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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.

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