Intersections of random walks. A direct renormalization approach. (English) Zbl 0652.60073

Duplantier, Bertrand

Commun. Math. Phys. 117, No. 2, 279-329 (1988).

Summary: Various intersection probabilities of independent random walks in d dimensions are calculated analytically by a direct renormalization method, adapted from polymer physics. This heuristic approach, based on S. F. Edwards’ continuum model [Proc. Phys. Soc. 85, 613 ff. (1965)], leads to a straightforward derivation and also to refinements of G. F. Lawler’s results [e.g.: Commun. Math. Phys. 97, 583-594 (1985; Zbl 0585.60069), and Contemp. Math. 41, 281-289 (1985; Zbl 0568.60066)] for the simultaneous intersections of two walks in \({\mathbb{Z}}^ 4,\) or three walks in \({\mathbb{Z}}^ 3.\) These results are generalized to P walks in \({\mathbb{Z}}^{d\quad *}\), d \(*=2P/(P-1)\), \(P\geq 2.\)
For \(d<4\), an infinite set of universal critical exponents \(\sigma_ L\), \(L\geq 1\), are derived. They govern the asymptotic probability \({\mathcal Z}_ L\sim S^{\sigma_ L}\) that L “star walks” in \({\mathbb{R}}^ d,\) with a common origin, do not intersect before time S. The \(\sigma_ L's\) are calculated up to order O(\(\epsilon\) 2), where \(d=4-\epsilon\). This information is used to calculate the probability \({\mathcal Z}({\mathcal G})\) that a set of independent random walks in \({\mathbb{R}}^ d \)or \({\mathbb{Z}}^ d,\) \(d\leq 4\), (respectively \(d\leq 3)\) form a given topological network \({\mathcal G}\) of multiple intersection points, in the absence of any other double point (respectively triple point). This is generalized to a network in \(d\leq 2P/(P-1)\) dimensions with the exclusion of P-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).

MSC:

60G50	Sums of independent random variables; random walks
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Keywords:

intersection probabilities of independent random walks; polymer physics; critical exponents; multiple intersection points

Citations:

Zbl 0585.60069; Zbl 0568.60066

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