Mean conservation of nodal volume and connectivity measures for Gaussian ensembles
We study in depth the nesting graph and volume distribution of the nodal domains of a Gaussian field, which have been shown in previous works to exhibit asymptotic laws. A striking link is established between the asymptotic mean connectivity of a nodal domain (i.e.\ the vertex degree in its nesting graph) and the positivity of the percolation probability of the field, along with a direct dependence of the average nodal volume on the percolation probability. Our results support the prevailing ansatz that the mean connectivity and volume of a nodal domain is conserved for generic random fields in dimension $d=2$ but not in $d \ge 3$, and are applied to a number of concrete motivating examples.
We present a novel technique for accurately estimating the proportions of packet losses arising from collisions and from other sources of loss (channel noise, hidden nodes etc). Our approach is robust, makes use of local station-level measurements and requires no message passing. The required measurements are available in standard hardware as they are required for CSMA/CA operation. No channel quality probing is required which ensures energy efficiency