1. offspring whose birth times are a point process ξx on (0, ∞),
• 2. a lifetime which is a non-negative random variable Lx,
• 3. a characteristic which is a (possibly random) càdlàg function φx on R.

We make no assumption on the joint distribution of (ξx, Lx, φx) and allow φx to depend on the progeny of x. Each individual evolves independently. Let ξ(t) = ξ((0, t]), ν(dt) = Eξ(dt), ξγ(dt) = e−γtξ(dt), νγ(dt) = Eξγ(dt).

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We assume that the GBP is super-critical in that ν(∞) > 1. Then there exists a Malthusian parameter γ ∈ (0, ∞) such that νγ(∞) = 1. Let µ = ∞ tνγ(dt). The individuals of the population are counted using the characteristic φ through the characteristic counting process Zφ defined by Zφ(t) =

• x∈T

φx(t − σx) = φ∅(t) +

ξ∅(∞)

• i=1

Zφ

i (t − σi),

where σx is the birth time of the individual x, T is the ancestral tree and Zφ

i are i.i.d. copies of Zφ.

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Counting with characteristics

The population size: φ(t) = I0≤t≤L, then Zφ(t) corresponds to the number of individuals in the population alive at time t. In the calculation of the Minkowski dimension for a fractal φ(t) = ξ(∞) − ξ(t), then φ(t) corresponds to the number of offspring born after time t to parents born up to time t. Later we will use characteristic functions whose corresponding counting process contains information about the Minkowski content, the spectral counting function or the heat content of the set.

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Random recursive fractals

A random recursive fractal is a random compact subset K of Rd. We take a family of IFSs A = {Φ(i) = (Φ(i)

1 , . . . , Φ(i) ni ), i ∈ I} where I is a possible

uncountable indexing set. We then choose randomly from A according to a probability measure P. This determines a random number N and set of contracting similitudes Φ1, . . . , ΦN, with contraction ratios R1, . . . , RN. The set K is such that K =

N

• i=1

Φi(Ki), P-a.s., where K1, . . . , KN are i.i.d. copies of K.

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Random recursive fractals

Theorem: Let K be a non-empty random recursive fractal with int(Ki) ∩ int(Kj) = ∅ for all i, j. Write (N, R1, . . . , RN) for the random variable of number of similitudes and their ratios, then a.s. dim K = α := inf

• s : E

N

• i=1

Rs

i

• ≤ 1
• .

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Connection with the GBP

The general branching process for a random recursive fractal has ξx =

Nx

• i=1

δ− log Rx,i. For the first generation of offspring this means that e−σi = Ri. The offspring x born around time t correspond to compact sets Kx of size around e−t. As E ∞ e−sxξ(dx) = E N

• i=1

Rs

i

• ,

the Malthusian parameter of the underlying general branching process is equal to the almost sure Hausdorff/Minkowski dimension of the set K.

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Renewals

We observe that Z has a (random) renewal property Zφ(t) =

• i

φi(t − σi) = φ∅(t) +

Nφ

• i=1

Zφ

i (t − σi)

where Zφ

i are iid copies of Zφ.

Thus the functions zφ(t) = e−γtEZφ(t) and uφ(t) = e−γtEφ(t), satisfy a classical renewal equation zφ(t) = uφ(t) + ∞ zφ(t − s)νγ(ds).

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Martingales

Let Fx = σ({(ξy, Ly) : σy ≤ σx}), Ft = σ(Fx, σx ≤ t) and Λt = {x ∈ T : x = yi for some y ∈ T , i ∈ N, and σy ≤ t < σx}. The process M defined by Mt =

• x∈Λt

e−γσx is a non-negative càdlàg Ft-martingale and hence converges to M∞ a.s. which is non-degenerate under an X log X condition.

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A strong law for GBP

Theorem (Nerman) Let (ξx, Lx, φx)x be a general branching process with Malthusian parameter γ, where φ ≥ 0 and φ(t) = 0 for t < 0. Assume that νγ is non-lattice. Assume there exist non-increasing bounded positive integrable càdlàg functions g and h on [0, ∞) such that E

• sup

t≥0

ξγ(∞) − ξγ(t) g(t)

• < ∞

and E

• sup

t≥0

e−γtφ(t) h(t)

• < ∞.

