Exotic local limit theorems at the phase transition in free products

2023/03/10 by Matthieu Dussaule, Marc Peigné, Dussaule, Matthieu +3 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2303.05818

openalex publication_date 2023/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that μ * n (e) ∼ CR --n n --5/3 if d = 5 and μ * n (e) ∼ CR --n n --3/2 log(n) --1/2 if d = 6, where μ * n is the nth convolution power of μ and R is the inverse of the spectral radius of μ. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a rst version of [11]. This also shows that the classication of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.

Citations

• A local limit theorem for convergent random walks on relatively hyperbolic groups
• Stability phenomena for Martin boundaries of relatively hyperbolic groups

Cited by

• Local limit theorems on relatively hyperbolic groups with respect to virtually nilpotent subgroups

Related

• Local limit theorems in relatively hyperbolic groups II : the non-spectrally degenerate case
• Phase Transitions for Random Walk Asymptotics on Free Products of Groups
• A local limit theorem for convergent random walks on relatively hyperbolic groups
• Local limit theorems in relatively hyperbolic groups I : rough estimates
• Local limit theorems on relatively hyperbolic groups with respect to virtually nilpotent subgroups