4 research outputs found
Dimensional Reduction for Directed Branched Polymers
Dimensional reduction occurs when the critical behavior of one system can be
related to that of another system in a lower dimension. We show that this
occurs for directed branched polymers (DBP) by giving an exact relationship
between DBP models in D+1 dimensions and repulsive gases at negative activity
in D dimensions. This implies relations between exponents of the two models:
(the exponent describing the singularity of the
pressure), and (the correlation length exponent of
the repulsive gas). It also leads to the relation ,
where is the Yang-Lee edge exponent. We derive exact expressions
for the number of DBP of size N in two dimensions.Comment: 7 pages, 1 eps figure, ref 24 correcte
Directed Branched Polymer near an Attractive Line
We study the adsorption-desorption phase transition of directed branched
polymer in dimensions in contact with a line by mapping it to a
dimensional hard core lattice gas at negative activity. We solve the model
exactly in 1+1 dimensions, and calculate the crossover exponent related to
fraction of monomers adsorbed at the critical point of surface transition, and
we also determine the density profile of the polymer in different phases. We
also obtain the value of crossover exponent in 2+1 dimensions and give the
scaling function of the sticking fraction for 1+1 and 2+1 dimensional directed
branched polymer.Comment: 19 pages, 4 figures, accepted for publication in J. Phys. A:Math. Ge
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.