Then, zφ(t) → zφ(∞) = µ−1 ∞ uφ(s)ds, e−γtZφ(t) → zφ(∞)M∞, a.s. as t → ∞, where M∞ is the almost sure limit of the fundamental martingale of the general branching process.

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Spectral asymptotics for self-similar sets

For the Sierpinski gasket (and other nested fractals) we have N(λ) = λdS/2(G(ln λ) + o(1)), as λ → ∞, where G is a periodic function (Fukushima-Shima, Barlow-Kigami). This is due to the symmetry and exact self-similarity of the set. We can construct strictly localized eigenfunctions on this set and use the self-similarity and symmetry to construct other eigenfunctions. Thus there are eigenvalues with very high multiplicity. For self-similar sets with less symmetry but finite ramification (p.c.f fractals), if the logarithms of the scaling ratios are not rationally related, then (Kigami-Lapidus) lim

λ→∞

N(λ) λds/2 = C.

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Spectral asymptotics for random gaskets

For the random recursive Sierpinski gasket, where each 2, 3 is independently chosen with probability p, 1 − p for each triangle lim

λ→∞

N(λ) λds/2 = W, a.s. where ds = 2α/(α + 1) and α satisfies p3( 3

5)α + (1 − p)6( 7 15)α = 1.

For the homogeneous random gasket, where each 2, 3 is independently chosen with probability p, 1 − p for each scale there are constants s.t. 0 < lim sup

λ→∞

N(λ) λds/2φ(λ)c1 lim sup

λ→∞

N(λ) λds/2φ(λ)c2 < ∞, a.s. where ds 2 = p log 3 + (1 − p) log 6 p log 5 + (1 − p) log 90/7, φ(t) = exp(

• log λ log log log λ).

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The continuum random tree

The continuum random tree, initially constructed by Aldous, arises as the scaling limit of uniform random trees on n vertices. a random real tree defined as the contour process of Brownian excursion. A third view is that it is a random recursive self-similar set. It is closely related to mean field limits for critical percolation on graphs, in particular high dimensional critical percolation on Zd and limit models arising in the critical window of the random graph model.

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Self-similar decomposition

Let Z1, Z2 be two µT -random vertices of T . There exists a unique branch-point bT (ρ, Z1, Z2) ∈ T of these three vertices. Let T1, T2 and T3 the components containing ρ, Z1 and Z2. For i = 1, 2, 3, we define a metric dTi and probability measure µTi on Ti by setting dTi := ∆−1/2

i

dT |Ti×Ti, µTi(·) := ∆−1

i µ(· ∩ Ti),

where ∆i := µT (Ti).

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Random recursive fractal

Lemma The collections (Ti, dTi, µTi, ρi, Z1

i , Z2 i ), i = 1, 2, 3, are independent copies

• f (T , dT , µT , ρ, Z1, Z2), and moreover, the entire family of random

variables is independent of (∆i)3

i=1, which has a Dirichlet-( 1 2, 1 2, 1 2)

distribution. The CRT is isomorphic to a deterministic self-similar set with a random metric

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The Dirichlet form

The natural Laplace operator on T is defined via its Dirichlet form. P-a.s. there exists a local regular Dirichlet form (ET , FT ) on L2(T , µ), which is associated with the Laplace operator LT via for f, g ∈ FT ET (f, g) = −(LT f, g). and the metric dT through, for every x = y, dT (x, y)−1 = inf{ET (f, f) : f ∈ FT , f(x) = 0, f(y) = 1}. A Neumann eigenvalue λ with eigenfunction u satisfies ET (f, u) = λ(f, u) for all f ∈ FT . We work with the eigenvalue counting function defined from (ET , FT , µ).

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Scalings

We have the following relationships: For the Dirichlet form ET (f, f) =

3

• i=1

∆−1/2

i

ETi(f ◦ φi, f ◦ φi), where φi is the map from T → Ti. For the measure

• T

f dµT =

3

• i=1

∆i

• Ti

f ◦ φi dµTi.

