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

Source abstract

Abstract 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 → ∞ $n \to \infty$ if 2 less than or equals d less than or equals 7 2 ≤ d ≤ 7 $2 \le d \le 7$ , but not if d greater than or equals 8 d ≥ 8 $d \ge 8$ , 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 n $n$ , leading to different limits as n n $n$ approaches infinity along evens or odds. For d greater than or equals 8 d ≥ 8 $d \ge 8$ , the essence of this phenomenon is preserved, which obstructs the convergence. For d less than or equals 7 d ≤ 7 $d \le 7$ , this phenomenon ceases to exist, and the law of the value at the root loses its connection with the parity of n n $n$ . Along the way, we also obtain an alternative proof of localization. The key idea is a fixed point convergence result for a related operator on script l Superscript normal infinity ℓ ∞ $\ell ^\infty$ and a procedure to show that the iterations get into a ‘basin of attraction’ of the fixed point. We also prove some accompanying analogous ‘even-odd phenomenon’ type results about upper M M $M$ -Lipschitz functions on general non-amenable graphs with high enough expansion (this includes for example the large d d $d$ case for regular trees). We also prove a convergence result for 1-Lipschitz functions with StartSet 0 comma 1 EndSet { 0 , 1 } $\{0,1\}$ boundary condition. This last result relies on an absolute value FKG for uniform 1-Lipschitz functions when shifted by 1 divided by 2 1 / 2 $1/2$ .