On the local convergence of integer-valued Lipschitz functions on regular trees

We study random integer-valued Lipschitz functions on regular trees. It was shown by Peled, Samotij, and Yehudayoff [22] that such functions are localized; however, finer questions about the structure of Gibbs measures remain unanswered. Our main result is that the weak limit of a uniformly chosen 1-Lipschitz function with 0 boundary condition on a d d d -ary tree of height n n n exists as n right arrow normal infinity n → ∞ n → ∞ if 2 less than or equals d less than or equals 7 2 ≤ d ≤ 7 2 ≤ d ≤ 7 , but not if d greater than or equals 8 d ≥ 8 d ≥ 8 , thereby partially answering a question posed by Peled, Samotij and Yehudayoff. For large d d d , the value at the root alternates between being almost entirely concentrated on 0 for even n n n and being roughly uniform on StartSet negative 1 comma 0 comma 1 EndSet { − 1 , 0 , 1 } \-1,0,1\ for odd n <mm