New transience bounds for long walks in weighted digraphs

International audienceWe consider the sequence of maximal weights of walks of lengt n between two fixed nodes in a weighted digraph. It is known that these sequences show a periodic behavior after an initial transient. We identify relevant graph parameters and propose a modular strategy to derive new upper bounds on the transient. To the best of our knowledge, our bounds are the first that are both asymptotically tight and potentially subquadratic. In particular, the new bounds show that the transient is linear in the number of nodes in bi-directional trees. Besides, our results enable a fine-grained performance analysis and give guidelines for the design of distributed systems based on max-plus recursions

An Overview of Transience Bounds in Max-Plus Algebra

We survey and discuss upper bounds on the length of the transient phase of max-plus linear systems and sequences of max-plus matrix powers. In particular, we explain how to extend a result by Nachtigall to yield a new approach for proving such bounds and we state an asymptotic tightness result by using an example given by Hartmann and Arguelles.Comment: 13 pages, 2 figure

Weak CSR expansions and transience bounds in max-plus algebra

This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: A^t=CS^tR + B^t. We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the CS^tR term to dominate. To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman-Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.Comment: 32 page

Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees

We develop a technique that provides a lower bound on the speed of transient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and regeneration time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic (2002) prove an upper bound of the form $b/(b+\delta)$ for the speed on the $b$-ary tree, where $\delta$ is the reinforcement parameter. For $\delta>1$ we provide a lower bound of the form $\gamma^2 b/(b+\delta)$, where $\gamma$ is the survival probability of an associated branching process.Comment: 21 page

Almost sure functional central limit theorem for ballistic random walk in random environment

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.Comment: Accepted to the Annales de l'Institut Henri Poincar