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In this paper, we prove that if
is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer
, such that, for any
, there exists an
-invariant Jordan curve
containing the postcritical set of
We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as an aligned union of independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.