UVictoria Probability and Dynamics Seminar: Kesav Krishnan

In this talk, I will be discussing joint work with Nathaniel Butler, Gourab Ray and Yinon Spinka on the behavior of uniformly chosen integer valued 1-lipschitz functions on regular trees, with prescribed boundary conditions on the nth generation. This falls under a much larger umbrella of the study of various gradient/height function models on lattices on which there is a vast collection of literature. In the context of regular trees, it was known that the height function is localized as n becomes large, that is the law at any given vertex is tight. Moreover, the heights have double exponential tail. We provide alternative proofs of this fact, and go further to prove that the heights locally converge in distribution if and only if the degree of the tree is less than or equal to 7. For larger degree, we establish an alternating pattern of the law on even and odd generations. Finally, with certain special boundary conditions, local convergence holds for all degrees.