Partial differential equations from integrable vertex models

In this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations. This mechanism is illustrated for the $U_q[\widehat{\mathfrak{sl}}(2)]$ invariant six-vertex model and the resulting partial differential equation is studied for particular values of the lattice length.Comment: 19 pages. v2: affiliation and references updated, minor changes, accepted for publication in J. Math. Phy

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

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

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

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

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

Double Potts chain and exact results for some two-dimensional models

A closed-form exact analytical solution for the q-state Potts model on a ladder 2 x oo with arbitrary two-, three-, and four-site interactions in a unit cell is presented. Using the obtained solution it is shown that the finite-size internal energy equation yields an accurate value of the critical temperature for the triangular Potts lattice with three-site interactions in alternate triangular faces. It is argued that the above equation is exact at least for self-dual models on isotropic lattices.Comment: 13 pages in latex and 2 ps figures. preprint ICTP IC/2000/176; ZhETF 120 (2001) (in press

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 $q_{c}$ 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

A possible combinatorial point for XYZ-spin chain

We formulate and discuss a number of conjectures on the ground state vectors of the XYZ-spin chains of odd length with periodic boundary conditions and a special choice of the Hamiltonian parameters. In particular, arguments for the validity of a sum rule for the components, which describes in a sense the degree of antiferromagneticity of the chain, are given.Comment: AMSLaTeX, 15 page