8,088 research outputs found
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
On kernel engineering via PaleyâWiener
A radial basis function approximation takes the form
where the coefficients a 1,âŠ,a n are real numbers, the centres b 1,âŠ,b n are distinct points in â d , and the function Ï:â d ââ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which Ï is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure ÎŒ for which the convolution Ï=ÎŒ Ï is a function of compact support, and when Ï is polyharmonic. The novelty of this construction is its use of the PaleyâWiener theorem to identify compact support via analysis of the Fourier transform of the new kernel Ï, so providing a new form of kernel engineering
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
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
The triangular Ising model with nearest- and next-nearest-neighbor couplings in a field
We study the Ising model on the triangular lattice with nearest-neighbor
couplings , next-nearest-neighbor couplings , and a
magnetic field . This work is done by means of finite-size scaling of
numerical results of transfer matrix calculations, and Monte Carlo simulations.
We determine the phase diagram and confirm the character of the critical
manifolds. The emphasis of this work is on the antiferromagnetic case , but we also explore the ferromagnetic regime for H=0.
For and H=0 we locate a critical phase presumably covering the
whole range . For , we locate a
plane of phase transitions containing a line of tricritical three-state Potts
transitions. In the limit this line leads to a tricritical model
of hard hexagons with an attractive next-nearest-neighbor potential
Protected Qubits and Chern Simons theories in Josephson Junction Arrays
We present general symmetry arguments that show the appearance of doubly
denerate states protected from external perturbations in a wide class of
Hamiltonians. We construct the simplest spin Hamiltonian belonging to this
class and study its properties both analytically and numerically. We find that
this model generally has a number of low energy modes which might destroy the
protection in the thermodynamic limit. These modes are qualitatively different
from the usual gapless excitations as their number scales as the linear size
(instead of volume) of the system. We show that the Hamiltonians with this
symmetry can be physically implemented in Josephson junction arrays and that in
these arrays one can eliminate the low energy modes with a proper boundary
condition. We argue that these arrays provide fault tolerant quantum bits.
Further we show that the simplest spin model with this symmetry can be mapped
to a very special Z_2 Chern-Simons model on the square lattice. We argue that
appearance of the low energy modes and the protected degeneracy is a natural
property of lattice Chern-Simons theories. Finally, we discuss a general
formalism for the construction of discrete Chern-Simons theories on a lattice.Comment: 20 pages, 7 figure
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
On spherical averages of radial basis functions
A radial basis function (RBF) has the general form
where the coefficients a 1,âŠ,a n are real numbers, the points, or centres, b 1,âŠ,b n lie in â d , and Ï:â d ââ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when Ï is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm Ï(x)=âxâ when d is an odd positive integer, the thin plate spline Ï(x)=âxâ2log ââxâ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243â264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the PaleyâWiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserlesâs study of geometric integration
Extended surface disorder in the quantum Ising chain
We consider random extended surface perturbations in the transverse field
Ising model decaying as a power of the distance from the surface towards a pure
bulk system. The decay may be linked either to the evolution of the couplings
or to their probabilities. Using scaling arguments, we develop a
relevance-irrelevance criterion for such perturbations. We study the
probability distribution of the surface magnetization, its average and typical
critical behaviour for marginal and relevant perturbations. According to
analytical results, the surface magnetization follows a log-normal distribution
and both the average and typical critical behaviours are characterized by
power-law singularities with continuously varying exponents in the marginal
case and essential singularities in the relevant case. For enhanced average
local couplings, the transition becomes first order with a nonvanishing
critical surface magnetization. This occurs above a positive threshold value of
the perturbation amplitude in the marginal case.Comment: 15 pages, 10 figures, Plain TeX. J. Phys. A (accepted
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
- âŠ