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M V Tamm1, V I Stadnichuk2, A M Ilyina2
1Physics Department, Moscow State University, 119991, Moscow, Russia and Department of Applied Mathematics, National Research University Higher School of Economics, 101000, Moscow, Russia.
Abstract:
We consider two random walkers starting at the same time t=0 from different points in space separated by a given distance R. We compute the average volume of the space visited by both walkers up to time t as a function of R and t and dimensionality of space d. For d<4, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable R/√t. We provide general integral formulas for scaling functions for arbitrary dimensionality d<4. In contrast, we show that no scaling function exists for higher dimensionalities d≥4.
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