A rigorous upper bound of a scalar self-propagator in a random lattice ensemble in higher dimensions

Abstract

Using an improved weight for a scalar field on a random lattice, it is rigorously proved that the self-propagator, averaged over an ensemble of random lattices with site density ρ, is bounded from above inD dimensions (D>2) i.e.:

$$\begin{array}{*{20}c} {\Delta _0 \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \left( {4 + \frac{D}{{2\left( {D - 2} \right)}}\omega _D^{{2 \mathord{\left/ {\vphantom {2 {D - 1}}} \right. \kern-\nulldelimiterspace} {D - 1}}} D!^{{2 \mathord{\left/ {\vphantom {2 {D - 1}}} \right. \kern-\nulldelimiterspace} {D - 1}}} D^{{D \mathord{\left/ {\vphantom {D {2 - 1}}} \right. \kern-\nulldelimiterspace} {2 - 1}}{{ - 2} \mathord{\left/ {\vphantom {{ - 2} D}} \right. \kern-\nulldelimiterspace} D}} (D + 1)^{{D \mathord{\left/ {\vphantom {D {2{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}{{ - 1} \mathord{\left/ {\vphantom {{ - 1} D}} \right. \kern-\nulldelimiterspace} D}}}} \right. \kern-\nulldelimiterspace} {2{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}{{ - 1} \mathord{\left/ {\vphantom {{ - 1} D}} \right. \kern-\nulldelimiterspace} D}}}} } \right)} \\ { \cdot D^{D - 2{{ + 2} \mathord{\left/ {\vphantom {{ + 2} D}} \right. \kern-\nulldelimiterspace} D}} \omega _D^{2{{ - 2} \mathord{\left/ {\vphantom {{ - 2} D}} \right. \kern-\nulldelimiterspace} D}} \Gamma \left( {D - 1 + \frac{2}{D}} \right)\rho ^{{{1 - 2} \mathord{\left/ {\vphantom {{1 - 2} D}} \right. \kern-\nulldelimiterspace} D}} ,} \\ \end{array}$$

where ω _D is the solid angle inD dimensions.