The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory (Q1062679)

A new approach is presented for the study of the probability that the random paths generated by two independent Brownian motions in \({\mathbb{R}}^ d\) intersect or, more generally, are within a short distance a of each other. The well known behavior of that function of a-above, below, and at the critical dimension \(d=4\), as well as further corrections, are derived here by means of a single renormalization group equation. The equation's derivation is expected to shed some light on the \(\beta\)-function of the \(\lambda \phi^ 4_ d\) quantum field theory.