Holographic Observables at Large $d$

We study holographically non-local observables in field theories at finite temperature and in the large $d$ limit. These include the Wilson loop, the entanglement entropy, as well as an extension to various dual extremal surfaces of arbitrary codimension. The large $d$ limit creates a localized potential in the near horizon regime resulting in a simplification of the analysis for the non-local observables, while at the same time retaining their qualitative physical properties. Moreover, we study the monotonicity of the coefficient $\alpha$ of the entanglement's area term, the so called area theorem. We find that the difference between the UV and IR of the $\alpha$-values, normalized with the thermal entropy, converges at large $d$ to a constant value which is obtained analytically. Therefore, the large $d$ limit may be used as tool for the study and (in)validation of the renormalization group monotonicity theorems. All the expectation values of the observables under study show rapid convergence to certain values as $d$ increases. The extrapolation of the large $d$ limit to low and intermediate dimensions shows good quantitative agreement with the numerical analysis of the observables.