Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions

We prove Barlow--Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant $\rho$ which is theoretically estimated to be in the interval $5/4\leqslant\rho\leqslant3/2$, with a numerical estimate $\rho\approx1.306$. This corresponds to the theoretical estimate of spectral dimension $d_S$ between 1.63 and 1.77, with a numerical estimate $d_S\approx1.74$. On the other hand, if the random walk jumps between the corners of triangles in the subdivision, then the resistance scales as a power of a constant $\rho^T=1/\rho$, which is theoretically estimated to be in the interval $2/3\leqslant\rho^T\leqslant4/5$. This corresponds to the spectral dimension between 2.28 and 2.38. The difference between $\rho$ and $\rho^T$ implies that the the limiting behavior of random walks on the repeated barycentric subdivisions is more delicate than on the generalized Sierpinski Carpets, and suggests interesting possibilities for further research, including possible non-uniqueness of self-similar Dirichlet forms.


I
The repeated barycentric subdivision of a simplex is a classical and fundamental notion that appears frequently in algebraic topology, see [18, and references therein]. Recently it was considered from probabilistic point of view in [9,12,10,11,36]. Understanding how resistance scales on finite approximating graphs is the first step to developing analysis on fractals and fractal-like structures, such as self-similar graphs and groups, see [5,15,22,34,33, and references therein]. For finitely ramified post-critically finite fractals, including nested fractals, the resistance scales by the same factor between any two levels of approximating graphs. Kigami, Linsdtrøm et al [24,25,30] used this fact to prove the existence of a Dirichlet form on the limiting fractal structures. In the infinitely ramified case, resistance estimates are more difficult to obtain, but are just as important F . Barycentric subdivision of a 2-simplex, the graphs G T 0 , G T 1 and G T 2 . to understanding diffusions on fractals. Barlow and Bass [1,2,3,4] proved such estimates for the Sierpinski carpet and its generalizations. These techniques were extended to understanding resistance estimates between more complicated regions of the Sierpinski carpet, see [32]. The paper [26] provides another technique for proving the existence of Dirichlet forms on non-finitely ramified self-similar fractals, which estimates the parameter ρ by studying the Poincaré inequalities on the the approximating graphs of the fractals.
Our work further develops existing techniques to obtain resistance scaling estimates for the 1skeleton of n-times iterated barycentric subdivisions of a triangle which we will denote G T n , and its weak dual the hexacarpet (introduced in [6]), which we will denote G n . Note that in the current work, hexacarpet graphs will mostly be used to refer to G H n , which is a modification of G n by adding a set of "boundary" verteces.
If R T n and R n is the resistance between the appropriate boundaries in G T n and G H n respectively (see Figures 1,2,3,4), then we prove that the resistance R T n and R n scale by constants ρ T and ρ respectively, and obtain estimates on these constants. Our main result is the following theorem. exist. Further, ρ T = 1/ρ and satisfies the estimates 2/3 ≤ ρ T ≤ 4/5 and 5/4 ≤ ρ ≤ 3/2. These estimates agree with the numerical experiments from [6], which suggest that there exists a limiting Dirichlet form on these fractals and estimates ρ ≈ 1.306, and hence ρ T ≈ 0.7655.
