Statistics of random optical field generated by a random walk with a finite number of steps

We show that a branching random walk that is supercritical on is also supercritical, in a rather strong sense, when restricted to a large enough finite ball of This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.

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Comparison of the semiclassical and quantum optical field dynamics in a pulse-excited optical cavity with a finite number of quantum emitters

Physical Review B ◽

10.1103/physrevb.104.205308 ◽

2021 ◽

Vol 104 (20) ◽

Author(s):

K. Jürgens ◽

F. Lengers ◽

D. Groll ◽

D. E. Reiter ◽

D. Wigger ◽

...

Keyword(s):

Finite Number ◽

Optical Cavity ◽

Optical Field ◽

Quantum Optical ◽

Quantum Emitters

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On a storage allocation model with finite capacity

European Journal of Applied Mathematics ◽

10.1017/s0956792516000048 ◽

2016 ◽

Vol 27 (5) ◽

pp. 738-755 ◽

Cited By ~ 2

Author(s):

EUNJU SOHN ◽

CHARLES KNESSL

Keyword(s):

Random Walk ◽

Steady State ◽

Finite Number ◽

Poisson Process ◽

Joint Distribution ◽

Marginal Distribution ◽

Finite Capacity ◽

Storage Allocation ◽

Allocation Model ◽

Numerical Studies

We consider a storage allocation model with a finite number of storage spaces. There are m primary spaces that are ranked {1,2,. . .,m} and R secondary spaces ranked {m + 1, m + 2,. . .,m + R}. Items arrive according to a Poisson process, occupy a space for a random exponentially distributed time, and an arriving item takes the lowest ranked available space. Letting N1 and N2 denote the numbers of occupied primary and secondary spaces, we study the joint distribution Prob[N1 = k, N2 = r] in the steady state. The joint process (N1, N2) behaves as a random walk in a lattice rectangle. We shall obtain explicit expressions for the distribution of (N1, N2), as well as the marginal distribution of N2. We also give some numerical studies to illustrate the qualitative behaviors of the distribution(s). The main contribution is to study the effects of a finite secondary capacity R, whereas previous studies had R = ∞.

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The speed of a random walk excited by its recent history

Advances in Applied Probability ◽

10.1017/apr.2015.14 ◽

2016 ◽

Vol 48 (1) ◽

pp. 215-234 ◽

Cited By ~ 1

Author(s):

Ross G. Pinsky

Keyword(s):

Random Walk ◽

Finite Number ◽

Threshold Level ◽

Recent History ◽

Threshold Levels ◽

One Step ◽

The Right ◽

Positive Integers

Abstract Let N and M be positive integers satisfying 1≤ M≤ N, and let 0< p0 < p1 < 1. Define a process {Xn}n=0∞ on ℤ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first N steps, the process behaves like a random walk that jumps to the right with probability p0 and to the left with probability 1-p0. At subsequent steps the jump mechanism is defined as follows: if at least M out of the N most recent jumps were to the right, then the probability of jumping to the right is p1; however, if fewer than M out of the N most recent jumps were to the right then the probability of jumping to the right is p0. We calculate the speed of the process. Then we let N→ ∞ and M/N→ r∈[0,1], and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number l of threshold levels, (Mi,pi) i=1l, above the pre-threshold level p0, as well as for one model with l=N such thresholds.

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Diffusional Coating of Nanoparticles

MRS Proceedings ◽

10.1557/proc-737-f3.13 ◽

2002 ◽

Vol 737 ◽

Author(s):

James P. Lavine

Keyword(s):

Random Walk ◽

Finite Number ◽

Concentration Dependence ◽

Time Dependent ◽

Initial Number ◽

Surface Absorption ◽

Absorption Probability ◽

Diffusional Coating ◽

Range Of Values ◽

Number Of Particles

ABSTRACTThe time-dependent diffusion of particles to an absorbing sphere is investigated with three random walk models. The first uses consecutive independent particles and finds the capture time distributions are exponential for a range of values of the surface absorption probability. The next two models are of the ensemble variety and assume that only a finite number of particles may be absorbed by the sphere. These models investigate depletion effects and concentration dependence. The latter is probed by varying the initial number of diffusing particles. It is found

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Geometric random walk of finite number of agents under constant variance

Journal of Statistical Mechanics Theory and Experiment ◽

10.1088/1742-5468/aa6ce6 ◽

2017 ◽

Vol 2017 (5) ◽

pp. 053204 ◽

Cited By ~ 1

Author(s):

Ryosuke Yano

Keyword(s):

Random Walk ◽

Finite Number ◽

Constant Variance ◽

Geometric Random Walk

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A Note on Branching Random Walks on Finite Sets

Journal of Applied Probability ◽

10.1017/s0021900200000243 ◽

2005 ◽

Vol 42 (01) ◽

pp. 287-294 ◽

Cited By ~ 2

Author(s):

Thomas Mountford ◽

Rinaldo B. Schinazi

Keyword(s):

Random Walk ◽

Finite Number ◽

Random Walks ◽

Ecological Model ◽

Critical Value ◽

Strong Sense ◽

Branching Random Walk ◽

Finite Sets ◽

Branching Random Walks ◽

Number Of Species

We show that a branching random walk that is supercritical on is also supercritical, in a rather strong sense, when restricted to a large enough finite ball of This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.

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Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100051669 ◽

1974 ◽

Vol 32 ◽

pp. 330-331

Author(s):

R. A. Crowther

Keyword(s):

Image Reconstruction ◽

Finite Number ◽

Density Distribution ◽

Electron Micrographs ◽

Mathematical Problem ◽

Three Dimensional ◽

Ideal Case ◽

Dimensional Image ◽

General Object ◽

The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

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