Random walks on barycentric subdivisions andStrichartz hexacarpet

M. Begue1, *D. J. Kelleher2, A. Nelson2, H. Panzo2,R. Pellico2, A. Teplyaev2

1Department of MathematicsUniversity of Maryland

2Department of MathematicsUniversity of Connecticut

Fractals 4 — AMS Eastern Sectional Meeting — Fall 2011

D. J. Kelleher Hexacarpet

Acknowledgments

M. Begue, G. Khan, DJK, D. Raisingh, UConn FractalsREU 2011.

Research supported in part by NSF grant DMS-0505622.

D. J. Kelleher Hexacarpet

Barycentric subdivision

We start off with a simplicial complex,

Take each 1-simplex (line segment) and add a vertex at themidpoint.

Take each 2-simplex (triangle) add a vertex at thebarycenter (average of corners), ad an edge connecting thebarycenter to each vertex adjacent to the original2-simplex.

. . . proceed inductively.

D. J. Kelleher Hexacarpet

Start off with a simplex

D. J. Kelleher Hexacarpet

Add midpoints to 1-simplexes

D. J. Kelleher Hexacarpet

Add edges to 2-simpleces

D. J. Kelleher Hexacarpet

...proceed inductively

D. J. Kelleher Hexacarpet

We approximate the hexacarpet with a graph that has asvertices the faces of the nth barycentric subdivision, which areconnect if the faces share and edge. (the second level is picturedabove)

D. J. Kelleher Hexacarpet

This is what the 4th level approximation looks like.

D. J. Kelleher Hexacarpet

To get a fractal out of this, we looks at the symbolic dynamicsystemLet X = {0, 1, . . . , 5} where x is any element in X∗ andv ∈ {0, 5}ω. Suppose i is odd and j = i+ 1 mod 6. Then,

xi3v ∼ xj3v and xi4v ∼ xj4v. (1)

If i is even (j is still i+ 1 mod 6), then

xi1v ∼ xj1v and xi2v ∼ xj2v. (2)

This produces a fractal with Cantor boundaries and hexagonalsymmetries. We call it the hexacarpet.

D. J. Kelleher Hexacarpet

(a) (ϕ2, ϕ3) (b) (ϕ2, ϕ4) (c) (ϕ2, ϕ5)

(d) (ϕ2, ϕ6) (e) (ϕ3, ϕ4) (f) (ϕ3, ϕ5)

(g) (ϕ3, ϕ6) (h) (ϕ4, ϕ5) (i) (ϕ4, ϕ6)

Figure 6.1. Two-dimensional eigenfunction coordinateseigfpics

12

D. J. Kelleher Hexacarpet

(a) (ϕ2, ϕ3, ϕ4) (b) (ϕ2, ϕ3, ϕ5) (c) (ϕ2, ϕ3, ϕ6)

(d) (ϕ2, ϕ3, ϕ7) (e) (ϕ2, ϕ4, ϕ5) (f) (ϕ2, ϕ4, ϕ6)

(g) (ϕ2, ϕ5, ϕ6) (h) (ϕ3, ϕ5, ϕ6) (i) (ϕ4, ϕ5, ϕ6)

Figure 6.2. Three-dimensional eigenfunction coordinates3Deigfpics

13

D. J. Kelleher Hexacarpet

Level n

c 1 2 3 4 5 6 7

1

2 1.2801 1.3086 1.3085 1.3069 1.3067 1.3065 1.3064

3 1.2801 1.3086 1.3079 1.3075 1.3066 1.3065 1.3064

4 1.1761 1.3011 1.3105 1.3064 1.3068 1.3065 1.3065

5 1.1761 1.3011 1.3089 1.3074 1.3073 1.3065 1.3065

6 1.0146 1.2732 1.3098 1.3015 1.3067 1.3065 1.3064

7 1.2801 1.3114 1.3055 1.3071 1.3066 1.3065

8 1.2801 1.3079 1.3086 1.3075 1.3067 1.3065

9 1.2542 1.3191 1.2929 1.3056 1.3065 1.3065

10 1.2542 1.3017 1.3089 1.3069 1.3066 1.3065

11 1.2461 1.3051 1.3063 1.3048 1.3065 1.3065

12 1.2461 1.3019 1.3075 1.3068 1.3066 1.3065

13 1.1969 1.6014 1.0590 1.3068 1.3066 1.3065

14 1.1969 1.2972 1.3063 1.3078 1.3066 1.3065

15 1.2026 1.3059 1.3020 1.3060 1.3066 1.3065

Table: Hexacarpet estimates for resistance coefficient c given by16

λnj

λn+1j.

