2017/08/08 by Gerard Letac, Letac, Gerard, Mauro Piccioni +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1708.02506
arxiv created 2017/08/08 · arxiv updated 2017/08/09
Consider G=SL2(ℤ)/\± I\ acting on the complex upper half plane H by hM(z)=(az+b)/(cz+d), for M ∈ G. Let D=\z ∈ H: |z|≥ 1, |\Re(z)|≤ 1/2\. We consider the set E ⊂ G with the 9 elements M, different from the identity, such that (MMT)≤ 3. We equip the tiling of H defined by \mathbbD=\hM(D), M ∈ G\ with a graph structure where the neighbours are defined by hM(D) ∩ hM'(D) ≠ ∅, equivalently M-1M' ∈ E. The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point X of the real line with the same distribution of S2 WS1, where S1,S2,W are independent with Pr (Si=± 1)=1/2 and where W is valued in (0,1) with distribution Pr(W<w)=?(w). Here ? is the Minkowski function. If K1, K2, … are i.i.d with distribution Pr (Ki=n)= 1/2n for n=1,2,…, then W= \frac1K1+\frac 1K2+…: this known result (Isola (2014)) is derived again here.