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Intersection properties of simple random walks: A renormalization group approach. (English) Zbl 0573.60065

We study estimates for the intersection probability, g(m), of two simple random walks on lattices of dimension \(d=4\), 4-\(\epsilon\) as a problem in Euclidean field theory. We rigorously establish a renormalization group flow equation for g(m) and bounds on the \(\beta\)-function which show that, in \(d=4\), g(m) tends to zero logarithmically as the killing rate (mass) m tends to zero, and that the fixed point, \(g^*\), in \(d=4- \epsilon\) is bounded by \(const'\epsilon \leq g^*\leq const \epsilon\). Our methods also yield estimates on the intersection probability of three random walks in \(d=3\), 3-\(\epsilon\). For \(\epsilon =0\), these results were first obtained by G. F. Lawler, ibid. 86, 539-554 (1982; Zbl 0502.60057).

MSC:

60G50 Sums of independent random variables; random walks
81P20 Stochastic mechanics (including stochastic electrodynamics)
81T17 Renormalization group methods applied to problems in quantum field theory

Citations:

Zbl 0502.60057
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References:

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