Spectral dimension from nonlocal dynamics on causal sets

We investigate the spectral dimension obtained from nonlocal continuum d’Alembertians derived from causal sets. We find a universal dimensional reduction to two dimensions, in all dimensions. We conclude by discussing the validity and relevance of our results within the broader context of quantum field theories based on these nonlocal dynamics.

Figure 1

A schematic diagram of the singular structure of $1/g(k^2)$ in the complex $k^0$ plane, including two contours, ${\mathrm{\Gamma }}_R$ and ${\mathrm{\Gamma }}_F$, which define the retarded and Feynman propagators, respectively. Note that before we are able to define the latter, we must first analytically continue $g(k^2)$ to the lower half the plane. The contour ${\mathrm{\Gamma }}_R$ will clearly encounter an obstruction under a Wick rotation, $k^0\to -i{k}^0$, while ${\mathrm{\Gamma }}_F$ will not.

Figure 2

From top to bottom we have the spectral dimension $d_s$ as a function of diffusion time $s$ for $\rho =1$ for $d=2$, 3, and 4, respectively. The dashed line represents the value of the Hausdorff dimension. In the $2d$ plot the spectral dimension can clearly be seen to interpolate between $d_s=2$ at short scales and $d_s=2$ at large scales, with a maximum occurring for $s\approx {\rho }^{1/2}$. The large scale asymptotic behavior and the maximum are not as clear in the three- and four-dimensional cases due to poorer numerics in $d>2$. Nonetheless one can see the short scale limit to $d_s=2$, a maximum for $s\approx 10{\rho }^{1/d}$, and a large scale limit which is very slowly asymptoting the Hausdorff dimension. These numerical results corroborate the analytical analysis of Sec. 3 and provide evidence for the behavior of $d_s$ interpolating between short and large scales.