Ziyu Neroli | Search arXiv

arXiv2026

Moiré Pattern of Rotated Sierpiński Gaskets

We study the moiré pattern formed by the intersection of a planar Sierpiński gasket and a copy rotated about its centre. At every resonant angle, we prove that the intersection is of finite type and compute its Hausdorff and Minkowski dimensions from an explicit finite matrix; for Lebesgue-almost every angle, we bound its upper Minkowski dimension by $\overline{\dim}_{\mathrm B}(\mathcal S_θ)\le 2\dim_{\mathrm H}(S)-2$. Finally, we conjecture that both dimensions equal this bound at every non-resonant angle. Sections 2 and 3 are formalised in Lean.

math.MG↗

arXiv2026

Iterated Graph Systems (I): random walks and diffusion limits

This paper investigates random walks and diffusion limits on a broad class of fractals generated by Edge Iterated Graph Systems (EIGS), a generalisation of hierarchical lattices. We prove that the rescaled simple random walks converge in the Gromov--Hausdorff--Prokhorov--Skorokhod topology to the limiting diffusion, which coincides with Brownian motion when the resistance dimension is positive. The graph analysis underlying this convergence identifies the degree dimension as the natural correction term for on-diagonal heat-kernel estimates, yielding a unified formulation in the locally finite and locally infinite (scale-free) regimes. Using this framework, we solve the open problem on the diamond hierarchical lattice (DHL) percolation cluster posed by Hambly and Kumagai [Commun. Math. Phys. 295 (2010), 29--69]; this suggests that the Alexander--Orbach-type conjecture fails for the natural Brownian motion on the cluster. Sections 2 to 4 have been formalised in Lean.

math.PR↗