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We obtain the determinant expression for the time dependent transition probabilities in the totally asymmetric exclusion process with parallel update on the infinite lattice and a ring. Using the Bethe ansatz, we constructed the resolution of the identity operator, which allowed us to calculate the matrix elements of the evolution operator and its powers. This construction implies finding the spectrum of the operator, which is continuous for the infinite lattice and discrete for the ring. In the latter case we developed the method of summation over the roots of Bethe equations based on the multidimensional Cauchy residue theorem. |
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