{"paper":{"title":"Diffusive estimates for random walks on stationary random graphs of polynomial growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"James R. Lee, Shirshendu Ganguly, Yuval Peres","submitted_at":"2016-09-13T20:30:12Z","abstract_excerpt":"Let $(G,\\rho)$ be a stationary random graph, and use $B^G_{\\rho}(r)$ to denote the ball of radius $r$ about $\\rho$ in $G$. Suppose that $(G,\\rho)$ has annealed polynomial growth, in the sense that $\\mathbb{E}[|B^G_{\\rho}(r)|] \\leq O(r^k)$ for some $k > 0$ and every $r \\geq 1$.\n  Then there is an infinite sequence of times $\\{t_n\\}$ at which the random walk $\\{X_t\\}$ on $(G,\\rho)$ is at most diffusive: Almost surely (over the choice of $(G,\\rho)$), there is a number $C > 0$ such that \\[ \\mathbb{E} \\left[\\mathrm{dist}_G(X_0, X_{t_n})^2 \\mid X_0 = \\rho, (G,\\rho)\\right]\\leq C t_n\\qquad \\forall n \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04040","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"}