Local convergence of grounded lipschitz functions on d-ary trees
| dc.contributor.author | Butler, Nathaniel | |
| dc.contributor.supervisor | Ray, Gourab | |
| dc.date.accessioned | 2026-01-07T21:25:31Z | |
| dc.date.available | 2026-01-07T21:25:31Z | |
| dc.date.issued | 2025 | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Master of Science MSc | |
| dc.description.abstract | We consider the uniform sampling of grounded M-Lipschitz functions on the d-ary tree with n levels, with special interest as n → ∞. In the case M = 1, it was shown in [2] that this sampling converges weakly (in the infinite d-ary tree) iff 2 ≤ d ≤ 7. We continue this work by putting the computations into a form that a computer can handle, and we use this to confirm convergence for several other values of M and d. As in [2], the main idea is use the recursive structure of the d-ary tree to reduce the problem to studying the fixed points of a certain function on ℓ∞(N). In [2], the authors also showed an even-odd phenomenon for M-Lipschitz functions on any infinite bipartite graph with ‘rapid expansion’ (i.e. with sufficiently large Cheeger constant). Specialized to our original problem of grounded M-Lipschitz functions on Tn d, this shows that the samplings for n even and n odd both converge, but to separate limits when d ≫ M logM. We reproduce this proof here. | |
| dc.description.scholarlevel | Graduate | |
| dc.identifier.bibliographicCitation | Nathaniel Butler, Kesav Krishnan, Gourab Ray, and Yinon Spinka. On the local convergence of integer-valued lipschitz functions on regular trees. arXiv preprint arXiv:2410.05542, 2024 | |
| dc.identifier.uri | https://hdl.handle.net/1828/23058 | |
| dc.language | English | eng |
| dc.language.iso | en | |
| dc.rights | Available to the World Wide Web | |
| dc.subject | Probability | |
| dc.subject | Random functions | |
| dc.subject | Mathematics | |
| dc.title | Local convergence of grounded lipschitz functions on d-ary trees | |
| dc.type | Thesis |