2017/10/22 by John Dever, Dever, John · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Combinatorics #Dimension (graph theory) #Exponent #FOS: Mathematics #Fractal #Hausdorff dimension #Hausdorff measure #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Measure (data warehouse) #Metric Geometry (math.MG) #Metric space #Random walk #Sierpinski carpet #Sierpinski triangle #Space (punctuation) #Statistics #Topological and Geometric Data Analysis #Type (biology) #math.MG
paper · pdf · doi:10.48550/arxiv.1710.07872
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2017/10/22 · arxiv created 2017/11/02 · arxiv updated 2017/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We provide a definition of a new critical exponent β that has the interpretation of a type of local walk dimension, and may be defined on any compact metric space. We then specialize to the case of random walks that jump uniformly in metric balls with respect to a given Borel measure of full support. We use the local exponent β as a local time scaling exponent to re-normalize the time scale and produce approximating continuous time walks. We show a Faber-Krahn type inequality λ1,r(B)≥ \fraccRβ(x0), where c is a constant independent of r and x0 and where λ1,r(B) is the bottom of the spectrum of the generator for the re-normalized continuous time walk at stage r killed outside of B=BR(x0). In addition, we examine the local Hausdorff dimension α. We show that any variable Ahlfors Q-regular measure is strongly equivalent to the local Hausdorff measure and that Q=α. We also provide new examples of variable dimensional spaces, including a variable dimensional Sierpinski carpet.