Dirichlet forms have many applications in geometry, analysis and probability, see [7,8,14]. In general terms, a Dirichlet form on a fractal is a bilinear form which is analogous to the classic Dirichlet energy on R d given by A Dirichlet form is equivalent, in a certain sense, to a symmetric Markov process on the base space. The potential theoretic properties of the Dirichlet form have implications for this stochastic process. In particular, the resistance between two boundary sets is related to to the crossing times of these sets. In our case, on the 1-skeleton G T n , the Markov process jumps between corners of the triangles in the subdivision. Our theoretical estimates correspond to a process with limiting spectral dimension between 2.28 and 2.38. The Markov process on the hexacarpet graphs (which . . The labeling B 2 (T 0 ) (left) and the graph isomorphism to G 2 ow that S and T are adjacent, so without loss of generality, let their common side BCS we see that S and T must have either [v 0 , b 01 ] or [v 1 , b 01 ] as their common side. oss of generality, assume that the common side of S and T is [v 0 , b 01 ]. Subdividing hat the common side of s and t must be either is a side of s and t we see that s and t with respect to their respective grandparent, S and T . Thus, the last letter in the h s and t must be 0 by construction. On the other hand, if [m, b 01 ] is the common e see that s and t are both side adjacent to their siblings who received a label of 0. tter in the addresses of s and t must be 5 by the standard labeling construction. n a position to define the desired isomorphism of graphs. et B n (T 0 ) be the vertex set of a graph where two level n offspring of T 0 are cone if and only if they are adjacent as level n offspring of T 0 . This graph is isomorphic omorphism given by Proposition 4.3. dy have that Φ n : B n (T 0 ) → X n is a bijection between the vertex sets of the two ins to show that Φ n preserves the adjacency structure of the two graphs. Most of as done in Lemma 4.6. We must now show in particular that if s, t ∈ B n (T 0 ) are es, then Φ n (s) and Φ n (t) satisfy an edge relation in F k for some 1 ≤ k ≤ n. s n−1 ⊂ · · · ⊂ s 1 ⊂ s 0 = T 0 and t = t n ⊂ t n−1 ⊂ · · · ⊂ t 1 ⊂ t 0 = T 0 be for s and t, respectively, and let m be maximal with respect to s m = t m . Assume (t m ) = x 1 x 2 · · · x m ∈ X m . First suppose that m = n − 1 and Φ n−1 (s n−1 ) = 1 x 2 · · · x n−1 ∈ X n−1 . Then, by part (1) of Lemma 4.6, without loss of generality = x 1 x 2 · · · x n−1 i and Φ n (t) = x 1 x 2 · · · x n−1 j where j = i + 1 mod 6. Thus, ∈ F 1 , as desired. that m = n − 2 and Φ n−2 (s n−2 ) = Φ n−2 (s n−2 ) = x ∈ X n−2 . We apply part .6 to s n−1 and t n−1 to obtain Φ n−1 (s n−1 ) = xi and Φ n−1 (t n−1 ) = xj, where 6 and i is either even or odd. We apply part (2) of Lemma 4.6 to s n and t n to see are denoted by G and G H later on) corresponds to a random walk which jumps between the centers of these triangles, with spectral dimension between 1.63 and 1.77 (≈ 1.74 using the numerical estimates in [6]). This is a substantial difference implying, in particular, that one Markov process is not recurrent, while the other is recurrent. From the point of view of fractal analysis, our results suggest that the corresponding self-similar diffusion is not unique, unlike [17,4]. The long term motivation for our work comes from [20,19,21,27] and related papers, and especially from the works on the heat kernel estimates [1,3,4,16,23,28,29] In the discrete setting, the Dirichlet form is the graph energy. In this case the resistance between two sets is determined using Kirchhoff's laws. In the appendix in Section 5 we collect basic results about graph energies and resistance which we need throughout the paper. For a more thorough introduction to these topics, one can see, for example, [13,31]. This paper is organized as follows. Section 2 lays out the formal definitions and shows how to take advantage of the duality of the graphs G H n and G T n to prove that R n = 1/R T n . In Section 3 we first prove sub-multiplicative estimates R m+n ≤ cR m R n for some constant c independent of m and n. This is done in a fashion generalized from [1]. Fekete's theorem implies that the limits ρ and ρ T exist. To prove that these limits are finite, in section 4 we establish upper and lower estimates on R T n and R n . This is done by comparing G T n and G H n to subgraphs and quotient graphs respectively.

