Explicit constants and convergence rates for resistances and time-scaling on generalized Sierpiński carpets

Cao and Chen (2024; J. Eur. Math. Soc. 2026) proved that for every generalized Sierpiński carpet (GSC) the time-scaling constants of the approximating reflected Brownian motions converge to a limit constant $c_0$ , identified as the limit of the mean boundary-hitting times, and that the renormalized effective resistances $R_n$ between opposite faces of the approximating domains converge to a positive limit $R$ — resolving a 1990 problem of Barlow and Bass. Both proofs are qualitative: no explicit formula, no computation with certified error, and no rate of convergence is known for either limit, in sharp contrast with finitely ramified fractals, where finite renormalization recursions determine the constants algebraically. This problem asks for the quantitative layer of those convergence theorems: an explicit or certified-computable determination of $c_0$ and $R$ for generalized Sierpiński carpets, and explicit-in- $n$ rates for $\alpha_n\to c_0$ and $R_n\to R$ , ideally uniform over families of carpets.

Background

A generalized Sierpiński carpet (GSC) is a self-similar fractal $F\subset[0,1]^d$ ( $d\ge 2$ ) built by subdividing the unit cube into $L_F^d$ closed sub-cubes of side $L_F^{-1}$ (integer $L_F\ge 3$ ), keeping a pattern of $m_F$ of them — the pattern must satisfy the standing regularity conditions of Barlow–Bass: symmetry under all isometries of the cube, connected interior, non-diagonality, and inclusion of the cube's border line segments — and repeating the same removal pattern on every surviving cube ad infinitum. The standard planar Sierpiński carpet is the case $L_F=3$ , $m_F=8$ , where the central ninth square is removed at each stage. Carpets are infinitely ramified — no finite set of points disconnects them — which is what separates their analysis from finitely ramified (post-critically finite, p.c.f.) fractals such as the Sierpiński gasket, where every quantity of interest is determined by a finite renormalization recursion on a finite graph.

Write $F_n$ for the stage- $n$ approximating domain, a union of $m_F^n$ cubes. Let $W^{(F_n)}$ be normally reflected Brownian motion on $F_n$ (Brownian motion confined to $F_n$ by instantaneous reflection at the boundary, i.e. with Neumann boundary conditions), and let $X^{(F_n)}$ be this process run at the natural carpet speed $\bar\rho_F^{\,n}$ , where $\bar\rho_F=\rho_F m_F/L_F^2$ and $\rho_F$ is the carpet's resistance scaling factor, i.e. the factor by which the unnormalized effective resistance is rescaled from one approximation level to the next; the walk dimension $d_w=\log(\rho_F m_F)/\log L_F>2$ governs the sub-Gaussian heat kernel behavior of the limiting diffusion. The Dirichlet form (the quadratic energy functional generating the process) of $X^{(F_n)}$ is denoted $\mathcal E^{(F_n)}$ , and the renormalized effective resistance between a pair of opposite faces of $F_n$ is

R_n^{-1}=\inf\Big\{\mathcal E^{(F_n)}(f,f):\ f\in W^{1,2}(F_n)\cap C(F_n),\ f=0\ \text{on one face},\ f=1\ \text{on the opposite face}\Big\}.

Let $\tau_n$ be the time $X^{(F_n)}$ started at the origin needs to hit the faces of the outer cube that do not contain the origin. Since Barlow and Bass constructed Brownian motion on carpets in 1989–1999 as subsequential scaling limits of the $X^{(F_n)}$ , two qualitative questions stayed open for three decades: whether the whole sequence of approximating processes converges (not merely subsequences), and whether $\lim_n R_n$ exists — the latter posed as Problem 1 by Barlow and Bass, who proved for the standard planar carpet that $1/4\le R_n\le 4$ and, with Sherwood, obtained only non-rigorous numerical estimates of the resistance exponent. Uniqueness of the limiting diffusion up to time change was settled by Barlow, Bass, Kumagai and Teplyaev (2010).

Cao and Chen resolved both questions. They proved (Theorem 1.1) that $X^{(F_n)}$ converges weakly to the canonical carpet diffusion slowed by a constant factor, $X^{(F)}_{t/c_0}$ , where $X^{(F)}$ is normalized so that its mean boundary-hitting time equals $1$ ; (Theorem 1.3) that the time-scaling constants $\alpha_n$ — a specific sequence, bounded between two positive constants, for which the time-changed processes $X^{(F_n)}_{\alpha_n t}$ converge weakly to $X^{(F)}$ — converge, $\lim_n\alpha_n=c_0$ , with the forms $\mathcal E^{(F_n)}$ Mosco-converging (Mosco convergence being the standard functional-analytic notion of convergence of quadratic forms on varying spaces, implying weak convergence of the associated processes) to $c_0^{-1}\mathcal E^{(F)}$ ; and (Remark 1.2) that $c_0=\lim_n\mathbb E^{(F_n)}_0[\tau_n]$ ; and finally (Theorem 1.4) that $R=\lim_n R_n$ exists and is positive. The proofs are qualitative throughout — they pass to subsequences, use trace theorems and polynomial decay of energy measures near cell boundaries — and produce no explicit constants and no rates.

Problem Statement

For a generalized Sierpiński carpet $F$ with its Cao–Chen approximation scheme (reflected Brownian motions $X^{(F_n)}$ on the stage- $n$ domains, run at the natural carpet speed), the limits

c_0=\lim_{n\to\infty}\alpha_n=\lim_{n\to\infty}\mathbb E^{(F_n)}_0[\tau_n]\qquad\text{and}\qquad R=\lim_{n\to\infty}R_n

are now known to exist (Cao–Chen, Theorems 1.3, 1.4 and Remark 1.2). Make these existence theorems quantitative. Two linked parts, serving one common objective — replacing the qualitative convergence of the resistance and time-scaling limits by usable quantitative information about the same limits:

(a) Determine $c_0$ and $R$ explicitly for generalized Sierpiński carpets: a closed-form expression in the carpet's defining data (the subdivision scale and hole pattern), or a computation with rigorously certified error. The determination in (a) is required in full generality over GSCs, of which the standard planar carpet ( $L_F=3$ , $m_F=8$ ) is an instance.

(b) Prove quantitative rates of convergence for arbitrary GSCs: explicit-in- $n$ upper bounds for $|\alpha_n-c_0|$ and for $|R_n-R|$ , ideally with constants uniform over families of GSCs.

The quantitative layer of the Cao–Chen convergence theorems consists of both (a) and (b).

References

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