ISSN:
1572-9613
Keywords:
Random walk
;
Coulomb gas
;
orthogonal polynomials
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP t (p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP t (p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limitt≫p.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01016779
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