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Hyperscaling violation and the shear diffusion constant

We consider holographic theories in bulk $(d+1)$-dimensions with Lifshitz and hyperscaling violating exponents $z,θ$ at finite temperature. By studying shear gravitational modes in the near-horizon region given certain self-consistent approximations, we obtain the corresponding shear diffusion constant on an appropriately defined stretched horizon, adapting the analysis of Kovtun, Son and Starinets. For generic exponents with $d-z-θ>-1$, we find that the diffusion constant has power law scaling with the temperature, motivating us to guess a universal relation for the viscosity bound. When the exponents satisfy $d-z-θ=-1$, we find logarithmic behaviour. This relation is equivalent to $z=2+d_{eff}$ where $d_{eff}=d_i-θ$ is the effective boundary spatial dimension (and $d_i=d-1$ the actual spatial dimension). It is satisfied by the exponents in hyperscaling violating theories arising from null reductions of highly boosted black branes, and we comment on the corresponding analysis in that context.