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Dirichlet-Neumann bracketing

There is a simple relationship between the Dirichlet and Neumann counting functions. For trees we have N D(λ) ≤ N N(λ) ≤ N D(λ) + 2. For the CRT we can compare eigenvalues and use the scalings: For a Neumann eigenvalue λ of T we have

3

• i=1

∆−1/2

i

ETi(f ◦ φi, u ◦ φi) = ET (f, u) = λ(f, u) = λ

3

• i=1

∆i(f ◦ φi, u ◦ φi). Thus λ∆3/2

i

is a Neumann eigenvalue of Ti. Hence

3

• i=1

N D

i (λ∆3/2 i

) ≤ N D(λ) ≤ N N(λ) ≤

3

• i=1

N N

i (λ∆3/2 i

).

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The renewal equation

The key tool for studying the behaviour is a renewal equation. Let X(t) = N D(et) and η(t) = N D(et) − 3

i=1 N D i (et∆3/2 i

). Then X(t) = η(t) +

3

• i=1

Xi(t + 3 2 log ∆i). If m(t) = e−2t/3EX(t), u(t) = e−2t/3Eη(t), then m(t) = u(t) + ∞ e−sm(t − s)ds.

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Theorem [Croydon + H (2008)] There exists a deterministic constant C0 = m(∞) = ∞

−∞

u(t)dt ∈ (0, ∞) such that λ−2/3EN(λ) → C0, as λ → ∞. λ−2/3N(λ) → C0, as λ → ∞, P-a.s. This second result is proved in a similar manner to Nerman’s proof of the almost sure convergence of the general branching process.

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The annealed heat kernel

We have been working directly with the eigenvalue counting function. We can use the trace theorem and Tauberian theorems to obtain results on the partition function. Firstly from the trace theorem and the property that the continuum random tree is invariant under random rerooting Ept(ρ, ρ) = E

• T

pt(x, x)µT (dx) = E ∞ e−stN N(ds).

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Thus an application of a Tauberian Theorem gives Corollary Let Γ be the standard gamma function, then t2/3Ept(ρ, ρ) → C0Γ(5/3) as t → 0, Note that Croydon obtained quenched and annealed heat kernel bounds for the CRT: C1 ≤ t2/3Ept(ρ, ρ) ≤ C2, 0 < t < 1. and P-a.s. C3| log t|a′ ≤ t2/3 sup

x∈T

pt(x, x) ≤ C4| log t|a, 0 < t < 1.

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CRT results

Theorem Suppose (NT (λ))λ∈R is the eigenvalue counting function for the natural Laplacian on the continuum random tree. As λ → ∞: ENT (λ) = C0λ2/3 + O(1).

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CRT results

Theorem Suppose (NT (λ))λ∈R is the eigenvalue counting function for the natural Laplacian on the continuum random tree. As λ → ∞: ENT (λ) = C0λ2/3 + O(1). P-a.s., for ǫ > 0, NT (λ) = C0λ2/3 + o(λ1/3+ǫ).

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CRT results

Theorem Suppose (NT (λ))λ∈R is the eigenvalue counting function for the natural Laplacian on the continuum random tree. As λ → ∞: ENT (λ) = C0λ2/3 + O(1). P-a.s., for ǫ > 0, NT (λ) = C0λ2/3 + o(λ1/3+ǫ). NT (λ) − C0λ2/3 λ1/3 → N(0, y(∞)), in distribution.

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CRT results

Theorem Suppose (NT (λ))λ∈R is the eigenvalue counting function for the natural Laplacian on the continuum random tree. As λ → ∞: ENT (λ) = C0λ2/3 + O(1). P-a.s., for ǫ > 0, NT (λ) = C0λ2/3 + o(λ1/3+ǫ). NT (λ) − C0λ2/3 λ1/3 → N(0, y(∞)), in distribution. Health warning... we have not yet proved y(∞) > 0!

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Percolation

For percolation clusters in high dimensions the Alexander-Orbach conjecture has been proved by Kozma and Nachmias. This shows that for random walk on the incipient infinite cluster at criticality we have ds = 4/3. This is established for the on-diagonal decay of the heat kernel on the graph. This scaling is observed in other mean field models including the critical random graph. The question of the spectral asymptotics for the CRG will be determined by the spectral asymptotics for random self-similar trees.

