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The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.

A thresholdlike behavior of the far-infrared photoconductivity due to cyclotron resonance and a drastic deviation of the cyclotron-resonance line shape from a Lorentzian has been observed in n-GaAs at low temperatures by applying a high-power cw far-infrared laser. Both effects may consistently be explained in terms of generation-recombination-induced nonequilibrium phase transitions showing that, besides impact ionization of impurities, cyclotron resonance can critically control the conductivity of the semiconductors.PACS numbers: 72.20.Jv, 76.40. + bThe most important autocatalytic process in extrinsic semiconductors is impact ionization of impurities, which is responsible for the breakdown observed in the current-voltage characteristics of many high-purity semiconductors. 1 " 4 At low temperatures almost all carriers are bound to shallow impurities, and this yields a low conductance of the sample. At a critical electric field strength free carriers gain sufficient energy so that the impact ionization rate of shallow impurities exceeds the recombination rate for low carrier concentration, which results in a rapid increase of the current. The transition from the low-conducting state to the high-conducting state has been recognized as a nonequilibrium phase transition governed by nonlinear generation-recombination rate equations. 5,6 The order parameter of the transition corresponds to the steadystate free-carrier concentration n. By increase of the electric field E, and thereby the impact ionization probability per electron, the system is driven from the phase of low carrier concentration into that of high concentration. Thus the electric field can be identified as a control parameter of the phase transition. Dependent upon the material parameters and the excitation conditions, the phase transition may be of first or second order, which corresponds to a discontinuity in n (E) or dn/dE, respectively. The first case is connected with hysteresis of the current-voltage characteristic, and spatial phase coexistence, 6 while the second case corresponds to passing through a critical point at a threshold field E c , analogous to the Curie point of a ferromagnetic phase transition. Other types of nonequilibrium phase transitions associated with impurity breakdown at low temperatures, in particular the onset of chaos, 7 " 9 have also been observed recently.In a previous investigation of A2-GaAs it was shown that photoconductivity due to low-power far-infrared (FIR) excitation of cyclotron resonance probes a generalized susceptibility of the second-order nonequilibrium phase transition following a classical Curie-Weiss law. 10 In the present work we investigate high-power FIR laser irradiation under cyclotron-resonance conditions, and establish that the optical excitation probability of cyclotron resonance crisis an additional control parameter of the nonequilibrium phase transition, where F is the photon flux density and cr is the cyclotron-resonance absorption cross section which depends upon the magnetic field and the F...

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.