From: Osher Doctorow, Ph.D. [log in to unmask], Monday Dec. 25, 2000 8:09PM

Some interesting papers relating to the uniform spanning tree are found in
the volume Perplexing Problems in Probability, Editors M. Bramson and R.
Durrett, Birkhauser: Boston 1999.  These papers include I. Benjamini's
"Large scale degrees and the number of spanning clusters for the uniform
spanning tree," pages 175-183, and "Loop-erased random walk," by G. F.
Lawler, pages 197-217.  Logic-based probability (LBP), which my wife Marleen
and I introduced in 1980, agrees with Shannon entropy methods in isolating
the uniform probability distribution as maximum entropy among continuous
distributions with two or less unknown parameters out of two - in fact, LBP
goes further and selects the uniform and other finite interval distributions
as the best maximum entropy distributions.  A detailed introduction to LBP
can be found in my recent paper "Magnetic monopoles, massive neutrinos and
gravitation via logic-experimental unification theory (LEUT) and
Kursunuglu's theory," pages 89-97 in the volume Quantum Gravity, Generalized
Theory of Gravitation, and Superstring Theory-Based Unification, Editors B.
N. Kursunuglu (Ph.D. Cambridge University under P. Dirac), S. L. Mintz, and
A. Perlmutter, Kluwer Academic/Plenum: New York, 2000.   Abstracts of 49 of
my papers can be found at http://www.logic.univie.ac.at, Institute for Logic
of the University of Vienna (select ABSTRACTS, then BY AUTHOR, then my
name).  The uniform spanning tree measure on a finite connected graph is the
uniform measure on the spanning tree space of the graph, while for infinite
graphs the measures are weak limits of uniform spanning tree measures from
subgraphs of finite type.  These measures arise as limits of random cluster
measures which generalize ising and Potts models of statistical physics and
Bernoulli percolation, but they appear to be applicable to various
operational research and computer problems and are closely related to
potential theory and random walks.  The uniform spanning tree has no
parameters but appears to behave like a critical model.   Loop-erased random
walk (LERW) is a process obtained by erasing loops from simple random walk
and exists on the integer lattice and has self-avoiding paths.  It is
closely related to uniform spanning trees, Q-state Potts models as Q --> 0,
and domino tilings.  It was thought to be originally in the same
universality class as self-avoiding random walk (the unifork measure on
random walk paths of fixed specified length given no self-intersection), but
this has been found to be incorrect.  Lawler and others are doing research
into loop-erased Brownian motion.  Brownian motion (BM), in turn, and its
close relatives fractional and fractal and reflecting and exursion- and
super-BM, tie in closely not only with the uniform distribution and uniform
empirical process but with the Gaussian and Poisson distributions and
processes in one of the most "explosive" areas of current research which
also covers applications ranging from estimating probability distribution
and density functions and change-of-distribution points of time series to
stochastic quantum theory.

Osher Doctorow
Doctorow Consultants, West Los Angeles College, etc.