{"paper":{"title":"Random walks in the hyperbolic plane and the question mark function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerard Letac, Mauro Piccioni","submitted_at":"2017-08-08T14:39:53Z","abstract_excerpt":"Consider $G=SL_2(\\mathbb{Z})/\\{\\pm I\\}$ acting on the complex upper half plane $H$ by $h_M(z)=\\frac{az+b}{cz+d},$ for $M \\in G$. Let $D=\\{z \\in H: |z|\\geq 1, |\\Re(z)|\\leq 1/2\\}$. We consider the set $\\mathcal{E} \\subset G$ with the $9$ elements $M$, different from the identity, such that $(MM^T)\\leq 3$. We equip the tiling of $H$ defined by $\\mathbb{D}=\\{h_M(D), M \\in G\\}$ with a graph structure where the neighbours are defined by $h_M(D) \\cap h_{M'}(D) \\neq \\emptyset$, equivalently $M^{-1}M' \\in \\mathcal{E}$.\n  The present paper studies several Markov chains related to the above structure. We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02506","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}