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.

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 (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.

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.

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.

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 ON A BOUNDARY VALUE PROBLEM FOR ONE OVERDETERMINED SYSTEM IN A HALF-PLANE

2021 ◽  
Vol 4 (1) ◽  
pp. 226-231
Author(s):  
Mikhail V. Urev ◽  
Kholmatzhon Kh. Imomnazarov ◽  
Ilham K. Iskandarov
Keyword(s):  
Boundary Value Problem ◽  
Half Plane ◽  
Boundary Value ◽  
Generalized Solution ◽  
Stationary System ◽  
Dimensional Case ◽  
Mass Force ◽  
Two Dimensional ◽  
Overdetermined System ◽  
Significant Difference

This paper considers a boundary value problem for an overdetermined system of equations in a half-plane. This problem arises in particular when solving a stationary system of the two-velocity hydrodynamics with one pressure and homogeneous divergent conditions and the Dirichlet boundary conditions for two phase velocities, as well as in problems of electrodynamics. The generalized solution to a stationary system of the two-velocity hydrodynamics in the case of two-dimensional unbounded domains, for instance, in a half-plane, has a significant difference from the three-dimensional case. Namely, in the two-dimensional case for the velocities it is impossible to satisfy the pre-set conditions at infinity and the condition of boundedness at infinity is imposed. In this case, the medium is considered to be homogeneous, and the energy dissipation occurs due to the shear viscosities of the phases of the subsystems, and other effects are not discussed in this paper. The mass transfer occurs due to the mass force. With an appropriate choice of functional spaces, the existence and uniqueness of a generalized solution with an appropriate stability estimate has been proved.

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.

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