Probability and Mathematical Physics Seminar

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Lipschitz functions on expanders

Speaker: Jinyoung Park, Courant Institute

Location: Warren Weaver Hall 1302

Date: Friday, September 27, 2024, 11:10 a.m.

Synopsis:

We will discuss the typical behavior of M-Lipschitz functions on d-regular expander graphs, where an M-Lipschitz function means any two adjacent vertices admit integer values differ by at most M. While it is easy to see that the maximum possible height of an M-Lipschitz function on an n-vertex expander graph is about C*M*log n, where C depends (only) on d and the expansion of the given graph, it was shown by Peled, Samotij, and Yehudayoff (2012) that a uniformly chosen random M-Lipschitz function has height at most C'*M*loglog n with high probability, showing that the typical height of an M-Lipschitz function is much smaller than the extreme case. Peled-Samotij-Yehudayoff's result holds under the condition that, roughly, subsets of the expander graph expand by the rate of about M*log(dM). We will show that the same result holds under a much weaker condition assuming that d is large enough. This is joint work with Robert Krueger and Lina Li.