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In recent years several approaches to quantum gravity have found evidence fora scale dependent spectral dimension of space-time varying from four at largescales to two at small scales of order of the Planck length. The first evidencecame from numerical results on four-dimensional causal dynamical triangulations(CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then littleprogress has been made in analytically understanding the numerical resultscoming from the CDT approach and showing that they remain valid when taking thecontinuum limit. Here we argue that the spectral dimension can be determinedfrom a model with fewer degrees of freedom obtained from the CDTs by "radialreduction". In the resulting "toy" model we can take the continuum limitanalytically and obtain a scale dependent spectral dimension varying from fourto two with scale and having functional behaviour exactly of the form which wasconjectured on the basis of the numerical results.

Dynamical dimensional reduction in toy models of 4D causal quantum gravity