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Topic

Local Convergence of Grounded Lipschitz Functions on d-ary Trees

Department of Mathematics and Statistics

Date & location

• Wednesday, December 10, 2025
• 9:00 A.M.
• Virtual Defence

Examining Committee

Supervisory Committee

• Dr. Gourab Ray, Department of Mathematics and Statistics, ӣ��Ӱ�� (Supervisor)
• Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)

External Examiner

• Dr. Omer Angel, Department of Mathematics, University of British Columbia

Chair of Oral Examination

• Dr. Steve Perlman, Department of Biology, UVic

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 𝕋^𝑛_𝑑, this shows that the samplings for 𝑛 even and 𝑛 odd both converge, but to separate limits when 𝑑≫𝑀 log 𝑀. We reproduce this proof here.