Wald's identity and absorption probabilities for two-dimensional random walks

1965 ◽  
Vol 61 (3) ◽  
pp. 747-762 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Double Series ◽  
Moment Generating Function ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Region ◽  
Discrete Random Variables ◽  
Image Position ◽  
Wald's Identity ◽  
Absorption Probabilities

SummarySuppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), whereand we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.

A two-dimensional random walk in the presence of a partially reflecting barrier

1974 ◽  
Vol 11 (01) ◽  
pp. 199-205
Author(s):  
Noel Cressie
Keyword(s):  
Random Walk ◽  
Two Dimensional ◽  
Absorption Probabilities

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.

A two-dimensional random walk in the presence of a partially reflecting barrier

1974 ◽  
Vol 11 (1) ◽  
pp. 199-205 ◽  
Author(s):  
Noel Cressie
Keyword(s):  
Random Walk ◽  
Two Dimensional ◽  
Absorption Probabilities

A general two-dimensional random walk is considered with a barrier along the y-axis. Absorption probabilities are derived when the barrier is absorbing, and when it is semi-reflecting.

THE HELIX-COIL TRANSITION IN A DOUBLE-STRANDED POLYNUCLEOTIDE AND THE TWO-DIMENSIONAL RANDOM WALK

2012 ◽  
Vol 26 (13) ◽  
pp. 1250083
Author(s):  
G. N. HAYRAPETYAN ◽  
V. F. MOROZOV ◽  
V. V. PAPOYAN ◽  
S. S. POGHOSYAN ◽  
V. B. PRIEZZHEV
Keyword(s):  
Random Walk ◽  
Hydrogen Bonds ◽  
Polymer Chains ◽  
Square Lattice ◽  
Two Dimensional ◽  
Competing Interactions ◽  
New Approach ◽  
Rich Phase ◽  
Coil Transition ◽  
The Rich

The helix-coil transition in a double-stranded homopolynucleotide is considered. The new approach to the melted loops account is proposed. The relative distance between the corresponding monomers of two polymer chains is modeled by the two-dimensional random walk on the square lattice. Returns of the random walk to the origin describe the formation of hydrogen bonds between complementary units. To take into account the interaction of monomers inside the chains, we consider various regimes of return to the origin. One of them involves two competing interactions and demonstrates a nontrivial sharp denaturation transition. The rich phase behavior of the double-stranded homopolynucleotide is discussed in terms of the proposed approach.

A random walk problem

1960 ◽  
Vol 56 (4) ◽  
pp. 390-392 ◽  
Author(s):  
J. Gillis
Keyword(s):  
Random Walk ◽  
Legendre Polynomial ◽  
Rectangular Lattice ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Image Position ◽  
The Right ◽  
Step Turn ◽  
Previous Step

We consider a random walk on a two-dimensional rectangular lattice in which steps are strictly between nearest neighbour points. The conditions of the walk are that the walker must, at each step, turn either to the right or to the left of his previous step with respective probabilities ½(1+α), ½(1−α), (≤ α ≤ 1). To fix the ideas it is assumed that he starts from the origin and the probability of each of the four possible starting directions is ¼. If Ar denotes the probability of return to the origin after r steps we shall show thatwhere β = ½(α + α−1) and Pn is the nth Legendre polynomial. It is clear that Ar is zero for r ≢ 0(mod 4).

Some explicit results for an asymmetric two-dimensional random walk

1963 ◽  
Vol 59 (2) ◽  
pp. 451-462 ◽  
Author(s):  
V. D. Barnett
Keyword(s):  
Random Walk ◽  
Lattice Point ◽  
General Interest ◽  
Two Dimensional ◽  
General Lattice ◽  
Methods Of Analysis ◽  
Starting Point ◽  
Accessible Point ◽  
Absorption Probabilities

AbstractThree distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

Two‐Dimensional V‐VI Binary Nanosheets with Square Lattice: A Theoretical Investigation

2021 ◽  
Author(s):  
Xin Qiao ◽  
Xiaodong Lv ◽  
Yinan Dong ◽  
Yanping Yang ◽  
Fengyu Li
Keyword(s):  
Theoretical Investigation ◽  
Square Lattice ◽  
Two Dimensional

scholarly journals Structural Disorder and Collective Behavior of Two-Dimensional Magnetic Nanostructures

Nanomaterials ◽  
2021 ◽  
Vol 11 (6) ◽  
pp. 1392
Author(s):  
David Gallina ◽  
G. M. Pastor
Keyword(s):  
Interaction Energy ◽  
Time Evolution ◽  
Magnetic Nanostructures ◽  
Thermal Quenching ◽  
Structural Disorder ◽  
Square Lattice ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Two Dimensional ◽  
Energy Landscapes

Structural disorder has been shown to be responsible for profound changes of the interaction-energy landscapes and collective dynamics of two-dimensional (2D) magnetic nanostructures. Weakly-disordered 2D ensembles have a few particularly stable magnetic configurations with large basins of attraction from which the higher-energy metastable configurations are separated by only small downward barriers. In contrast, strongly-disordered ensembles have rough energy landscapes with a large number of low-energy local minima separated by relatively large energy barriers. Consequently, the former show good-structure-seeker behavior with an unhindered relaxation dynamics that is funnelled towards the global minimum, whereas the latter show a time evolution involving multiple time scales and trapping which is reminiscent of glasses. Although these general trends have been clearly established, a detailed assessment of the extent of these effects in specific nanostructure realizations remains elusive. The present study quantifies the disorder-induced changes in the interaction-energy landscape of two-dimensional dipole-coupled magnetic nanoparticles as a function of the magnetic configuration of the ensembles. Representative examples of weakly-disordered square-lattice arrangements, showing good structure-seeker behavior, and of strongly-disordered arrangements, showing spin-glass-like behavior, are considered. The topology of the kinetic networks of metastable magnetic configurations is analyzed. The consequences of disorder on the morphology of the interaction-energy landscapes are revealed by contrasting the corresponding disconnectivity graphs. The correlations between the characteristics of the energy landscapes and the Markovian dynamics of the various magnetic nanostructures are quantified by calculating the field-free relaxation time evolution after either magnetic saturation or thermal quenching and by comparing them with the corresponding averages over a large number of structural arrangements. Common trends and system-specific features are identified and discussed.

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