THE SPEED OF RANDOM WALKS ON TREES AND ELECTRIC NETWORKS

2011 ◽  
Vol 26 (1) ◽  
pp. 105-116 ◽  
Author(s):  
Mokhtar Konsowa ◽  
Fahimah Al-Awadhi
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Trees ◽  
Hitting Times ◽  
Electric Networks ◽  
Starting Point ◽  
The Way

The speed of the random walk on a tree is the rate of escaping its starting point. It depends on the way that the branching occurs in the sense that if the average number of branching is large, the speed is more likely to be positive. The speed on some models of random trees is calculated via calculating the hitting times of the consecutive levels of the tree.

COMMUTE TIMES AND THE EFFECTIVE RESISTANCES OF RANDOM TREES

2009 ◽  
Vol 23 (4) ◽  
pp. 649-660 ◽  
Author(s):  
Fahimah Al-Awadhi ◽  
Mokhtar Konsowa ◽  
Zainab Najeh
Keyword(s):  
Random Walks ◽  
Random Trees ◽  
Hitting Times ◽  
Spherically Symmetric ◽  
Electric Networks

In this article we study the commute and hitting times of simple random walks on spherically symmetric random trees in which every vertex of levelnhas outdegree 1 with probability 1−qnand outdegree 2 with probabilityqn. Our argument relies on the link between the commute times and the effective resistances of the associated electric networks when 1 unit of resistance is assigned to each edge of the tree.

On a Result of Aleliunas et al. Concerning Random Walks on Graphs

1990 ◽  
Vol 4 (4) ◽  
pp. 489-492 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Proof ◽  
Hitting Times ◽  
Initial State ◽  
Random Walks On Graphs ◽  
Minimum Number

Aleliunas et al. [3] proved that for a random walk on a connected raph G = (V, E) on N vertices, the expected minimum number of steps to visit all vertices is bounded by 2|E|(N - 1), regardless of the initial state. We give here a simple proof of that result through an equality involving hitting times of vertices that can be extended to an inequality for hitting times of edges, thus obtaining a bound for the expected minimum number of steps to visit all edges exactly once in each direction.

On overshoots and hitting times for random walks

1999 ◽  
Vol 36 (2) ◽  
pp. 593-600
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Hitting Time ◽  
Hitting Times ◽  
First Hitting Time ◽  
Fixed Integer ◽  
Ladder Heights ◽  
Regenerative Sets

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

RANDOM WALKS AND DIMENSIONS OF RANDOM TREES

2010 ◽  
Vol 13 (04) ◽  
pp. 677-689
Author(s):  
MOKHTAR H. KONSOWA
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Trees ◽  
The Relationship

We study the relationship between the type of the random walk on some random trees and the structure of those trees in terms of fractal and resistance dimensions. This paper generalizes some results of Refs. 8–10.

Simple random walk statistics. Part I: Discrete time results

1996 ◽  
Vol 33 (2) ◽  
pp. 311-330 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Order Statistics ◽  
Discrete Time ◽  
Rank Order ◽  
Simple Random Walk ◽  
Alternative Method ◽  
Starting Point ◽  
Famous Paper ◽  
Rank Order Statistics

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

scholarly journals On the Moments of Hitting Times for Random Walks on Trees

2009 ◽  
Vol 2009 ◽  
pp. 1-4 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Hitting Times ◽  
Natural Numbers ◽  
Ergodic Markov Chain

Using classical arguments we derive a formula for the moments of hitting times for an ergodic Markov chain. We apply this formula to the case of simple random walk on trees and show, with an elementary electric argument, that all the moments are natural numbers.

Simple random walk statistics. Part I: Discrete time results

1996 ◽  
Vol 33 (02) ◽  
pp. 311-330 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Order Statistics ◽  
Discrete Time ◽  
Rank Order ◽  
Simple Random Walk ◽  
Alternative Method ◽  
Starting Point ◽  
Famous Paper ◽  
Rank Order Statistics

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

On overshoots and hitting times for random walks

1999 ◽  
Vol 36 (02) ◽  
pp. 593-600
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Hitting Time ◽  
Hitting Times ◽  
First Hitting Time ◽  
Fixed Integer ◽  
Ladder Heights ◽  
Regenerative Sets

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

Appreciate the process, by taking a random walk

2020 ◽  
pp. 241-244
Author(s):  
Susan D'Agostino
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Life ◽  
Mathematics Students ◽  
E Coli ◽  
Starting Point ◽  
Mathematical Formalization ◽  
Biased Random Walks

“Appreciate the process, by taking a random walk” offers a basic introduction to random walks—a mathematical formalization for a path as an object wanders away from a starting point—with a particular focus on biased random walks, including a real-life example of a random walk of an E. coli bacterium. The discussion is illustrated with hand-drawn sketches. Mathematics students and enthusiasts are encouraged to build in a productive bias in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

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