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We study the mean number of encounters up to time t, E_N(t), taking place ina subspace with dimension d* of a d-dimensional lattice, for N independentrandom walkers starting simultaneously from the same origin. E_N is firstevaluated analytically in a continuum approximation and numerically throughMonte Carlo simulations in one and two dimensions. Then we introduce the notionof the intersection of strongly anisotropic fractals and use it to calculatethe long-time behaviour of E_N.

Mean number of encounters of N random walkers and intersection of strongly anisotropic fractals