B ,
We define the 2-simplicial complexes T n = (E 0 n , E 1 n , E 2 n ) where E i n are the i-simplexes of the complex, starting with a 2-simplex (a triangle) T 0 with 0-simplexes (vertexes) Here, if q 0 , q 1 , q 2 ∈ E 0 n , then [q 0 , q 1 ] will refer to the 1-simplex with q 0 and q 1 as endpoints (which may or may not be in E 1 n ), and similarly for [q 0 , q 1 , q 2 ]. We will only be considering simple simplicial complexes (without multiple edges/triangles) and we shall not be considering orientation. Thus [q 0 , q 1 ] = [q 1 , q 0 ] determines a unique 1-simplex. We shall also use the notation < to denote containment of simplexes, i.e. q 0 < [q 0 , q 1 ] < [q 0 , q 1 , q 2 ].

F
. Adjacency (dual) graphs G 4 and G T 4 for barycentric subdivision of a 2-simplex.
T n+1 is defined inductively from T n by barycentric subdivision, pictured in figure 1. That is E 0 n+1 is E 0 n along with the barycenters of simplexes in E i n , i = 1, 2, which we shall refer to as b(e) for e ∈ E i n , i = 1, 2. 1-and 2-simplexes of T n+1 are formed from barycenters of nested simplexes. More concretely put, elements of E 1 n+1 are either of the form n . We will refer to this as the 1-skeleton of the T n . Definition 2.2. Following [6], we define the graph G n = (V n , E n ) with vertex set V n = E 2 n and edge relation [q 0 , q 1 , q] ∼ [q 0 , q 1 , q ]. That is, the vertexes of G n are the 2-simplexes of T n and they are connected by an edge if these simplexes share a 1-simplex. Remark 1. G T n is classically known to be a planar graph, as seen in figure 1, although throughout this work we will refer to the hexagonal embedding from figure 4 more often. With either of these embeddings, G n is the weak planar dual, that is each of its vertexes correspond to a plane region carved out by the embedding of G T n with the exception of the unbounded component. We will need explicit names for the elements of E 0 . This is convenient for recursively defining functions on T n . Define self-similarity maps F i : However, this is not true for G n and G n+1 , since not every edge of G n+1 is covered (2) and B

T /H
(2) on the hexagonal embedding of G We want to take advantage of this self-similarity throughout the current work, thus we shall define a modified hexacarpet graph.
We define the graph energy and energy dissipation on these graphs as in the appendix in section 5, so it is sufficient to define the conductance of edges. For G T n , We take E T n to be the graph energy defined with the above conductance. The advantage of these conductance values is the resulting self-similarity relation Similarly if we define c H q,q = 1/2 if q ∼ H q and 0 otherwise, then the resulting energy function E H n satisfies the following relation Both of these relations are also true for energy dissipation of functions on edges of these graphs.
It will often be useful to think of these graphs as embedded in the plane R 2 . It is typical to think of T n as a subdivided triangle in the plane, but we shall embed it as a hexagon, as T n , n > 0, has symmetry group D 6 , the dihedral group on 6 elements. As such, for n > 0 we define a map F T : E 0 n = V T n → R 2 by F T (p ) = (0, 0), F T (p k ) = (cos(2kπ/3), sin(2kπ/3)) and F T (p k ) = (cos((2k + 1)π/6), sin((2k + 1)π/6)) for k = 0, 1, 2. Thus E 0 1 is mapped to the corners and midpoint of a regular hexagon centered at (0, 0), see figure 4. We extend to E 0 n by taking averages: We embed G H n by the map Thus the vertexes of the embedded G H n are the centers of the triangles and edges of the embedding of G T n . If e = [q 0 , q 1 ] ∈ E 1 n then we define the geometric realization of e , |e|, to be the convex hull of F T (p 0 ) and F T (p 1 ). That is To define the resistance problem on these graphs, we shall need to define the boundary of these graphs. |e| ⊂ |f |, for e ∈ E i k and f ∈ E i n for k ≤ n, i ≤ j if the geometric realization of e is a subset of the geometric realization of f , e.g.