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Scaling limit of the CRG

Let G(N, p) be the Erdös-Renyi random graph. The critical window is p = N −1 + νN −4/3 for a fixed ν ∈ (−∞, ∞). Addario-Berry, Broutin and Goldschmidt construct the scaling limit: Conditioned on the number of connections J = j we have (for j ≥ 2) that M is constructed by taking a random 3 regular graph on 2(j − 1) vertices generate (α1, . . . , α3(j−1)) according to a Dirichlet ( 1

2, . . . , 1 2)

distribution. construct 3(j − 1) size αj CRTs with root plus a randomly chosen vertex. replace the edges in the graph with the trees linked at the roots and randomly chosen vertices.

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Dirichlet-Neumann bracketing allows us to compare eigenvalues of M, T . Theorem Suppose (NM(λ))λ∈R is the eigenvalue counting function for the natural Laplacian on the scaling limit of the giant component of the critical random graph M, and Z1 is the mass of M with respect to its canonical measure µM. Then, as λ → ∞: ENM(λ) = C0EZ1λ2/3 + O(1). λ−2/3NM(λ) → C0Z1. P − a.s.

NM(λ)−aZ1λ2/3 Z1/2

1

λ1/3

→ N(0, y(∞)) in distribution.

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The diamond hierarchical lattice

A recursively constructed graph.

• 1. D0 = (V0, E0), where V0 consists of two vertices and E0 an edge

between them.

• 2. Dn+1 = (Vn+1, En+1) with each edge in En replaced by a diamond:

two sets of two edges in series in parallel.

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Some previous work on this lattice Berker-Ostlund, Tremblay and Southern, Derrida et al early 80s. Derrida, Hakim and Vanninenius 92, Giacomin, Lacoin, Tonninelli 10, Lacoin 10 for pinning/wetting Cook and Derrida 89, Lacoin and Moreno 08 for polymer models Some simple remarks on the geometry: En has 4n edges and has scaling properties of Z2 The local geometry is quite different to that of Z2 Each x ∈ V0 is in 2n edges in Dn, Each x ∈ Vn\Vn−1 is in 2 edges in Dn.

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Scaling limit

The scaling limit for the diamond lattice is a self-similar set. Let (K, d) be a compact metric space with two points labelled 0,1. Let {ψi : i = 1, 2, 3, 4} be a set of 1/2-contractions ψi : K → K, with contraction factor 1/2 with respect to the metric d, and the following properties: ψ1(0) = ψ2(0) = 0, ψ3(1) = ψ4(1) = 1, ψ1(1) = ψ4(0), ψ2(1) = ψ3(0) ψi(int (K)) ∩ ψj(int (K)) = ∅ for all i = j, where int K = K\{0, 1}. This defines the scaling limit of the diamond hierarchical lattice K as a self-similar set K =

4

• i=1

ψi(K). The set is not finitely ramified. Its Hausdorff dimension is 2.

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The diffusion on the scaling limit

Let µ be the Hausdorff measure on K so µ(ψi(K)) = 4−1. It is not volume doubling. Theorem 1 There is a local regular Dirichlet form (E, F) on L2(K, µ) with the following self-similarity, E(f, g) =

4

• i=1

E(f ◦ ψi, g ◦ ψi), ∀f, g ∈ F. The corresponding non-negative self-adjoint operator HN on L2(K, µ) has compact resolvent. There is a corresponding continuous, strong Markov process X on K

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Properties

Theorem 2 Let N(x) be the Neumann eigenvalue counting function, then 0 < lim inf

x→∞

N(x) x < lim sup

x→∞

N(x) x < ∞.

(1)

Recall that for a bounded domain in R2, N(x)/x converges. Theorem 3 1. There exists a jointly continuous heat kernel pt(x, y), for all t ∈ (0, 1), x, y ∈ K.

• 2. For µ-a.a x ∈ K

c1t−1| log t|−2−ǫ ≤ pt(x, x) ≤ c2t−1 The global lower bound is ct−1/2.

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Percolation on Dn

Fix p ∈ [0, 1]. For e ∈ En, let µe be independent random variables with P(µe = 1) = p, P(µe = 0) = 1 − p. Let Ep

n be the open edges in Dn.

We say percolation occurs if there is a connected component of Ep

n

containing 0 and 1 as n gets large. Let f be the map on the percolation probability obtained by considering whether a single diamond is open f(p) = 1 − (1 − p2)2 = 2p2 − p4.

