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Electron-Electron scattering and resistivity of ballistic multimode channels

We show that electron--electron scattering gives a positive contribution to the resistivity of ballistic multimode wires whose width is much smaller than their length. This contribution is not exponentially small at low temperatures and therefore may be experimentally observable. It scales with temperature as $T^2$ for three-dimensional channels and as $T^{5/2}$ for two-dimensional ones.