143 research outputs found
Combinatorics of Hard Particles on Planar Graphs
We revisit the problem of hard particles on planar random tetravalent graphs
in view of recent combinatorial techniques relating planar diagrams to
decorated trees. We show how to recover the two-matrix model solution to this
problem in this purely combinatorial language.Comment: 35 pages, 20 figures, tex, harvmac, eps
Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions
We address the general problem of hard objects on random lattices, and
emphasize the crucial role played by the colorability of the lattices to ensure
the existence of a crystallization transition. We first solve explicitly the
naive (colorless) random-lattice version of the hard-square model and find that
the only matter critical point is the non-unitary Lee-Yang edge singularity. We
then show how to restore the crystallization transition of the hard-square
model by considering the same model on bicolored random lattices. Solving this
model exactly, we show moreover that the crystallization transition point lies
in the universality class of the Ising model coupled to 2D quantum gravity. We
finally extend our analysis to a new two-particle exclusion model, whose
regular lattice version involves hard squares of two different sizes. The exact
solution of this model on bicolorable random lattices displays a phase diagram
with two (continuous and discontinuous) crystallization transition lines
meeting at a higher order critical point, in the universality class of the
tricritical Ising model coupled to 2D quantum gravity.Comment: 48 pages, 13 figures, tex, harvmac, eps
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
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