Random walks on hyperbolic spaces: Concentration inequalities and probabilistic Tits alternative

Area of Science:

• Probability Theory
• Geometric Group Theory
• Ergodic Theory

Background:

• Non-elementary random walks on hyperbolic spaces exhibit complex behavior.
• Concentration inequalities are crucial for understanding the deviation of random processes.
• Previous work focused on simpler settings, necessitating extensions to hyperbolic geometries.

Purpose of the Study:

• To derive Azuma-Hoeffding type concentration inequalities for random walks on hyperbolic spaces.
• To provide quantitative estimates for the probability of random walks generating free subgroups.
• To extend concentration bounds to random matrix products.

Main Methods:

• Proving Azuma-Hoeffding type concentration inequalities for random walk displacement.
• Developing explicit bounds dependent on the hyperbolic space and the driving measure.
• Applying concentration inequalities to estimate probabilities of subgroup generation.

Main Results:

• Explicit concentration bounds for random walks on proper hyperbolic spaces.
• Uniform bounds for hyperbolic groups and effective bounds for rank-one linear groups.
• Quantitative finite-time estimates for random walks generating free non-abelian subgroups.

Conclusions:

• The derived concentration inequalities provide a powerful tool for analyzing random walks in hyperbolic settings.
• The results offer new insights into the probabilistic generation of free subgroups by random walks.
• The methodology extends to provide subgaussian concentration bounds for random matrix products.