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f has 3 fixed points in [0, 1]. 0 and 1 are attracting pc = ( √ 5 − 1)/2 is repulsive. Lemma 1 If the lattice Dn is subject to Bernoulli bond percolation with p = pc, then there is percolation in the sense that the end points of the lattice, 0 and 1, are connected with probability pc. As n → ∞ If p > pc, then P(0 and 1 are connected in Dn) → 1. If p < pc, then P(0 and 1 are connected in Dn) → 0. Thus we can let n → ∞ when p = pc to obtain an ‘infinite lattice’ under critical percolation Dpc

∞. Easier to think about the scaling limit.

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‘Random walk on Dpc

∞’

H and Kumagai (2010): We can define the scaling limit of the critical percolation cluster on D∞ as a random recursive fractal and analyse the properties of the diffusion on this scaling limit. We have an explicit formula for the spectral exponent in that for P-a.e. ω for µω − a.e.x ∈ C we have lim

λ→∞

log Nω(λ) log λ = lim

t→0

log qω

t (x, x)

− log t = θ θ + 1 where θ = 5.2654.... Thus ds = 1.6807... Sharper results mimic those on the diamond hierarchical lattice itself. The analysis can be extended to the random cluster model on the diamond lattice.

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A tree description

We build a branching tree model of (Dpc

n )∞ n=0. For any graph Dn we label

each edge by either c for connected or d for disconnected. To produce Dn+1 we use the following reproduction rule:

• 1. If we have a c, the replacement graph is one of the 7 possible

connected graph structures.

• 2. If we have a d, the replacement graph for that non-edge is chosen

from the 9 possible disconnected configurations. Thus we view our sequence of percolation configurations (Gn) as starting from the initial edge G0, that is D0 labelled with a c, and then each graph Gn is the subgraph of the labelled graph Dn where we only keep the edges with labels c.

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The configurations

c d

For Dp

1 we have 7 combinations of edges which give a connection across

D0 and 9 which give a disconnection.

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Let I = {1, 2, 3, 4} label each edge of D1. Let I0 = ∅, Tn = ∪n

i=0Ii and T = ∪nTn.

Let U = {c, d} denote the possible types for each vertex in the tree. A c at vertex i ∈ Tn means the corresponding edge in Dn is present after percolation and a d at the vertex means the corresponding edge is absent. We can construct a probability space (ΩT , P) with ΩT = U T , and P the probability measure for a multitype branching process with types in U. There are always 4 offspring with types given by the set of configurations. Lemma 2 At p = pc, the sequence (Gn)0≤n≤N has the same distribution as that of the decimated sequence of random cluster graphs (Dpc

n,N)0≤n≤N

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c

level 0

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c c c c d

level 1

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c c c d cccc cc d d c c c c d d c d c

level 2

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level 3

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The critical cluster

The critical cluster is the connected component of Dpc

∞ containing 0

and 1. The existence of the critical cluster has positive probability, thus we can condition on its existence and work on a subset Ωc ⊂ Ω of our probability space. The construction of (Dpc

n,N)0≤n≤N can be extended to describe the infinite

cluster at criticality. This is a sequence of subgraphs of (Dpc

n,N)0≤n≤N

which we label (Cn)0≤n≤N where we remove all the edges of the graph Dpc

N that are not connected to the vertices 0 and 1 and then apply the

percolation construction.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 48

The infinite cluster at level 3

Diffusion on fractals:Branching Processes andRandom Fractals – p. 49

Constructing the critical cluster

The disconnected edge case is now split up into two types. d(1) for those disconnected edges which have one end connected to C∞ and d(2) for those disconnected edges which have two ends connected to C∞. We now have the following replacement rules:

• 1. If c use one of the 7 possible connected graph structures.
• 2. If d(1), then use one of the 4 possible disconnected configurations

which only have one vertex in the infinite cluster.

