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Register a new userRate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups
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Authors
Lorenz A. Gilch
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We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.
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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.
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.
If $G$ is a free product of finite groups, let $Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti{e}rrez-Krsti{c} [M. Guti{e}rrez and S. Krsti{c}, {em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krsti{c} [W. Bogley and S. Krsti{c}, {em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $underline{E} Sigma Aut_1(G)$-space for these 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.
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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