Random walks on fractal graphs exhibit anomalous diffusion — the walker spreads slower than on regular lattices because the fractal geometry creates bottlenecks and dead ends. Quantifying the anomaly requires the resistance exponent, which governs how electrical resistance scales with distance on the fractal. Computing this exponent for percolation clusters on hierarchical lattices has been an open problem.
Croydon, Hambly, and Kumagai solve it. They introduce iterated graph systems — a general framework for fractals built by recursive graph substitution — and prove that random walks on these systems converge to limiting diffusion processes. The convergence is in the Gromov-Hausdorff-Prohorov-Skorokhod topology, meaning the geometry, measure, and stochastic process all converge simultaneously.
For the diamond hierarchical lattice percolation cluster, they determine the quenched resistance exponent: the exponent that governs typical (not averaged) resistance growth. The quenched and annealed (averaged) exponents differ — a phenomenon known as strong disorder — and their framework computes both.
The technical achievement is handling the quenched case. Annealed exponents can often be computed by averaging over the randomness in the graph. Quenched exponents require controlling the fluctuations of the random graph and showing that the walk's behavior concentrates around the typical case. The iterated graph system framework provides the recursive structure needed for this concentration argument.
Random walks on fractals, converging to diffusions. The exponent that controls the speed of the walk, determined exactly for the first time on a class of random fractals.