Necessity of additional smallness hypotheses for convergence in dimensions ≥3

Ascertain whether Theorem A (Convergence of random walk) on orthogonal tilings in R^d for d≥3 holds under the smallness condition that the maximum tile diameter and the vertex approximation error go to zero, without assuming either (II) the exponential lower bound on the minimal tile volume or (III) the diameter bound in terms of incident edge lengths and facet measures. Specifically, either prove that the convergence of the linearly interpolated random walk to Brownian motion modulo time parameterization still holds without assumptions (II) and (III), or construct an explicit orthogonal tiling in dimension d≥3 showing that at least one of these hypotheses is necessary.

Background

Theorem A establishes convergence of random walk on sequences of orthogonal tilings to Brownian motion modulo time change under a core smallness condition on tile diameters and vertex approximation, together with one of two additional regularity hypotheses: (II) a lower bound on minimal tile volume and (III) a bound on each tile’s diameter in terms of incident edge lengths and facet measures. These hypotheses are designed to control highly irregular tilings and ensure uniform transfer from L^2 to L^∞ estimates needed for the convergence.

The authors emphasize that these assumptions are quite mild and are satisfied by many natural models, including Voronoi tessellations sampled from Gaussian multiplicative chaos measures. Nonetheless, it is unknown whether either hypothesis is intrinsically required in higher dimensions (d≥3). The paper explicitly poses the problem of removing these assumptions or proving their necessity by counterexample.