• 3. If d(2), then use one of the 9 possible replacement disconnected

graphs available.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 50

The probability distribution for the evolution of the types is given by c →        (2c, 2d(1), 0d(2)) 2 configs p2(1 − p)2q2/Z1 (3c, 0d(1), 1d(2)) 4 configs p3(1 − p)/Z1 (4c, 0d(1), 0d(2)) 1 config p4/Z1 d(1) →        (0c, 2d(1), 0d(2)) 1 config (1 − p)2/Z2 (1c, 2d(1), 0d(2)) 2 configs p(1 − p)2/Z2 (2c, 2d(1), 0d(2)) 1 config p2(1 − p)2/Z2 d(2) →        (0c, 4d(1), 0d(2)) 1 config (1 − p)4/Z3 (1c, 2d(1), 1d(2)) 4 configs p(1 − p)3/Z3 (2c, 0d(1), 2d(2)) 4 configs p2(1 − p)2/Z3

Diffusion on fractals:Branching Processes andRandom Fractals – p. 51

Lemma 3 The sequence of graphs (Gn)0≤n≤N has the same distribution as (Cn)0≤n≤N, the sequence of graphs which grow to be the infinite cluster in the random cluster model on the diamond hierarchical lattice conditional upon connecting the vertices 0 and 1.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 52

The critical cluster

The scaling limit for the critical random cluster itself will be a random recursive graph directed fractal. As for the diamond hierarchical lattice we define the limit as a self sufficient metric space and we take the same contraction maps as for the diamond hierarchical lattice. We write ω ∈ Ω as ω = {ui : i ∈ T} for a labelled tree. For ω ∈ Ω the cluster C(ω), often denoted C(u∅) to indicate the type u∅

• f the set, will be a random self-similar set

C(u∅) =

4

• i=1

ψi(C(ui)) =

4

• i=1

ψi(C(σiω)). where σi is the shift on the tree down the branch labelled i.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 53

Diffusion on the critical cluster

The key is to understand the electrical resistance. To ensure the effective resistance across the essentially two different configurations is one we take edge weights ρui =    1 ui = c, uij = c, j = 1, 2, 3, 4 2

• therwise

(2)

1 1 1 1 2 2 2

The electrical resistance can be used as a metric and we can describe the cluster’s properties in the resistance metric by a multitype branching random walk.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 54

The Dirichlet form

We put resistances on each cell to ensure that the global resistance remains at 1. Thus, for ω = {ui}i∈T , we set E(ω)

n (f, g) = 4

• i=1

E(σiω)

n−1 (f ◦ ψi, g ◦ ψi)ρu∅,

where E0(f, g) is the form for the single edge labelled c. These are compatible resistance networks, hence let E(ω)(f, f) = lim

n E(ω) n (f, f), ∀f ∈ F(ω) = {f : sup n E(ω)(f, f) < ∞}.

and define the resistance metric as Rω(x, y) = [inf{Eω(f, f) : f(x) = 0, f(y) = 1}]−1.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 55

The Dirichlet Form

Theorem 4 1. There exists a Dirichlet form (E(ω), Fω) on L2(C(ω), µω) for all ω ∈ Ω.

• 2. It satisfies the self-similarity condition

E(ω)(f, g) =

4

• i=1

E(σiω)(f ◦ ψi, g ◦ ψi)ρu∅ ∀f, g ∈ Fω.

• 3. The form (E(ω), Fω) is a resistance form P-almost surely.

The Hausdorff dimension in the resistance metric dr

f = θ, the Malthusian

parameter of our branching process. The spectral dimension ds = 2dr

f/(dr f + 1) < 2.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 56

Spectral Properties

As before the Dirichlet and Neumann Laplacians have compact resolvent and we have an eigenvalue counting function N(λ) for either. Using multidimensional renewal theory we have: Theorem 5 There is oscillation for the high frequency asymptotics of either Dirichlet or Neumann eigenvalues in that for each λ ∈ [1, 2(1+θ)), we have lim

n→∞

• N u∅(λ2(1+θ)n)

(λ2(1+θ)n)θ/(θ+1) − mu∅

∞(log λ)W

• = 0, a.s.

where mu∅

∞(t) is the limit of the renewal equation for the mean.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 57

The heat kernel

Let θǫ = 2(2θ + 3)(θ + 2) + ǫ and qω

t (x, y) the heat kernel.