We will suppress arguments of the superscript when there is no danger of confusion. . Further, define R T n to be the effective resistance with respect to E T n between A T (n) and B T (n) . Theorem 2.1. The resistances R n and R T n are related by Proof. The main tool of this proof is proposition 9.4 from [31]. Define the weighted graph G T n = ( V T n , E T n ) as in Proposition 5.3 where V T n is V T n modulo the relation which identifies all elements of A T (n) and B T (n) into two vertexes called a T and b T respectively. From proposition 5.3, effective resistance between a T and b T with respect toẼ T n is R T n . Further we define G H n by, not only identifying A H (n) and B H (n) into single vertexes a H and b H respectively, but also to replace the sequential edges of the form [q 0 , q 1 , q] ∼ H [q 0 , q 1 ] ∼ H [q 0 , q 1 , q ] using Kirchoff's laws. Thus the sequential connections with resistance 1/2 are replaced with one connection [q 0 , q 1 , q] ∼ H [q 0 , q 1 , q ] with resistance 1. It is easy to see that the resistance between a H and b H is R n . Also, remove the vertexes contained in A T (n) and B T (n) and associated edgessince these vertexes are connected to the graph by only 1 edge,removing them has no impact on the resistance.
.G H 2 andG T 2 without the additional edges.
If we define the graphs (G H n ) † and (G T n ) † to be theG H n andG T n with an additional edge connecting a H to b H and a T to b T respectively, then (G H n ) † is the planar dual of (G T n ) † (see figure 5), and thus by Proposition 9.4 in [31], R n = 1/R T n .

E
This section is dedicated to the proof of Theorem 3.1. R m+n ≤ 4 3 R n R m , and equivalently R T m+n ≥ 3 4 R T m R T n . This theorem implies that there are constants c 0 and c 1 such that c 0 + log R T n and c 1 + log R n are superadditive/subadditive positive sequences. Thus the limits exist, and we have the following corollary. Note that this corollary does not rule out the possibility that ρ = 0 or ∞. In section 4, we establish positive upper and lower estimates on ρ and ρ T .
We give two proofs of theorem 3.1, the constants differ in the proofs, but the exact value of the constant does not effect the existence of ρ and ρ T . This establishes the result independently of the duality of the graphs -indeed variations on these proof work for different choices of boundary.
One version proves the upper estimate on R m+n directly, and uses flows on G H n . The direct proof of the lower estimate on R T m+n is proven using potentials on G T n . The two proofs mirror the upper and lower bounds for the resistance of the pre-carpet approximations for the Sierpinski carpet in [1]. The two versions of our proof highlight the importance of duality in proving Barlow-Bass style resistance estimates, and suggest possible directions in which their proofs can be generalized. is depicted in figure 6. Using the embedding F H , this symmetry is the reflection through the line which makes a π/2 angle with the x-axis. Because I n subjected to a π-rotation is −I n , H n 02 is a flow.
Thus H n 02 is a flow on G H n between L Take a 0 (x), a 1 (x), a 2 (x) to refer to the outward flow of I m restricted to each of these edges in the Y-network associated to x with orientation such that a 0 (x) is of a different sign then a 1 (x) and a 2 (x). i.e. if a 0 (x) < 0, then a 1 (x), a 2 (x) ≥, if a 0 (x) > 0, a 1 (x), a 2 (x) ≤ 0, and in the case when a 0 (x) = a 1 (x) = a 2 (x) = 0 then the choice is arbitrary. Notice that Take F ω G H n ⊂ G H n+m such that F ω ([p 0 , p 1 , p 2 ]) = x to be the subgraph of G H n+m isomorphic to G H n corresponding to x ∈ E 2 m . We shall assume that the labeling of the sides F ω (L n 2 ∪ L n 3 ) be contained in the edge which corresponds to the values a 1 (x), and F ω (L n 4 ∪L n 5 ) corresponds to a 2 (x).
