Random walks on free solvable groups

TL;DR

This paper determines the asymptotic behavior of return probabilities for symmetric simple random walks on free solvable groups, providing insights into their probabilistic and geometric properties.

Contribution

It explicitly computes the return probability function for free solvable groups of derived length d, advancing understanding of random walks on these algebraic structures.

Findings

Asymptotic return probability functions are derived for free solvable groups.

Results apply to groups with specific algebraic properties, such as derived length and number of generators.

Provides new formulas for probabilistic behavior of random walks on complex algebraic groups.

Abstract

For any finitely generated group G, let n ---> \Phi_G(n) be the function that describes the rough asymptotic behavior of the probability of return to the identity element at time 2n of a symmetric simple random walk on G (this is an invariant of quasi-isometry). We determine this function when G is the free solvable group S_{d,r} of derived length d on r generators and some other related groups.