Hyperscaling violation and the shear diffusion constant

Motivated by the analysis of Kovtun, Son and Starinets, we study the diffusion constant for theories with Lifshitz and hyperscaling violating exponents z and Θ. We study shear gravitational perturbations in the near-horizon region imposing certain self-consistent approximations on a suitably defined stretched horizon. This is effectively done by compactifying the hyperscaling violating Lifshitz theory along a spatial direction thus mapping shear gravitational perturbations to gauge field perturbations in the compactified theory. Through appropriately defined currents on the stretched horizon, we set up a diffusion equation and hence calculate the shear diffusion constant for these hyperscaling violating theories. It is observed for a certain class of hyperscaling violating theories i.e when z<4-Θ, we see the diffusion constant scales as a power law with respect to temperature suggesting universal behaviour in relation to the viscosity bound. When z≥4-Θ, diffusion constant scales logarithmically with temperature, suggesting a breakdown of our analysis in the near horizon regime.