. The function u, v and w.
The set of subgraphs F ω G H n x∈E 2 n ,ω∈{0,1,...,5} m cover G H m+n and define the flow J on G H m+n by its values on these subgraphs J is a well defined flow because H n 02 is obtained by a reflection of H n 01 which I n is symmetric with respect to, so aH n 01 (y, z) + bH n 02 (y, z) = (a + b)I n (y, z) for all y ∈ L n 0 ∪ L n 1 . Now we see, Note that the first inequality holds because of Cauchy-Schwartz inequality and the fact that, by our labeling convention, a 1 (x)a 2 (x) ≥ 0, and the last inequality holds because a 2 1 (x) + a 2 2 (x) ≥ (a 1 (x) + a 2 (x)) 2 /2 = a 2 0 (x)/2.

Lower estimate and potentials on G T
n . Let φ n be the harmonic potential on G T n with boundary values 0 on A H (n) and 1 on B T (n) . On G T n−1 , define u = φ n • F 0 , v = φ n • F 1 and w = φ n • F 5 . φ n • F 2 , φ n • F 3 , and φ n • F 4 can be written in terms of u, v, w and the constant 1 function as illustrated in figure 7.
Notice that the function w is v • σ where σ : G T n−1 → G T n−1 is the symmetry which exchanges p 0 and p 2 , fixing p 0 , and is extended to the rest of G T n−1 by averages ( with respect to F T , σ is the flip about the horizontal axis). Also, for all f because σ is a graph isometry with respect to the conductances. However, the function u symmetric about the horizontal axis, i.e. u(x) = u • σ(x), and v and w is anti-symmetric about this axis, i.e.
For each x ∈ E 2 n−1 , i ≡ i + 1 mod 6 define a x = b(x) to be the barycenter of x. Also, define a x i , i = 0, 1, . . . , 5 to be vertexes and barycenters of edges contained in x ordered in such at way that [a x i , a x i ] is an edge in E 1 n , and that the vectors for ω ∈ {0, 1, . . . , 5} n−1 such that F ω ([p 0 , p 1 , p 2 ]) = x. Notice that for all x ∈ E 2 n−1 , a x and a x i are contained in E 0 n , and that, for any function f on G T n , then where i ≡ i + 1 mod 6. We now define a function f n+m on G T n+m such that f n+m | G T m = φ m as follows: if F ω : G T n−1 → G T m+n is the contraction mapping which takes G T n−1 to the [a x , a x i , a x i ], then

E
In this section we prove estimates on the constants ρ and ρ T to prove that they are finite. Proof. Figure 8 shows G 1 = (V H 1 , E 1 ), which is obtained from G H 1 by removing all edges which have the vertexes contained in |[p 0 , p ]| and |[p 2 , p ]|. Figure 8 also illustrates how six G n graphs can be glued together to form one G n+1 graph. By induction, we see that each G n is made up of 2 n paths between L in G n can be encoded in a sequence of 2 n integers. Let's call this sequence l n and write l n = {l n,1 , l n,2 , ..., l n,2 n } where l n,j is the length of the path which has initial (or terminal) point j th closest to p 0 (in graph or Euclidean distance with our embedding). Since G n+1 is 6 copies of G n glued together, 2 n k=1 l n,k = 6 2 n−1 k=1 l n−1,k = 6 n , because l 1 = 2, 4. of the G n graph is the resistance of 2 n paths connected in parallel so, by Kirchhoff's laws, Recalling that x → 1 x is convex on (0, ∞), this allows us to use Jensen's inequality to write To obtain an upper bound on R n , we glue together 6 copies of G n−1 to produce a graph which has an edge set contained in E H n consisting of 2 n paths from A H n to B H n . Then, using proposition 5.2 and Kirchhoff's laws in addition to the above argument, we obtain that we are connecting A H n to B H n with 2 parallel connections of three sequential wires of resistance R n . Thus we have that R n ≤ 3R n−1 /2 = (3/2) n .
