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Scaling indentity for crossing Brownian motion in a Poissonian potential. (English) Zbl 0938.60099
Consider \(d\)-dimensional \((d\geq 2)\) Brownian motion in a truncated Poissonian potential. If Brownian motion starts at the original and ends in the closed ball with center \(y\) and radius 1, then the transverse fluctuation of the path is expected to be of order \(|y|^\xi\), whereas the distance fluctuation is of order \(|y|\chi\). Physics literature tells us that \(\xi\) and \(\chi\) should satisfy a scaling identity \(2\xi-1=\chi\). The author studies mathematically the conjecture and presents a result which is quite close to the conjecture.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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