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 Diffusion on fractals:Branching Processes andRandom Fractals – p. 2
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 of 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. Diffusion on fractals:Branching Processes andRandom Fractals – p. 3
Percolation on Z d Fix p ∈ [0 , 1] . For each edge e in Z d , let µ e be independent random variables with P ( µ e = 1) = p , P ( µ e = 0) = 1 − p . The edges such that µ e = 1 are called open. Let η be the set of open edges. The connected components of the graph ( Z d , η ) are called open clusters. There exists p c ∈ (0 , 1) such that, a.s., -if p < p c , all clusters are finite, -if p > p c , then there exists a unique infinite cluster, C ∞ . -if p = p c , no infinite cluster. Diffusion on fractals:Branching Processes andRandom Fractals – p. 4
p = 0 . 5 Diffusion on fractals:Branching Processes andRandom Fractals – p. 5
p = 0 . 5 Diffusion on fractals:Branching Processes andRandom Fractals – p. 5
Random walk on C ∞ The supercritical case p > p c 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 Y t be the continuous time random walk on ( C ∞ ( ω ) , µ ( ω )) . Its transition density is q ω t ( x, y ) µ y ( ω ) = P x ω ( Y t = y ) . There is an invariance principle There are Gaussian heat kernel bounds There is a local CLT Diffusion on fractals:Branching Processes andRandom Fractals – p. 6
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: d s = 4 / 3 . Jarai has some estimates on resistance in d = 2 . Diffusion on fractals:Branching Processes andRandom Fractals – p. 7
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. Diffusion on fractals:Branching Processes andRandom Fractals – p. 8
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(3) F(3) F(3) F(2) F(2) F(2) F(2) The fractal dimension is d f = p log 3 + (1 − p ) log 6 p log 2 + (1 − p ) log 3 . Diffusion on fractals:Branching Processes andRandom Fractals – p. 9
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. Diffusion on fractals:Branching Processes andRandom Fractals – p. 10
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 ξ x on (0 , ∞ ) , 2. a lifetime which is a non-negative random variable L x , 3. a characteristic which is a (possibly random) càdlàg function φ x on R . We make no assumption on the joint distribution of ( ξ x , L x , φ x ) and allow φ x to depend on the progeny of x . Each individual evolves independently. Let ξ ( t ) = ξ ((0 , t ]) , ν ( dt ) = E ξ ( dt ) , ξ γ ( dt ) = e − γt ξ ( dt ) , ν γ ( dt ) = E ξ γ ( dt ) . Diffusion on fractals:Branching Processes andRandom Fractals – p. 11
We assume that the GBP is super-critical in that ν ( ∞ ) > 1 . Then there exists a Malthusian parameter γ ∈ (0 , ∞ ) such that ν γ ( ∞ ) = 1 . � ∞ Let µ = tν γ ( dt ) . 0 The individuals of the population are counted using the characteristic φ through the characteristic counting process Z φ defined by ξ ∅ ( ∞ ) � � Z φ Z φ ( t ) = φ x ( t − σ x ) = φ ∅ ( t ) + i ( t − σ i ) , x ∈ T i =1 where σ x is the birth time of the individual x , T is the ancestral tree and Z φ i are i.i.d. copies of Z φ . Diffusion on fractals:Branching Processes andRandom Fractals – p. 12
Counting with characteristics The population size: φ ( t ) = I 0 ≤ t ≤ L , then Z φ ( 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. Diffusion on fractals:Branching Processes andRandom Fractals – p. 13
Random recursive fractals A random recursive fractal is a random compact subset K of R d . We take a family of IFSs A = { Φ ( i ) = (Φ ( i ) 1 , . . . , Φ ( i ) n i ) , 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 Φ 1 , . . . , Φ N , with contraction ratios R 1 , . . . , R N . The set K is such that N � K = Φ i ( K i ) , P -a.s. , i =1 where K 1 , . . . , K N are i.i.d. copies of K . Diffusion on fractals:Branching Processes andRandom Fractals – p. 14
Random recursive fractals Theorem: Let K be a non-empty random recursive fractal with int ( K i ) ∩ int ( K j ) = ∅ for all i, j . Write ( N, R 1 , . . . , R N ) for the random variable of number of similitudes and their ratios, then a.s. � N � � � � R s dim K = α := inf s : E ≤ 1 . i i =1 Diffusion on fractals:Branching Processes andRandom Fractals – p. 15
Connection with the GBP The general branching process for a random recursive fractal has N x � ξ x = δ − log R x,i . i =1 For the first generation of offspring this means that e − σ i = R i . The offspring x born around time t correspond to compact sets K x of size around e − t . As � N � ∞ � � e − sx ξ ( dx ) = E R s , E i 0 i =1 the Malthusian parameter of the underlying general branching process is equal to the almost sure Hausdorff/Minkowski dimension of the set K . Diffusion on fractals:Branching Processes andRandom Fractals – p. 16
Renewals We observe that Z has a (random) renewal property � Z φ ( t ) = φ i ( t − σ i ) i N φ Z φ � i ( t − σ i ) = φ ∅ ( t ) + i =1 where Z φ i are iid copies of Z φ . Thus the functions z φ ( t ) = e − γt E Z φ ( t ) and u φ ( t ) = e − γt E φ ( t ) , satisfy a classical renewal equation � ∞ z φ ( t ) = u φ ( t ) + z φ ( t − s ) ν γ ( ds ) . 0 Diffusion on fractals:Branching Processes andRandom Fractals – p. 17
Martingales Let F x = σ ( { ( ξ y , L y ) : σ y ≤ σ x } ) , F t = σ ( F x , σ x ≤ t ) and Λ t = { x ∈ T : x = yi for some y ∈ T , i ∈ N , and σ y ≤ t < σ x } . The process M defined by � e − γσ x M t = x ∈ Λ t is a non-negative càdlàg F t -martingale and hence converges to M ∞ a.s. which is non-degenerate under an X log X condition. Diffusion on fractals:Branching Processes andRandom Fractals – p. 18
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