The one-dimensional hard-core gas and three-dimensional branched polymers

Last time I talked about an elementary method for computing the Green's function
for the one-dimensional hard-core gas, and discussed its form near the critical
activity at z=-1/e.  This time I will discuss its relation with the Green's 
function for branched polymers in three dimensions.  I will derive the critical
exponents in three dimensions and give the scaling form of the branched polymer
Green's function.