Non-critical strings at high energy

We consider scattering amplitudes in non-critical string theory of $N$ external states in the limit where the energy of all external states is large compared to the string tension. We argue that the amplitudes are naturally complex analytic in the matter central charge $c$ and we propose to define the amplitudes for arbitrary value of $c$ by analytic continuation. We show that the high energy limit is dominated by a saddle point that can be mapped onto an equilibrium electro-static energy configuration of an assembly of $N$ pointlike (Minkowskian) charges, together with a density of charges arising from the Liouville field. We argue that the Liouville charges accumulate on segments of curves, and produce quadratic branch cuts on the worldsheet. The electro-statics problem is solved for string tree level in terms of hyper-elliptic integrals and is given explicitly for 3- and 4-point functions. We show that the high energy limit should behave in a string-like fashion with exponential dependence on the energy scale for generic values of $c$.