Local convergence of grounded lipschitz functions on d-ary trees
Date
2025
Authors
Butler, Nathaniel
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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.
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Keywords
Probability, Random functions, Mathematics