scholarly journals The Fundamental Matrix of the Simple Random Walk with Mixed Barriers

2016 ◽  
Vol 8 (6) ◽  
pp. 128
Author(s):  
Yao Elikem Ayekple ◽  
Derrick Asamoah Owusu ◽  
Nana Kena Frempong ◽  
Prince Kwaku Fefemwole
Keyword(s):  
Random Walk ◽  
Fundamental Matrix ◽  
Transition Matrix ◽  
Transition Probabilities ◽  
Simple Random Walk ◽  
Identity Matrix ◽  
The Matrix

The simple random walk with mixed barriers at state $ 0 $ and state $ n $ defined on non-negative integers has transition matrix $ P $ with transition probabilities $ p_{ij} $. Matrix $ Q $ is obtained from matrix $ P $ when rows and columns at state $ 0 $ and state $ n $ are deleted . The fundamental matrix $ B $ is the inverse of the matrix $ A = I -Q $, where $ I $ is an identity matrix. The expected reflecting and absorbing time and reflecting and absorbing probabilities can be easily deduced once $ B $ is known. The fundamental matrix can thus be used to calculate the expected times and probabilities of NCD's.

The fundamental matrix for a certain random walk

1999 ◽  
Vol 36 (02) ◽  
pp. 320-333
Author(s):  
Howard M. Taylor
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Explicit Formula ◽  
Fundamental Matrix ◽  
Simple Random Walk ◽  
Limiting Behavior ◽  
Point Of Entry ◽  
Discrete Lattices ◽  
The Green Function ◽  
The Right

Consider the random walk {Sn} whose summands have the distributionP(X=0) = 1-(2/π), andP(X= ±n) = 2/[π(4n2−1)], forn≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean timexkfor the random walk starting fromS0=kto exit the interval. The explicit formula yields the limiting behavior ofxkasN→ ∞ withkfixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By lettingN→ ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting fromk= 0 results. The distributions for some related random variables are also discovered.Applications to stress concentration calculations in discrete lattices are briefly reviewed.

scholarly journals TRANSITION MATRIX MONTE CARLO

1999 ◽  
Vol 10 (08) ◽  
pp. 1563-1569 ◽  
Author(s):  
ROBERT H. SWENDSEN ◽  
BRIAN DIGGS ◽  
JIAN-SHENG WANG ◽  
SHING-TE LI ◽  
CHRISTOPHER GENOVESE ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Transition Matrix ◽  
Transition Probabilities ◽  
Simulation Technique ◽  
Efficient Approach ◽  
Complementary Approach ◽  
Combining Data ◽  
The Matrix

Although histogram methods have been extremely effective for analyzing data from Monte Carlo simulations, they do have certain limitations, including the range over which they are valid and the difficulties of combining data from independent simulations. In this paper, we describe a complementary approach to extracting information from Monte Carlo simulations that uses the matrix of transition probabilities. Combining the Transition Matrix with an N-fold way simulation technique produces an extremely flexible and efficient approach to rather general Monte Carlo simulations.

scholarly journals Uniform asymptotic estimates of transition probabilities on combs

2003 ◽  
Vol 75 (3) ◽  
pp. 325-354 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Local Limit ◽  
Simple Random Walk ◽  
Space Time ◽  
Asymptotic Estimates ◽  
Asymptotical Behaviour ◽  
Better Than ◽  
Gaussian Estimates

AbstractWe investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular, we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting sub-Gaussian estimates involving the spectral and walk dimensions of the graph.

The fundamental matrix for a certain random walk

1999 ◽  
Vol 36 (2) ◽  
pp. 320-333 ◽  
Author(s):  
Howard M. Taylor
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Explicit Formula ◽  
Fundamental Matrix ◽  
Simple Random Walk ◽  
Limiting Behavior ◽  
Point Of Entry ◽  
Discrete Lattices ◽  
The Green Function ◽  
The Right

Consider the random walk {Sn} whose summands have the distribution P(X=0) = 1-(2/π), and P(X = ± n) = 2/[π(4n2−1)], for n ≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean time xk for the random walk starting from S0 = k to exit the interval. The explicit formula yields the limiting behavior of xk as N → ∞ with k fixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By letting N → ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting from k = 0 results. The distributions for some related random variables are also discovered.Applications to stress concentration calculations in discrete lattices are briefly reviewed.

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

scholarly journals Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 729
Author(s):  
Miquel Montero
Keyword(s):  
Random Walks ◽  
Stereographic Projection ◽  
Transition Probabilities ◽  
Simple Random Walk ◽  
General Introduction ◽  
Elliptic Case ◽  
One Step ◽  
Underlying Geometry ◽  
Step Transition ◽  
Wide Family

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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