Topics in scaling limits on some Sierpinski carpet type fractals
| dc.contributor.author | Cao, Shiping | |
| dc.contributor.chair | Saloff-Coste, Laurent Pascal | |
| dc.contributor.committeeMember | Sosoe, Philippe | |
| dc.contributor.committeeMember | Muscalu, Camil | |
| dc.date.accessioned | 2022-10-31T16:19:50Z | |
| dc.date.available | 2022-10-31T16:19:50Z | |
| dc.date.issued | 2022-08 | |
| dc.description | 136 pages | |
| dc.description.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. | |
| dc.identifier.doi | https://doi.org/10.7298/m7z7-fm05 | |
| dc.identifier.other | Cao_cornellgrad_0058F_13104 | |
| dc.identifier.other | http://dissertations.umi.com/cornellgrad:13104 | |
| dc.identifier.other | bibid: 15578740 | |
| dc.identifier.uri | https://hdl.handle.net/1813/111928 | |
| dc.language.iso | en | |
| dc.relation.localuri | https://newcatalog.library.cornell.edu/catalog/15578740 | |
| dc.subject | Dirichlet form | |
| dc.subject | loop-erased random walk | |
| dc.subject | random walk | |
| dc.subject | resistance | |
| dc.subject | scaling limit | |
| dc.subject | Sierpinski carpet | |
| dc.title | Topics in scaling limits on some Sierpinski carpet type fractals | |
| dc.type | dissertation or thesis | |
| dcterms.license | https://hdl.handle.net/1813/59810.2 | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Cornell University | |
| thesis.degree.level | Doctor of Philosophy | |
| thesis.degree.name | Ph. D., Mathematics |
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