3,483 research outputs found
On the trace of branching random walks
We study branching random walks on Cayley graphs. A first result is that the
trace of a transient branching random walk on a Cayley graph is a.s. transient
for the simple random walk. In addition, it has a.s. critical percolation
probability less than one and exponential volume growth. The proofs rely on the
fact that the trace induces an invariant percolation on the family tree of the
branching random walk. Furthermore, we prove that the trace is a.s. strongly
recurrent for any (non-trivial) branching random walk. This follows from the
observation that the trace, after appropriate biasing of the root, defines a
unimodular measure. All results are stated in the more general context of
branching random walks on unimodular random graphs.Comment: revised versio
Recurrence of random walk traces
We show that the edges crossed by a random walk in a network form a recurrent
graph a.s. In fact, the same is true when those edges are weighted by the
number of crossings.Comment: Published at http://dx.doi.org/10.1214/009117906000000935 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Scalable Verification of Markov Decision Processes
Markov decision processes (MDP) are useful to model concurrent process
optimisation problems, but verifying them with numerical methods is often
intractable. Existing approximative approaches do not scale well and are
limited to memoryless schedulers. Here we present the basis of scalable
verification for MDPSs, using an O(1) memory representation of
history-dependent schedulers. We thus facilitate scalable learning techniques
and the use of massively parallel verification.Comment: V4: FMDS version, 12 pages, 4 figure
Quantum Markov chains associated with open quantum random walks
In this paper we construct (nonhomogeneous) quantum Markov chains associated
with open quantum random walks. The quantum Markov chain, like the classical
Markov chain, is a fundamental tool for the investigation of the basic
properties such as reducibility/irreducibility, recurrence/transience,
accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on
the discussion of the reducibility and irreducibility of open quantum random
walks via the corresponding quantum Markov chains. Particularly we show that
the concept of reducibility/irreducibility of open quantum random walks in this
approach is equivalent to the one previously done by Carbone and Pautrat. We
provide with some examples. We will see also that the classical Markov chains
can be reconstructed as quantum Markov chains.Comment: 30 page
The Embedding Capacity of Information Flows Under Renewal Traffic
Given two independent point processes and a certain rule for matching points
between them, what is the fraction of matched points over infinitely long
streams? In many application contexts, e.g., secure networking, a meaningful
matching rule is that of a maximum causal delay, and the problem is related to
embedding a flow of packets in cover traffic such that no traffic analysis can
detect it. We study the best undetectable embedding policy and the
corresponding maximum flow rate ---that we call the embedding capacity--- under
the assumption that the cover traffic can be modeled as arbitrary renewal
processes. We find that computing the embedding capacity requires the inversion
of very structured linear systems that, for a broad range of renewal models
encountered in practice, admits a fully analytical expression in terms of the
renewal function of the processes. Our main theoretical contribution is a
simple closed form of such relationship. This result enables us to explore
properties of the embedding capacity, obtaining closed-form solutions for
selected distribution families and a suite of sufficient conditions on the
capacity ordering. We evaluate our solution on real network traces, which shows
a noticeable match for tight delay constraints. A gap between the predicted and
the actual embedding capacities appears for looser constraints, and further
investigation reveals that it is caused by inaccuracy of the renewal traffic
model rather than of the solution itself.Comment: Sumbitted to IEEE Trans. on Information Theory on March 10, 201
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