Growth Tight Actions of Product Groups

A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight. Examples of non-growth tight actions are product groups acting on the $L^1$ products of Cayley graphs of the factors. In this paper we consider actions of product groups on product spaces, where each factor group acts with a strongly contracting element on its respective factor space. We show that this action is growth tight with respect to the $L^p$ metric on the product space, for all $1<p\leq \infty$. In particular, the $L^\infty$ metric on a product of Cayley graphs corresponds to a word metric on the product group. This gives the first examples of groups that are growth tight with respect to an action on one of their Cayley graphs and non-growth tight with respect to an action on another, answering a question of Grigorchuk and de la Harpe.Comment: 13 pages v2 15 pages, minor changes, to appear in Groups, Geometry, and Dynamic

Asymptotic normality of extreme value estimators on C[0,1]C[0,1]

Consider $n$ i.i.d. random elements on $C[0,1]$. We show that, under an appropriate strengthening of the domain of attraction condition, natural estimators of the extreme-value index, which is now a continuous function, and the normalizing functions have a Gaussian process as limiting distribution. A key tool is the weak convergence of a weighted tail empirical process, which makes it possible to obtain the results uniformly on $[0,1]$. Detailed examples are also presented.Comment: Published at http://dx.doi.org/10.1214/009053605000000831 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Game-Theoretic Framework for Medium Access Control

In this paper, we generalize the random access game model, and show that it provides a general game-theoretic framework for designing contention based medium access control. We extend the random access game model to the network with multiple contention measure signals, study the design of random access games, and analyze different distributed algorithms achieving their equilibria. As examples, a series of utility functions is proposed for games achieving the maximum throughput in a network of homogeneous nodes. In a network with n traffic classes, an N-signal game model is proposed which achieves the maximum throughput under the fairness constraint among different traffic classes. In addition, the convergence of different dynamic algorithms such as best response, gradient play and Jacobi play under propagation delay and estimation error is established. Simulation results show that game model based protocols can achieve superior performance over the standard IEEE 802.11 DCF, and comparable performance as existing protocols with the best performance in literature

Improved transfer matrix method without numerical instability

A new improved transfer matrix method (TMM) is presented. It is shown that the method not only overcomes the numerical instability found in the original TMM, but also greatly improves the scalability of computation. The new improved TMM has no extra cost of computing time as the length of homogeneous scattering region becomes large. The comparison between the scattering matrix method(SMM) and our new TMM is given. It clearly shows that our new method is much faster than SMM.Comment: 5 pages,3 figure

Growth Tight Actions

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$--space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic, and, conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph-manifold groups and many Right Angled Artin Groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non-relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmueller space with the Teichmueller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements.Comment: 29 pages, 4 figures v2 added references v3 40 pages, 6 figures, expanded preliminary sections to make paper more self-contained, other minor improvements v4 updated bibliography, to appear in Pacific Journal of Mathematic