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Register a new userAsymptotic entropy of random walks on Fuchsian buildings and Kac-Moody groups
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In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac-Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotropic random walk onto an apartment of the building, to prove existence of the asymptotic entropy for this retracted walk, and to `lift this in order to deduce the existence of the entropy for the random walk on the building.
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Given an irreducible non-spherical non-affine (possibly non-proper) building $X$, we give sufficient conditions for a group $G < Aut(X)$ to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies to all irreducible (non-spherical and non-affine) Kac-Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov from the Kourovka notebook. Independently of these considerations, we also include a discussion of rank one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous.
Random walks on a group $G$ model many natural phenomena. A random walk is defined by a probability measure $p$ on $G$. We are interested in asymptotic properties of the random walks and in particular in the linear drift and the asymptotic entropy. If the geometry of the group is rich, then these numbers are both positive and the way of dependence on $p$ is itself a property of $G$. In this note, we review recent results about the regularity of the drift and the entropy for free groups, free products and hyperbolic groups.
We study the asymptotic behaviour of random walks on topological abelian groups $G$. Our main result is a sufficient condition for one random walk to overtake another in the stochastic order induced by any suitably large positive cone $G_+ subseteq G$, assuming that both walks have Radon distributions and compactly supported steps. We explain in which sense our sufficient condition is very close to a necessary one. Our result is a direct application of a recently proven theorem of real algebra, namely a Positivstellensatz for preordered semirings. It is due to Aubrun and Nechita in the one-dimensional case, but new already for $R^n$ with $n > 1$. We use our result to derive a formula for the rate at which the probabilities of a random walk decay relative to those of another, again for walks on $G$ with compactly supported Radon steps. In the case where one walk is a constant, this formula specializes to a version of Cramers large deviation theorem.
We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.
This is a summary (in French) of my work about brownian motion and Kac-Moody algebras during the last seven years, presented towards the Habilitation degree.
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