Exact results for the Potts model in two dimensions

By considering the zeros of the partition function, we establish the following results for the Potts model on the square, triangular, and honeycomb lattices: (i) We show that there exists only one phase transition; (ii) we give an exact determination of the critical point; (iii) we prove the exponential decay of the correlation functions, in one direction at least, for all temperatures above the critical point. The results are established forq 4, whereq is the number of components.Work supported by the Fond. National Suisse de la Recherche Scientifique.Work supported in part by NSF Grant No. DMR 76-20643.     

Exact results for curvature-driven coarsening in two dimensions

We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than A has, for large time t, the scaling form Nh(A,t)=2c/(A+lambdat), demonstrating the validity of dynamical scaling in this system, where c=1/8pisquare root 3 is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of A/lambdat. Identical forms are obtained for coarsening from a critical initial state, but with c replaced by c/2.     

Self-avoiding walk in 5 or more dimensions

Using an expansion based on the renormalization group philosophy we prove that for aT step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT.Dedicated to the memory of Kurt Symanzik whose profound contributions have guided and inspired usWork partially supported by N.S.F. Grant DMR 81-00417A. P. Sloan Foundation Fellow. Work partially supported by N.S.F. Grant MCS 82-02115     

Analytically modelling the effects of interaction between two or more rectangular nanomagnets

In a recent paper (Goode and Rowlands J. Magn. Magn. Mater. 295 (2005) 197–218), the micromagnetic equations which govern the magnetization distribution have been studied for rectangular nano-sized magnets where the magnetization is nearly uniform. Analytical solutions to these equations have been obtained in the form of Fourier series in which only the first few terms in the series are necessary to give results accurate to a few percent. In this paper, the above method is extended to include the effects of interaction between two or more rectangular nanomagnets. The near-uniform nature of the magnetization distributions is shown to change depending on the distance the nanomagnets are apart from each other. To estimate at what distance between the nanomagnets this interaction becomes important and therefore must be included in the analysis, the demagnetizing and interaction energies are compared for an array of uniformly magnetized rectangular nanomagnets.     

Some considerations of fluid interfaces in two dimensions

Basic assumptions of the capillary wave theory of fluid interfaces are examined critically as a function of space dimensionalityd. When the predictions of capillary wave theory are compared with those of the nonclassical Maxwell-van der Waals theory, agreement is found ind=3 and 4, but strong disagreement occurs ind=2. It is shown that the total effective mass density obtained from the Hamiltonian describing the collective capillary wave excitations has a logarithmic divergence ind=2. This result suggests the possibility of anomalous behavior for fluid interfaces ind=2.     

Statistical properties of chaotic wavefunctions in two and more dimensions

Starting with Berry's hypothesis for fixed energy waves in a classically chaotic system, and casting it in a Green function form, we derive wavefunction correlations and density matrices for few or many particles. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N↦∞. This arises through a little known asymptotic limit of Bessel functions. The Berry random wave hypothesis in many dimensions may be viewed as an alternative approach to quantum statistical mechanics, when extended to include constraints and potentials.     

Some fractal properties of the percolating backbone in two dimensions

A new algorithm is presented, based on elements of artificial intelligence theory, to determine the fractal properties of the backbone of the incipient infinite cluster. It is found that the fractal dimensionality of the backbone isd _f ^BB =1.61±0.01, the chemical dimensionality isd _t=1.40±0.01, and the fractal dimension of the minimum pathd _min=1.15 ± 0.02 for the two-dimensional triangular lattice.     

p-adic analyticity and virasoro algebras for conformal theories in more than two dimensions

The field of p-adic complex numbers has a much richer structure than the field of ordinary complex numbers. This is used in order to extend the powerful tools of two-dimensional conformal field theories to higher dimensions. It is thus proposed that critical systems in more than two dimensions be first studied over the p-adics and then, if possible, recovered by the adelic construction. It is further argued that this higher-dimensional p-adic analyticity may be the key to membrane theories. A natural ansatz for three-brane tree-scattering amplitudes, where p-adic analyticity is instrumental, is given as an explicit example.     

Some exact results for the Kondo lattice with infinite exchange interaction

Using an exact equivalence between the Kondo lattice with infinite J and the Hubbard model with infinite U, we show that the ground state of the Kondo lattice is non-magnetic for concentrations of conduction electrons close to 1, but there are still some magnetic regions even for J → ∞.     

Self-avoiding walk in five or more dimensions I. The critical behaviour

We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford5. We prove that the numberc _n ofn-step self-avoiding walks satisfiesc _n ~A ⁿ , where is the connective constant (i.e. =1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forc _n(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (^––Z)^1/2. The critical two-point function is shown to decay at least as fast as x^–2, and its Fourier transform is shown to be asymptotic to a multiple ofk ^–2 ask0 (i.e. =0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.Supported by the Nishina Memorial Foundation and NSF grant PHY-8896163.Supported by NSERC grant A9351     

Localization and interaction in two dimensions: The disordered Tomonaga model

The effect of electron-electron interactions on the localization behavior of two-dimensional disordered systems is investigated in the framework of a generalized version of Tomonaga's model. The main result is a closed expression for the dynamical conductivity which allows to predict the critical electron concentration below which localization sets in as a function of microscopic parameters only. Experimental verification is possible, although difficult.     

Exact entropy of dimer coverings for a class of lattices in three or more dimensions

We construct a class of lattices in three and higher dimensions for which the number of dimer coverings can be determined exactly using elementary arguments. These lattices are a generalization of the two-dimensional kagome lattice, and the method also works for graphs without translational symmetry. The partition function for dimer coverings on these lattices can be determined also for a class of assignments of different activities to different edges.     

Spin correlations in electron liquids: Analytical results for the local-field correction in two and three dimensions

We study the spin correlations in two- and three-dimensional electron liquids within the sum-rule version of the self-consistent field approach of Singwi, Tosi, Land, and Sj?lander. Analytic expressions for the spin-antisymmetric static structure factor and the corresponding local-field correction are obtained with density dependent coefficients. We calculate the spin-dependent pair-correlation functions, paramagnon dispersion, and static spin-response function within the present model, and discuss the spin-density wave instabilities in double-layer electron systems. Received: 22 September 1997 / Revised: 1 December 1997 / Accepted: 4 December 1997     

Finite-size corrections for inclined interfaces in two dimensions: Exact results for Ising and solid-on-solid models

Analysis of finite-size corrections for the surface tension and surface stiffness coefficients in two-dimensional models with inclined interfaces is presented. We obtain a universal leading contribution proportional to (lnL)/L for the 2D system of sizeL. By explicit calculations for restricted and unrestricted solid-on-solid models and the square lattice Ising model, we demonstrate the Gaussian nature of rough interfaces with fixed ends, and derive the leading 1/L-type corrections for appropriate surface quantities.