Theorem 6 There is a constant c1(ω) > 0 and for ǫ > 0 constants c2(ω), c3(ω) > 0 such that for P-a.e. ω for µω − a.e.x ∈ C and for t < 1, c2| log t|−θǫt−θ/(θ+1) ≤ qω

t (x, x)

and qω

t (x, x) ≤ c1t−θ/(θ+1)| log | log t||(θ−1)/(θ+1).

We can also get more precise results at the point 0. Theorem 7 For P-a.e. ω and for t < 1 there are constants c1(ω), c2(ω) and an explicit ν such that c1t−(θ−ν)/(θ−ν+1) ≤ qω

t (0, 0) ≤ c2t−(θ−ν)/(θ−ν+1).

Diffusion on fractals:Branching Processes andRandom Fractals – p. 58

Higher order terms

We try to examine the higher order terms in the spectral asymptotics.

• 1. The case of the continuum random tree is difficult due to the difficulty

in proving the non-triviality of the variance.

• 2. The random recursive Sierpinski gaskets have a random limit and this

makes the application of a suitable central limit theorem more challenging.

• 3. We look at a simpler model - random fractal strings.

A fractal string is a union of intervals with a boundary that is typically a Cantor set. The problem of the spectral asymptotics is more straightforward as we understand very well the eigenvalues for the interval.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 59

The Brownian string

A natural random fractal string can be generated by Brownian motion. Take Brownian motion started from 0 in R run for unit time. The path can be viewed as a sequence of excursions away from 0. The zero set is a Cantor set (perfect and nowhere dense) and so divides the time axis into a countable number of intervals. Thus we have a decomposition of the unit interval - a fractal string. For the Dirichlet counting function N(λ) = 1 π λ1/2 − Lζ(1/2)λ1/4 + o(λ1/8+ǫ). where L is the local time at 0 of the Brownian motion and ζ is the Riemann zeta function (H-Lapidus). We look at a family of examples

Diffusion on fractals:Branching Processes andRandom Fractals – p. 60

Dirichlet strings

Let T1, T2 be chosen from a Beta(α, α) distribution. Let γ ∈ (0, 1). Now divide [0, 1] into three pieces ψ1([0, 1]) = [0, T 1/γ

1

], ψ2([0, 1]) = [T γ

2 , 1] and

S1, the middle open interval, the first piece of string of length 1 − T 1/γ

1

− T 1/γ

2

. By choosing ψ1, ψ2 independently according to the distribution we generate a random cantor set K defined by K = ∪iψi(K). This is the boundary of the string S and has dimension γ a.s.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 61

The counting function

For the boundary term in the asymptotics, using the GBP Theorem: For the fractal string S, for all α ∈ N, γ ∈ (0, 1) we have λ−γ/2 π−1λ1/2 − N(λ)

• → C, a.s.,

as λ → ∞, for some positive constant C.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 62

The counting function

For the boundary term in the asymptotics, using the GBP Theorem: For the fractal string S, for all α ∈ N, γ ∈ (0, 1) we have λ−γ/2 π−1λ1/2 − N(λ)

• → C, a.s.,

as λ → ∞, for some positive constant C. For the second term using a Central Limit Theorem for the GBP (Charmoy, Croydon, H 2015) Theorem: For the string S, we have, provided α ≤ 59 λγ/4{λ−γ/2 1 π λ1/2 − N(λ)

• − C} → N(0, σ2), in distribution,

as λ → ∞, for some positive constant σ.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 62

Theorem: (Charmoy, Croydon, H 2015) There exists an ˜ α ≥ 80 and a γ ∈ (0, 1) such that: if 59 < α ≤ ˜ α, then there exists a constant c1(γ, α) > 0 such that ENγ,α(λ) = 1 π λ1/2 − C(γ, α)λγ/2 + c1(γ, α)λγη(α)/2 + o(λη(α)), where η(α) = max{ℜ(θ0) ∈ (−∞, 1) : P(θ0) = 0}, P(θ) :=

α−1

• i=0

(α + θ + i) − (2α)! α! and, for this range of α we have 1/2 < η(α) < 1. In particular λγ/4

• λ−γ/2

1 π λ1/2 − Nγ,α(λ)

• − C(γ, α)
• does not converge in distribution as λ → ∞.

Diffusion on fractals:Branching Processes andRandom Fractals – p. 63