10,066 research outputs found
Random walks on graphs: new bounds on hitting, meeting, coalescing and returning
We prove new results on lazy random walks on finite graphs. To start, we
obtain new estimates on return probabilities and the maximum
expected hitting time , both in terms of the relaxation time. We
also prove a discrete-time version of the first-named author's ``Meeting time
lemma"~ that bounds the probability of random walk hitting a deterministic
trajectory in terms of hitting times of static vertices. The meeting time
result is then used to bound the expected full coalescence time of multiple
random walks over a graph. This last theorem is a discrete-time version of a
result by the first-named author, which had been previously conjectured by
Aldous and Fill. Our bounds improve on recent results by Lyons and
Oveis-Gharan; Kanade et al; and (in certain regimes) Cooper et al.Comment: First draf
The Cover Time of Deterministic Random Walks for General Transition Probabilities
The deterministic random walk is a deterministic process analogous to a
random walk. While there are some results on the cover time of the rotor-router
model, which is a deterministic random walk corresponding to a simple random
walk, nothing is known about the cover time of deterministic random walks
emulating general transition probabilities. This paper is concerned with the
SRT-router model with multiple tokens, which is a deterministic process coping
with general transition probabilities possibly containing irrational numbers.
For the model, we give an upper bound of the cover time, which is the first
result on the cover time of deterministic random walks for general transition
probabilities. Our upper bound also improves the existing bounds for the
rotor-router model in some cases
Total Variation Discrepancy of Deterministic Random Walks for Ergodic Markov Chains
Motivated by a derandomization of Markov chain Monte Carlo (MCMC), this paper
investigates deterministic random walks, which is a deterministic process
analogous to a random walk. While there are several progresses on the analysis
of the vertex-wise discrepancy (i.e., discrepancy), little is known
about the {\em total variation discrepancy} (i.e., discrepancy), which
plays a significant role in the analysis of an FPRAS based on MCMC. This paper
investigates upper bounds of the discrepancy between the expected number
of tokens in a Markov chain and the number of tokens in its corresponding
deterministic random walk. First, we give a simple but nontrivial upper bound
of the discrepancy for any ergodic Markov chains, where
is the number of edges of the transition diagram and is the mixing
time of the Markov chain. Then, we give a better upper bound for non-oblivious deterministic random walks, if the
corresponding Markov chain is ergodic and lazy. We also present some lower
bounds
Context-free pairs of groups. II - cuts, tree sets, and random walks
This is a continuation of the study, begun by Ceccherini-Silberstein and
Woess, of context-free pairs of groups and the related context-free graphs in
the sense of Muller and Schupp. Instead of the cones (connected components with
respect to deletion of finite balls with respect to the graph metric), a more
general approach to context-free graphs is proposed via tree sets consisting of
cuts of the graph, and associated structure trees. The existence of tree sets
with certain "good" properties is studied. With a tree set, a natural
context-free grammar is associated. These investigations of the structure of
context free pairs, resp. graphs are then applied to study random walk
asymptotics via complex analysis
Deterministic Random Walks for Rapidly Mixing Chains
The rotor-router model is a deterministic process analogous to a simple
random walk on a graph. This paper is concerned with a generalized model,
functional-router model, which imitates a Markov chain possibly containing
irrational transition probabilities. We investigate the discrepancy of the
number of tokens at a single vertex between the functional-router model and its
corresponding Markov chain, and give an upper bound in terms of the mixing time
of the Markov chain
Perturbations of the Voter Model in One-Dimension
We study the scaling limit of a large class of voter model perturbations in
one dimension, including stochastic Potts models, to a universal limiting
object, the continuum voter model perturbation. The perturbations can be
described in terms of bulk and boundary nucleations of new colors (opinions).
The discrete and continuum (space) models are obtained from their respective
duals, the discrete net with killing and Brownian net with killing. These
determine the color genealogy by means of reduced graphs. We focus our
attention on models where the voter and boundary nucleation dynamics depend
only on the colors of nearest neighbor sites, for which convergence of the
discrete net with killing to its continuum analog was proved in an earlier
paper by the authors. We use some detailed properties of the Brownian net with
killing to prove voter model perturbations convergence to its continuum
counterpart. A crucial property of reduced graphs is that even in the
continuum, they are finite almost surely. An important issue is how vertices of
the continuum reduced graphs are strongly approximated by their discrete
analogues.Comment: 13 figure
The rotor-router group of directed covers of graphs
A rotor-router walk is a deterministic version of a random walk, in which the
walker is routed to each of the neighbouring vertices in some fixed cyclic
order. We consider here directed covers of graphs (called also periodic trees)
and we study several quantities related to rotor-router walks on directed
covers. The quantities under consideration are: order of the rotor-router
group, order of the root element in the rotor-router group and the connection
with random walks.Comment: 17 pages, 2 figure
Kesten's theorem for uniformly recurrent subgroups
We prove an inequality on the difference between the spectral radius of the
Cayley graph of a group and the spectral radius of the Schreier graph
for any subgroup . As an application we extend Kesten's
theorem on spectral radii to uniformly recurrent subgroups and give a short
proof that the result of Lyons and Peres on cycle density in Ramanujan graphs
holds on average. More precisely, we show that if is an infinite
deterministic Ramanujan graph, then the time spent in short cycles by a random
walk of length is .Comment: 8 page
Local limit theorems for sequences of simple random walks on graphs
In this article, local limit theorems for sequences of simple random walks on
graphs are established. The results formulated are motivated by a variety of
random graph models, and explanations are provided as to how they apply to
supercritical percolation clusters, graph trees converging to the continuum
random tree and the homogenisation problem for nested fractals. A subsequential
local limit theorem for the simple random walks on generalised Sierpinski
carpet graphs is also presented
Deterministic approximation for the cover time of trees
We present a deterministic algorithm that given a tree T with n vertices, a
starting vertex v and a slackness parameter epsilon > 0, estimates within an
additive error of epsilon the cover and return time, namely, the expected time
it takes a simple random walk that starts at v to visit all vertices of T and
return to v. The running time of our algorithm is polynomial in n/epsilon, and
hence remains polynomial in n also for epsilon = 1/n^{O(1)}. We also show how
the algorithm can be extended to estimate the expected cover (without return)
time on trees
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