 
Summary: Percolationlike Scaling Exponents for Minimal
Paths and Trees in the Stochastic Mean Field
Model
David J. Aldous
Department of Statistics
367 Evans Hall # 3860
U.C. Berkeley CA 94720
aldous@stat.berkeley.edu
August 6, 2004
Abstract
In the mean field (or random link) model there are n points and
interpoint distances are independent random variables. For 0 < # < #
and in the n # # limit, let #(#) = 1/n× (maximum number of steps
in a path whose average steplength is # #). The function #(#) is
analogous to the percolation function in percolation theory: there is
a critical value # # = e 1 at which #(·) becomes nonzero, and (pre
sumably) a scaling exponent # in the sense #(#) # (#  # # ) # . Recently
developed probabilistic methodology (in some sense a rephrasing of
the cavity method developed in the 1980s by M’ezard and Parisi) pro
vides a simple albeit nonrigorous way of writing down such functions
