TREES WITH RANDOM CONDUCTIVITIES AND THE (RECIPROCAL) INVERSE GAUSSIAN DISTRIBUTION

Equipping the edges of a finite rooted tree with independent resistanc es that are inverse Gaussian far interior edges and reciprocal inverse Gaussian for terminal edges makes it possible, for suitable constella tions of the parameters, to show that the total resistance is reciproc al inverse Gaussian (Barndorff-Nielsen 1994). This result is extended to infinite trees. Also, a connection to Brownian diffusion is establi shed and, for the case of finite trees, an exact distributional and in dependence result is derived for the conditional model given the total resistance.