Topics in scaling limits on some Sierpinski carpet type fractals

Cao, Shiping

Topics in scaling limits on some Sierpinski carpet type fractals

Abstract

We consider two topics in scaling limits on Sierpi\'nski carpet type fractals. First, we construct local, regular, irreducible, symmetric, self-similar Dirichlet forms on unconstrained Sierpi\'nski carpets, which are natural extension of planar Sierpi\'nski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be a line segment of irrational length, so the class contains some irrationally ramified self-similar sets. In addition, in this strongly recurrent setting, we can drop the non-diagonal condition, which was always assumed in previous works. We also give an example showing that a good Dirichlet form may not exist if we weaken more geometric conditions. Second, on any planar Sierpi\'nski carpet, we prove the existence of the scaling limit of loop-erased random walks on Sierpi\'nski carpet graphs. In addition, we have a more general result showing that the loop-erased Markov chains induced by the resistance form on a sequence of finite sets approximating the resistance space converge weakly in the Hausdorff metric. To prove this, we introduce partial loop-erasing operators, and prove a surprising result that, by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we will get a process equivalent to the loop-erased Markov chain. All the limit probabilities are shown to be supported on the set of simple paths.

Description

136 pages

Date Issued

2022-08

Keywords

Dirichlet form; loop-erased random walk; random walk; resistance; scaling limit; Sierpinski carpet

Committee Chair

Saloff-Coste, Laurent Pascal

Committee Member

Sosoe, Philippe
Muscalu, Camil

Degree Discipline

Mathematics

Degree Name

Ph. D., Mathematics

Degree Level

Doctor of Philosophy

Types

dissertation or thesis