7,550 research outputs found
Fisher Zeroes and Singular Behaviour of the Two Dimensional Potts Model in the Thermodynamic Limit
The duality transformation is applied to the Fisher zeroes near the
ferromagnetic critical point in the q>4 state two dimensional Potts model. A
requirement that the locus of the duals of the zeroes be identical to the dual
of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of
specific heat to internal energy discontinuity at criticality and the
relationships between the discontinuities of higher cumulants and (ii)
identifies duality with complex conjugation. Conjecturing that all zeroes
governing ferromagnetic singular behaviour satisfy the latter requirement gives
the full locus of such Fisher zeroes to be a circle. This locus, together with
the density of zeroes is then shown to be sufficient to recover the singular
form of the thermodynamic functions in the thermodynamic limit.Comment: 10 pages, 0 figures, LaTeX. Paper expanded and 2 references added
clarifying duality relationships between discontinuities in higher cumulant
Tax evasion dynamics and Zaklan model on Opinion-dependent Network
Within the context of agent-based Monte-Carlo simulations, we study the
well-known majority-vote model (MVM) with noise applied to tax evasion on
Stauffer-Hohnisch-Pittnauer (SHP) networks. To control the fluctuations for tax
evasion in the economics model proposed by Zaklan, MVM is applied in the
neighborhood of the critical noise to evolve the Zaklan model. The
Zaklan model had been studied recently using the equilibrium Ising model. Here
we show that the Zaklan model is robust because this can be studied besides
using equilibrium dynamics of Ising model also through the nonequilibrium MVM
and on various topologies giving the same behavior regardless of dynamic or
topology used here.Comment: 14 page, 4 figure
Critical and Multicritical Semi-Random (1+d)-Dimensional Lattices and Hard Objects in d Dimensions
We investigate models of (1+d)-D Lorentzian semi-random lattices with one
random (space-like) direction and d regular (time-like) ones. We prove a
general inversion formula expressing the partition function of these models as
the inverse of that of hard objects in d dimensions. This allows for an exact
solution of a variety of new models including critical and multicritical
generalized (1+1)-D Lorentzian surfaces, with fractal dimensions ,
k=1,2,3,..., as well as a new model of (1+2)-D critical tetrahedral complexes,
with fractal dimension . Critical exponents and universal scaling
functions follow from this solution. We finally establish a general connection
between (1+d)-D Lorentzian lattices and directed-site lattice animals in (1+d)
dimensions.Comment: 44 pages, 15 figures, tex, harvmac, epsf, references adde
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
New critical frontiers for the Potts and percolation models
We obtain the critical threshold for a host of Potts and percolation models
on lattices having a structure which permits a duality consideration. The
consideration generalizes the recently obtained thresholds of Scullard and Ziff
for bond and site percolation on the martini and related lattices to the Potts
model and to other lattices.Comment: 9 pages, 5 figure
A model-framed evaluation of elephant effects on tree and fire dynamics in African savannas
There is a concern that high densities of elephants in southern Africa could lead to the overall reduction of other forms of biodiversity. We present a grid-based model of elephant-savanna dynamics, which differs from previous elephant-vegetation models by accounting for woody plant demographics, tree-grass interactions, stochastic environmental variables (fire and rainfall), and spatial contagion of fire and tree recruitment. The model projects changes in height structure and spatial pattern of trees over periods of centuries. The vegetation component of the model produces long-term tree-grass coexistence, and the emergent fire frequencies match those reported for southern African savannas. Including elephants in the savanna model had the expected effect of reducing woody plant cover, mainly via increased adult tree mortality, although at an elephant density of 1.0 elephant/km(2), woody plants still persisted for over a century. We tested three different scenarios in addition to our default assumptions. (1) Reducing mortality of adult trees after elephant use, mimicking a more browsing-tolerant tree species, mitigated the detrimental effect of elephants on the woody population. (2) Coupling germination success (increased seedling recruitment) to elephant browsing further increased tree persistence, and (3) a faster growing woody component allowed some woody plant persistence for at least a century at a density of 3 elephants/km(2). Quantitative models of the kind presented here provide a valuable tool for exploring the consequences of management decisions involving the manipulation of elephant population densities
Selfduality for coupled Potts models on the triangular lattice
We present selfdual manifolds for coupled Potts models on the triangular
lattice. We exploit two different techniques: duality followed by decimation,
and mapping to a related loop model. The latter technique is found to be
superior, and it allows to include three-spin couplings. Starting from three
coupled models, such couplings are necessary for generating selfdual solutions.
A numerical study of the case of two coupled models leads to the identification
of novel critical points
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
Two-dimensional O(n) model in a staggered field
Nienhuis' truncated O(n) model gives rise to a model of self-avoiding loops
on the hexagonal lattice, each loop having a fugacity of n. We study such loops
subjected to a particular kind of staggered field w, which for n -> infinity
has the geometrical effect of breaking the three-phase coexistence, linked to
the three-colourability of the lattice faces. We show that at T = 0, for w > 1
the model flows to the ferromagnetic Potts model with q=n^2 states, with an
associated fragmentation of the target space of the Coulomb gas. For T>0, there
is a competition between T and w which gives rise to multicritical versions of
the dense and dilute loop universality classes. Via an exact mapping, and
numerical results, we establish that the latter two critical branches coincide
with those found earlier in the O(n) model on the triangular lattice. Using
transfer matrix studies, we have found the renormalisation group flows in the
full phase diagram in the (T,w) plane, with fixed n. Superposing three
copies of such hexagonal-lattice loop models with staggered fields produces a
variety of one or three-species fully-packed loop models on the triangular
lattice with certain geometrical constraints, possessing integer central
charges 0 <= c <= 6. In particular we show that Benjamini and Schramm's RGB
loops have fractal dimension D_f = 3/2.Comment: 40 pages, 17 figure
Non-Universal Critical Behaviour of Two-Dimensional Ising Systems
Two conditions are derived for Ising models to show non-universal critical
behaviour, namely conditions concerning 1) logarithmic singularity of the
specific heat and 2) degeneracy of the ground state. These conditions are
satisfied with the eight-vertex model, the Ashkin-Teller model, some Ising
models with short- or long-range interactions and even Ising systems without
the translational or the rotational invariance.Comment: 17 page
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