This also implies a lower bound on ρ T ≥ 2/3. However, this could also have been achieved by considering the graphG T n with vertex setṼ T n where x, y ∈ V T n (the vertex set of G T n ) are identified as one vertex if an edge connecting x and y was deleted in the construction of G n . One attains a graph as in figure 9, with "end" points P and Q, which correspond to the points in A T and B T .

4.2.
Lower bound on ρ by shorting graph. In this section we define a graph G H n , such that V n is V H n modulo an equivalence relation. Thus, resistance in G H n between two sets is less than the resistance between the fibers of these sets.
Proof. We define two vertexes in V H n to be equivalent if they are both in contained in F ω ([p i , p j ]) where j ≡ i+1 mod 3 for some i and some ω ∈ {0, 1, . . . , 5} k . Figure 10 shows G 2 . These graphs appeared in [35], as an examples of non-p.c.f. Sierpinski gaskets, where it was determined that their resistance scaling factor is 5 4 . This implies that R n+1 = 5 4 R n and subsequently R n = R 1 ( 5 4 ) n−1 . From this, and a gluing argument as in the previous subsection, it follows that R n ≥ c( 5 4 ) n for some c > 0 which is independent of n.
Using a similar argument, G T n can be obtained by identifying points in graph approximations of another non-p.c.f. Sierpinski gasket which appears at the end of [35], and which is pictured in figure 11. The resistance between the corner points of these graphs is some constant times (4/5) n , and thus the resistance between the corner points of G T n is greater than this value. This proves that ρ T < 4/5 because including more points in the boundary decreases resistance, so the resistance between the corner points is greater than the resistance between A and B.
Alternatively, if you connect the boundary points of the graph in figure 10 to a -network, it is dual to the network attained by connecting the corner points in the graph in figure 11 to a Y -network. This explains why the resistances are reciprocal.

A
: , , In this section we collect the basic definitions and results concerning graph energies and resistances which are used in the current work. For more detailed expositions on this subject see, for instance, [1,13,31]. Let G = (V, E) be a graph with finite vertex set V , and edge set E, which is a symmetric subset of V × V . We define the set (V ) = {f : V → R} to be the set of real valued functions on V which we will sometimes refer to as functions or potentials on the graph G.
If G is a weighted graph there are conductances (weights) c p,q = c q,p for all p, q ∈ V such that c p,q > 0 if (p, q) ∈ E and c p,q = 0 when (p, q) / ∈ E. Resistances between points p and q for any (p, q) ∈ E will be defined r p,q := 1/c p,q . To a graph with associated conductances, define the graph energy and write E (f ) := E (f, f ). The vector space (V ) is given the inner product Functions (with orientation) on the edge set will be denoted , and J(p, q) = 0 if (p, q) / ∈ E} . Define the energy dissipation E : (E) × (E) → R to be the inner product on (E). The discrete gradient ∇ : (V ) → (E) is given by ∇f (p, q) = c p,q (f (p) − f (q)), and discrete divergence div : (E) → (V ), defined by div J(p) = − q : (p,q)∈E J(p, q). We adopt the notation ∇ and div because can be interpreted as ∆ = div ∇. Further, it is elementary to check that the graph energy can be represented as  The effective resistance between sets A and B is defined by . Energy is minimized by the function φ such that φ| A ≡ 0, φ| B ≡ 1 and ∆φ(p) = 0 for p / ∈ A ∪ B, and thus E (φ) = 1/R(A, B). We shall refer to such a function φ as the harmonic function with boundary A and B. The only functions which satisfies ∆f ≡ 0 (i.e. harmonic without boundary) and f | A∪B ≡ 0 is the constant 0 function, thus the φ is unique.