Diffusion Approximation for Random Walks on Anisotropic Lattices

1998 ◽  
Vol 35 (1) ◽  
pp. 206-212 ◽  
Author(s):  
Lajos Horváth
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Approximation ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Anisotropic Lattices ◽  
Anisotropic Lattice

We show that the horizontal position of a random walk on a two-dimensional anisotropic lattice converges weakly to a diffusion process.

Diffusion Approximation for Random Walks on Anisotropic Lattices

1998 ◽  
Vol 35 (01) ◽  
pp. 206-212
Author(s):  
Lajos Horváth
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Approximation ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Anisotropic Lattices ◽  
Anisotropic Lattice

We show that the horizontal position of a random walk on a two-dimensional anisotropic lattice converges weakly to a diffusion process.

Exploring activity-driven network with biased walks

2017 ◽  
Vol 28 (09) ◽  
pp. 1750111
Author(s):  
Yan Wang ◽  
Ding Juan Wu ◽  
Fang Lv ◽  
Meng Long Su
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Stationary Distribution ◽  
Significant Role ◽  
Search Strategy ◽  
Transition Probability ◽  
Edge Weighting ◽  
Coverage Function ◽  
Analytical Expressions

We investigate the concurrent dynamics of biased random walks and the activity-driven network, where the preferential transition probability is in terms of the edge-weighting parameter. We also obtain the analytical expressions for stationary distribution and the coverage function in directed and undirected networks, all of which depend on the weight parameter. Appropriately adjusting this parameter, more effective search strategy can be obtained when compared with the unbiased random walk, whether in directed or undirected networks. Since network weights play a significant role in the diffusion process.

On entrance times of a homogeneous N-dimensional random walk: an identity

1988 ◽  
Vol 25 (A) ◽  
pp. 321-333 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Performance Evaluation ◽  
Random Walk ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Mathematical Problem ◽  
Lattice Points ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Computer Performance ◽  
Hilbert Type

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

scholarly journals Law of large numbers for the drift of the two-dimensional wreath product

2021 ◽  
Author(s):  
Anna Erschler ◽  
Tianyi Zheng
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Wreath Product ◽  
Law Of Large Numbers ◽  
Abelian Groups ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Lamplighter Group ◽  
Large Numbers ◽  
Asymptotic Geometry

AbstractWe prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

STATIONARY PROCESSES THAT LOOK LIKE RANDOM WALKS— THE BOUNDED RANDOM WALK PROCESS IN DISCRETE AND CONTINUOUS TIME

2002 ◽  
Vol 18 (1) ◽  
pp. 99-118 ◽  
Author(s):  
João Nicolau
Keyword(s):  
Time Series ◽  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Interest Rates ◽  
Continuous Time ◽  
Financial Time Series ◽  
Finite Limit ◽  
Closed Expression ◽  
Continuous Time Process

Several economic and financial time series are bounded by an upper and lower finite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to infinity). Yet, some of these time series behave just like random walks. In this paper we propose a new approach that takes into account these ideas. We propose a discrete-time and a continuous-time process (diffusion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower finite limit. We derive for both cases the ergodic conditions, and for the diffusion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behavior can in fact be stationarity processes.

Deterministic Random Walks on the Two-Dimensional Grid

2009 ◽  
Vol 18 (1-2) ◽  
pp. 123-144 ◽  
Author(s):  
BENJAMIN DOERR ◽  
TOBIAS FRIEDRICH
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Walk Model ◽  
Two Dimensional ◽  
Expected Number ◽  
Fixed Order ◽  
The Difference ◽  
Propp Machine

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

On entrance times of a homogeneous N-dimensional random walk: an identity

1988 ◽  
Vol 25 (A) ◽  
pp. 321-333
Author(s):  
J. W. Cohen
Keyword(s):  
Performance Evaluation ◽  
Random Walk ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Mathematical Problem ◽  
Lattice Points ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Computer Performance ◽  
Hilbert Type

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2 N -ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (04) ◽  
pp. 717-726
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10 m steps, for m = 2, 3, 4, 5, 6, 7.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

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