ISSN:
1572-9613
Keywords:
Random walks
;
prescribed return expectance
;
Ornstein-Zernike equation in random walks theory
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract We consider the problem of random flights in Euclidean space defined by a series of displacements, ri, the magnitude and direction of each being independent of all the preceding ones. The displacements are not restricted to prescribed lattice sites. We begin with some new results generalizing a well-known latticewalk relation between the probability of return to the origin and the expected number of times the origin is visited in the course of a random walk. We go on to consider flights for which the hit expectancy is prescribed within a hypersphere of radiusR centered at the flight origin. For uniform hit expectance density (i.e., hit expectancy proportional to volume size) within the sphere, we solve the problem in three and five dimensions for a certain class of displacement probability densities that are prescribed only for displacement distancesr greater thanR. For each such displacement probability τ(r), we find both the value of the hit expectance density and the form of the displacement probability density forr〈R that are dictated by the constraint of uniform hit expectance density within the sphere of radiusR. In an appendix, we show the way the common appearance of integral equations of Ornstein-Zernike type in problems of random flight, liquid and lattice-gas structure, and percolation theory yield certain corresponding results in all three areas.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01018840
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