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I apply the newly developed dynamical cluster approximation (DCA) to thecalculation of the electron and phonon dispersions in the two dimensionalHolstein model. In contrast to previous work, the DCA enables the effects ofspatial fluctuations (non-local corrections) to be examined. Approximationsneglecting and incorporating lowest-order vertex corrections are investigated.I calculate the phonon density of states, the renormalised phonon dispersion,the electron dispersion and electron spectral functions. I demonstrate howvertex corrections stabilise the solution, stopping a catastrophic softening ofthe $(\pi,\pi)$ phonon mode. A kink in the electron dispersion is found in thenormal state along the $(\zeta,\zeta)$ symmetry direction in both the vertex-and non-vertex-corrected theories for low phonon frequencies, correspondingdirectly to the renormalised phonon frequency at the $(\pi,0)$ point. This kinkis accompanied by a sudden drop in the quasi-particle lifetime. Vertex andnon-local corrections enhance the effects at large bare phonon frequencies.Comment: I am posting reprints of the final submitted versions of previous  articles to improve access. Here ARPES "kinks" are discussed. Article was  published in 2003. 17 pages, 9 figure

Electron and phonon dispersions of the two-dimensional Holstein model: effects of vertex and non-local corrections