Percolation on hierarchical lattices

Abstract: We consider independent Bernoulli percolation on top of sequences of hierarchical graphs. Given a graph $G_{1}$ with two distinguished vertices $a_{1}$ and $b_{1}$, the hierarchical graph with seed $G_{1}$ is the sequence $\big( G_{k} \big)<em>{k \geq 1}$ resulting from the inductive procedure, where the graph $G</em>{k+1}$ is obtained from $G_{k}$ by replacing each of its edges with a copy of $G_{1}$, attached by the vertices $a_{1}$ and $b_{1}$. We prove that, under sharp hypotheses, percolation on these graphs presents a unique phase transition. Second, we establish the existence of several critical exponents in this context, such as the critical exponents for the correlation length $ν$, the surface tension $μ$, the one-arm exponent $α<em>{1}$. Several results are also obtained for their infinite counterpart $G</em>\infty$, which is the Benjamini-Schramm limit of $G_k$: uniqueness of the infinite cluster, continuity of $θ(p)$, existence of the percolation-probability exponent $β$ and scaling relations for the critical exponents $α<em>1$, $ν$ and $β$. Furthermore, we analyze noise sensitivity for crossing functions in $G</em>{k}$ and establish sharp noise sensitivity in this setting. Finally, we propose a setup where it is possible to verify the locality hypothesis, stating that the critical threshold for percolation is a local property, while critical exponents are determined by the global geometry of the graph. As a consequence of the techniques developed here, we also provide a necessary and sufficient condition for the existence of a unique fixed point for the map $p \mapsto \mathbb{E}_p[g]$ in $(0,1)$, where $g:{0,1}ⁿ \to {0,1}$ is a nontrivial monotone Boolean function.