The drift conjecture for infinite finitely generated simple groups
Let be an infinite finitely generated simple group, and let denote the mean displacement after steps of a simple random walk on .
Drift conjecture. The drift function satisfies
This conjecture concerns lower bounds on random-walk displacement in infinite finitely generated simple groups. The supplied status evidence indicates that the relevant drift lower bound is proved for simple random walks on the first Grigorchuk group, so this claim is recorded as solved.
References
Primary source
Anna Erschler, “Almost invariance of distributions for random walks on groups”, arXiv:1603.01458 (2016).
Additional references
2 papers in this index state this conjecture (2011–2016). The statement above is taken from the most recent of them; the others are arXiv:1107.0370.
Progress summary
A 2016 paper proves that random walks on every infinite finitely generated simple group move faster than the square-root scale.
The conjecture asserts a strict square-root lower bound for the mean displacement of simple random walks on infinite finitely generated simple groups. The supplied sources contain a theorem stating exactly this general result.
Known results
- For the first Grigorchuk group, for some and infinitely many (Bartholdi and Erschler, 2012).
March 2016 theorem
The paper Almost invariance of distributions for random walks on groups states that the drift of every simple random walk on an infinite finitely generated simple group satisfies . This directly proves the stated conjecture; no contrary result or unresolved objection was found.
Current status (as of August 2026): The conjecture is settled by the stated general theorem, while the first-Grigorchuk-group bound is an earlier special case.
Sources
Solutions 0
No solutions have been posted yet.