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We construct Holographic Space-time models that reproduce the dynamics of $1+ 1$ dimensional string theory. The necessity for a dilaton field in the $1 +1$ effective Lagrangian for classical geometry, the appearance of fermions, andeven the form of the universal potential in the canonical $1$ matrix model,follow from general HST considerations. We note that 't Hooft's ansatz for theleading contribution to the black hole S-matrix, accounts for the entireS-matrix in these models in the limit that the string scale coincides with thePlanck scale, up to transformations between near horizon and asymptoticcoordinates. These $1 + 1$ dimensional models are describable as decouplinglimits of the near horizon geometry of higher dimensional extremal black holesor black branes, and this suggests that deformations of the simplest model areequally physical. After proposing a notion of "relevant deformations", wedescribe deformations, which contain excitations corresponding to lineardilaton black holes, some of which can be considered as UV completions of theCGHS model. We study the question of whether the AMPS paradox can be formulatedin those models. It cannot, because the classical in-fall time to thesingularity of linear dilaton black holes, is independent of the black holemass. This result is reproduced by our HST models. We argue that it is relatedto the absence of quasi-normal modes of these black hole solutions, which isitself related to the fact that the horizon has zero area. This is compatiblewith the resolution of the AMPS paradox proposed in previous work withFischler, according to which the compatibility conditions of HST identify thelong non-singular sojourn of observers behind the horizon, with the dynamics ofequilibration on the horizon as seen by a detector which has not yet fallenthrough the horizon.

Holographic Space-time Models in $1 + 1$ Dimensions