Intersections of random walks. A direct renormalization approach

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 ℤ4, or three walks in ℤ3. These results are generalized toP walks in ℤ d *, $$d^* = \frac{{2P}}{{P - 1}}$$ ,P≧2. Ford<4, an infinite set of universal critical exponents σ L ,L≧1, are derived. They govern the asymptotic probability $$Z_L \sim S^{\sigma _L }$$ thatL “star walks” in ℝ d , with a common origin, do not intersect before timeS. The σ L 's are calculated up to orderO(e2), whered=4−e. This information is used to calculate the probability $$Z(G)$$ that a set of independent random walks in ℝ d or ℝ d ,d≦4, (respectivelyd≦3) form a given topological networks $$G$$ of multiple intersection points, in the absence of any other double point (respectively triple point). This is generalized to a network in $$d \leqq \frac{{2P}}{{P - 1}}$$ 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).