Renewal theory for random walks on surface groups

Abstract

Abstract. We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as an aligned union of independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

Regularity of biased 1D random walks in random environment

“…Finally, to prove the dichotomy, it is enough to check that E[F (λ)] = ∞ whenever λ ≥ 0. From (19) we clearly see that, for λ ≥ 0, x := r(x, x± 1). Then its asymptotic velocity v Y (λ) is discontinuous at λ + .…”

Buildings, Groups of Lie Type and Random Walks

“…Thus the representation theoretic techniques that have worked so nicely for the spherical and affine cases do not seem to help. Instead the arguments rely much more heavily on the underlying hyperbolic geometry of the building and the planarity of its apartments, with the general ideas adapted from the work of Haïssinski, Mathieu and Müller [25]. In this work the planarity of the and hyperbolicity of the Cayley graph of a surface group are exploited to develop a 'renewal theory' related to the automata structure of the group.…”

Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups

“…More recently Björklund [6] proved a central limit theorem for hyperbolic groups with respect to the Green metric, and this was pushed forward by Benoist and Quint [5] for random walks on hyperbolic groups with respect to the word metric under the optimal second moment condition. Another approach to the central limit theorem for surface groups has been developed by Haissinski, Mathieu, and Müller [21], where the planarity and hyperbolicity of the Cayley graph are employed to develop a renewal theory for random walks on these groups. The resulting central limit theorem comes complete with formulae for the speed and variance of the walk.…”

“…From the point of view of Lie theory, this is a natural next step in the progression from 'spherical-type' Lie groups (the semisimple real Lie groups) and 'affine-type' Lie groups (the p-adic case) to a theory for random walks on buildings and Kac-Moody groups of arbitrary type. From the hyperbolic point of view, the buildings that we consider contain many copies of the hyperbolic disc tessellated using a 'Fuchsian Coxeter group', and thus while the buildings are certainly not planar, some of the renewal theory techniques from the planar surface group case [21] can be pushed through.…”

“…For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in [21]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building.…”