The standard planar Sierpiński carpet carries a canonical Brownian motion whose associated self-similar local Dirichlet form is renormalized by a resistance-scaling factor ρ>1\rho>1. Existence of ρ\rho is proved, and the rigorous shorting/cutting bounds 7/6≤ρ≤3/27/6\le\rho\le 3/2 together with the numerical estimates ρ∈[1.25147,1.25149]\rho\in[1.25147,1.25149] both date from 1990; the walk dimension is dw=log⁡(8ρ)/log⁡3d_w=\log(8\rho)/\log 3. The problem, explicitly acknowledged as unresolved in Grigor'yan–Yang's construction of the local Dirichlet form on the carpet, is the exact analytic determination of ρ\rho (equivalently of dwd_w). The problem remains open.

Background

The standard Sierpiński carpet KK is the self-similar subset of the unit square obtained by repeatedly deleting the open middle ninth of every retained square: divide [0,1]2[0,1]^2 into a 3×33\times 3 grid of congruent subsquares, remove the open central square, and iterate on the 8 remaining closed squares. Unlike finitely ramified fractals (those that can be disconnected by removing finitely many points, such as the Sierpiński gasket), the carpet is infinitely ramified, and this is the source of the problem's difficulty.

Brownian motion on KK was first constructed by Barlow and Bass by taking scaled limits of reflected Brownian motions on approximating square domains, and later characterized intrinsically: Barlow, Bass, Kumagai, and Teplyaev proved uniqueness of the Brownian motion on (generalized) Sierpiński carpets, so its invariants are canonical. Analytically, the process is generated by a self-similar local regular Dirichlet form (Eloc,Floc)(\mathcal{E}_{\mathrm{loc}},\mathcal{F}_{\mathrm{loc}}) — a closed Markovian quadratic form on L2(K;ν)L^2(K;\nu), where ν\nu is the normalized Hausdorff measure — constructed purely analytically by Grigor'yan and Yang via approximation by stable-like non-local forms. Self-similarity here means Eloc(u,u)=ρ∑i=07Eloc(u∘Fi,u∘Fi),\mathcal{E}_{\mathrm{loc}}(u,u)=\rho\sum_{i=0}^{7}\mathcal{E}_{\mathrm{loc}}(u\circ F_i,u\circ F_i), where F0,…,F7F_0,\dots,F_7 are the eight contracting similarities generating KK, and the constant ρ>1\rho>1 is the resistance-scaling (renormalization) factor: it is the exponential rate at which the effective electrical resistance between opposite sides of the level-nn carpet approximation networks grows, Rn≍ρnR_n\asymp\rho^n. Effective resistance is the total resistance of the approximation network when each retained level-nn cell is treated as a unit resistor and potentials are applied on the boundary; ρ\rho is equivalently the constant whose existence is asserted in the resistance estimates of Barlow–Bass reproduced as Theorems 5.1 and 5.4 of Grigor'yan–Yang.

The constant ρ\rho determines the walk dimension dw=log⁡(8ρ)/log⁡3d_w=\log(8\rho)/\log 3, the anomalous-diffusion exponent in the two-sided sub-Gaussian heat kernel estimate of the carpet's Brownian motion, pt(x,y)≍t−α/β∗exp⁡(−c (∣x−y∣/t1/β∗)β∗/(β∗−1)),α=log⁡8/log⁡3, β∗=dw,p_t(x,y)\asymp t^{-\alpha/\beta^*}\exp\bigl(-c\,(|x-y|/t^{1/\beta^*})^{\beta^*/(\beta^*-1)}\bigr),\qquad \alpha=\log 8/\log 3,\ \beta^*=d_w, and hence the spectral dimension ds=2α/dwd_s=2\alpha/d_w — the exponent governing the eigenvalue counting asymptotics N(λ)≍λds/2N(\lambda)\asymp\lambda^{d_s/2} of the carpet Laplacian — and all sharp scaling constants of the diffusion. For the Sierpiński gasket, an exact renormalization factor (ρ=5/3\rho=5/3) is obtained by spectral decimation — the reduction of the renormalization to a finite-dimensional linear-algebraic fixed point, possible because the gasket is post-critically finite (the self-similar analogue of finite ramification: only finitely many points are shared between its pieces). For the carpet no such finite reduction exists: renormalization acts on boundary data along interfaces whose complexity grows with the approximation level.

What is known: Barlow and Bass proved by the shorting technique (shorting nodes can only decrease network resistance) and the cutting technique (cutting bonds can only increase it) that ρ∈[7/6,3/2]\rho\in[7/6,3/2]; Barlow, Bass, and Sherwood computed numerically ρ∈[1.25147,1.25149]\rho\in[1.25147,1.25149]; McGillivray extended the same estimates to higher-dimensional carpets. Grigor'yan and Yang, whose construction of the local Dirichlet form depends on these resistance estimates, state that the exact value of ρ\rho "remains still unknown." No exact expression, and no rigorous improvement of [7/6,3/2][7/6,3/2], has been obtained since 1990.

Problem Statement

Determine, exactly and rigorously, the resistance-scaling (renormalization) factor ρ\rho of the standard middle-thirds Sierpiński carpet K⊂[0,1]2K\subset[0,1]^2 — the unique constant ρ>1\rho>1 in the self-similarity relation Eloc(u,u)=ρ∑i=07Eloc(u∘Fi,u∘Fi)\mathcal{E}_{\mathrm{loc}}(u,u)=\rho\sum_{i=0}^{7}\mathcal{E}_{\mathrm{loc}}(u\circ F_i,u\circ F_i) of the canonical self-similar local regular Dirichlet form on KK (equivalently, the constant with Rn≍ρnR_n\asymp\rho^n for the effective resistances of the level-nn carpet approximation networks). Equivalently, determine the exact walk dimension of Brownian motion on KK, dw=log⁡(8ρ)log⁡3,d_w=\frac{\log(8\rho)}{\log 3}, whose current rigorous confinement is log⁡(8⋅7/6)/log⁡3≤dw≤log⁡(8⋅3/2)/log⁡3\log(8\cdot 7/6)/\log 3\le d_w\le\log(8\cdot 3/2)/\log 3, with numerics indicating ρ≈1.25148\rho\approx 1.25148.

References

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