We study the statistics of encounters of Lévy flights by introducing the concept of vicious Lévy flights--distinct groups of walkers performing independent Lévy flights with the process terminating upon the first encounter between walkers of different groups. We show that the probability that the process survives up to time t decays as t-α at late times. We compute α up to the second order in ε expansion, where ε=σ-d, σ is the Lévy exponent, and d is the spatial dimension. For d=σ, we find the exponent of the logarithmic decay exactly. Theoretical values of the exponents are confirmed by numerical simulations. Our results indicate that walkers with smaller values of σ survive longer and are therefore more effective at avoiding each other.

-We have studied the modification of the near-surface layers of a copper foil under the action of a volume gas discharge, which was generated in air at atmospheric pressure by nanosecond highvoltage pulses of both negative and positive polarity applied between the foil and an electrode with a small radius of curvature. It is established that the surface layer of the discharge-treated copper foil in the central region is cleaned from carbon contaminations, while oxygen atoms penetrate in depth of the foil. The depth of a cleaned layer depends on the polarity of voltage pulses. For the positive voltage polarity on the foil, the cleaning takes place up to a depth exceeding 50 nm, while oxygen penetrates up to a depth of about 25 nm.

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The asymptotic behavior of the survival or reunion probability of vicious walks with short-range interactions is generally well studied. In many realistic processes, however, walks interact with a long-ranged potential that decays in d dimensions with distance r as r(-d-σ). We employ methods of renormalized field theory to study the effect of such long-range interactions. We calculate the exponents describing the decay of the survival probability for all values of parameters σ and d to first order in the double expansion in ε=2-d and δ=2-d-σ. We show that there are several regions in the σ-d plane corresponding to different scalings for survival and reunion probabilities. Furthermore, we calculate the leading logarithmic corrections.

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