Intersections of random walks. A direct renormalization approach |
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Authors: | Bertrand Duplantier |
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Institution: | (1) Service de Physique Théorique CEN Saclay, F-91191 Gif-sur-Yvette Cedex, France |
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Abstract: | Various intersection probabilities of independent random walks ind dimensions are calculated analytically by a direct renormalization method, adapted from polymer physics. This heuristic approach, based on Edwards' continuum model, leads to a straightforward derivation and also to refinements of Lawler's results for the simultaneous intersections of two walks in 4, or three walks in 3. These results are generalized toP walks in
d
*,
,P 2. Ford<4, an infinite set of universal critical exponents
L
,L 1, are derived. They govern the asymptotic probability
thatL star walks in
d
, with a common origin, do not intersect before timeS. The
L
's are calculated up to orderO( 2), whered=4– . This information is used to calculate the probability
that a set of independent random walks in
d
or
d
,d 4, (respectivelyd 3) form a given topological networks
of multiple intersection points, in the absence of any other double point (respectively triple point). This is generalized to a network in
dimension with exclusion ofP-tuple points. The method is quite general and can be used to calculate any critical intersection probability, and provides the probabilist with a large variety of exact results (yet to be proven rigorously). |
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Keywords: | |
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