Oe. Barndorffnielsen et Ae. Koudou, TREES WITH RANDOM CONDUCTIVITIES AND THE (RECIPROCAL) INVERSE GAUSSIAN DISTRIBUTION, Advances in Applied Probability, 30(2), 1998, pp. 409-424
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.