D. J. Kelleher Hexacarpet

Higher dimensions

We can perform a construction analogous to the hexacarpet onthe 3-simplex. This time, our approximating graphs havetetrahedra as verteces, connected if they share a face.

0.20.4

0.60.8

1

−0.4

−0.2

0

0.2

0.4

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

−0.4

−0.2

0

0.2

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure: First and second level graph approximations to the3-simplectic sponge.

D. J. Kelleher Hexacarpet

Eigenfunction Video - Level 4

D. J. Kelleher Hexacarpet

eigf234movie.mp4Media File (video/mp4)

Eigenfunction Pictures - Level 4

D. J. Kelleher Hexacarpet

Numerics

Original Shape Triangle Tetrahedron

Subdivisions (N) 6 24

Hausdorff dimensionlog(6)

log(2)≈ 2.58 log(24)

log(2)≈ 4.585

Resistance scaling (ρ) 1.304 2.035

Spectral dimension 1.74 2.45

D. J. Kelleher Hexacarpet

Prospects

For each of the barycentric fractals

1 there exists a unique self-similar local regular conservativeDirichlet form E with resistance scaling factor ρ and theLaplacian scaling factor τ = 6ρ.

2 the simple random walks on the repeated barycentricsubdivisions of a triangle, with the time renormalized byτn, converge to the diffusion process, which is thecontinuous symmetric strong Markov processcorresponding to the Dirichlet form E .

3 the spectral zeta function has a meromorphic continuationto C.

D. J. Kelleher Hexacarpet

Prospects

For each of the barycentric fractals

1 there exists a unique self-similar local regular conservativeDirichlet form E with resistance scaling factor ρ and theLaplacian scaling factor τ = 6ρ.

2 the simple random walks on the repeated barycentricsubdivisions of a triangle, with the time renormalized byτn, converge to the diffusion process, which is thecontinuous symmetric strong Markov processcorresponding to the Dirichlet form E .

3 the spectral zeta function has a meromorphic continuationto C.

D. J. Kelleher Hexacarpet

Prospects

For each of the barycentric fractals

1 there exists a unique self-similar local regular conservativeDirichlet form E with resistance scaling factor ρ and theLaplacian scaling factor τ = 6ρ.

2 the simple random walks on the repeated barycentricsubdivisions of a triangle, with the time renormalized byτn, converge to the diffusion process, which is thecontinuous symmetric strong Markov processcorresponding to the Dirichlet form E .

3 the spectral zeta function has a meromorphic continuationto C.

D. J. Kelleher Hexacarpet

Heat kernel estimates

Barlow and Bass used a probabilaistical interperation,comparing transition density of a random walk to the diffusionof heat, to prove, in some cases, that the heat kernel can beapproximated for time 0 < t ≤ 1

p(t, x, y) ≈ tds/2exp

(−cR(x, y)

dwdw−1

t1

dw−1

)

Where R is the effective resistance metric, and

dh = Hausdorff dimension

ds = Spectral dimension

dw =2dhds

= Walk “dimension” ≤ 2

D. J. Kelleher Hexacarpet

Heat kernel

p(t, x, y) ≈ tds/2exp

(−cR(x, y)

dwdw−1

t1

dw−1

)

Where R is the effective resistance metric, and

dh = Hausdorff dimension

ds = Spectral dimension

dw =2dhds

= Walk “dimension” ≤ 2

However, the Hexacarpet and higher dimensional analogues donot quite fit this frame. There may be some logarithmiccorrections...

D. J. Kelleher Hexacarpet

If we look at the graph approximation, the inner “circle” of thegraph has length 3 · 2n at the nth level. The perimeter of thisgraph will have 3n2n length.How do we resize the graph?

D. J. Kelleher Hexacarpet

Another fractal

What if we consider the triangle, and then perform barycentricsubdivision. This time, we keep the simplicial complex, butrenormalize each triangle to be equilateral.

In the case of the first subdivision, we get a hexagon.

D. J. Kelleher Hexacarpet

If we renormalize so that each line segment has length 2−n onthe nth level, the induced geodesic metric converges to a metricon the triangle.

1 The Hausdorff (and self-similarity) dimension of this objectis log2(6) ≈ 2.58.

2 Topologically we still have the same object! i.e. the limit ishomeomorphic to the Euclidean triangle.

3 The distance from a vertex of an nth level simplex to theany point on the opposite side is 2−n.

4 There are infinitely many geodesics between points.

The same construction can be done when starting with anyn-simplex.

D. J. Kelleher Hexacarpet