Percolation and the Potts model

The Kasteleyn-Fortuin formulation of bond percolation as a lattice statistical model is rederived using an alternate approach. It is shown that the quantities of interest arising in the percolation problem, including the critical exponents, can be obtained from the solution of the Potts model. We also establish the Griffith inequality for critical exponents for the bond percolation problem.Work supported in part by NSF Grant No. D MR 76-20643.     

The stochastic geometry of invasion percolation

Invasion percolation, a recently introduced stochastic growth model, is analyzed and compared to the critical behavior of standardd-dimensional Bernoulli percolation. Various functions which measure the distribution of values accepted into the dynamically growing invaded region are studied. The empirical distribution of values accepted is shown to be asymptotically unity above the half-space threshold and linear below the point at which the expected cluster size diverges for the associated Bernoulli problem. An acceptance profile is defined and shown to have corresponding behavior. Quantities related to the geometry of the invaded region are studied, including the surface to volume ratio and the volume fraction. The former is shown to have upper and lower bounds in terms of the above defined critical points, and the latter is bounded above by the probability of connection to infinity at the half-space threshold. Provided that the critical regimes of Bernoulli percolation possess their anticipated properties, as is known to be the case in two dimensions, these results verify numerical predictions on the acceptance profile, establish the existence of a sharp surface to volume ratio and show that the invaded region has zero volume fraction. Large-time asymptotics are analyzed in terms of the probability that the invaded region accepts a value greater thanx at timen. This quantity is shown to be bounded below byh(x)exp[–c(x)n ^(d-1)/d ] forx above threshold, and to have an upper bound of the same form forx larger than a particular value (which coincides with the threshold ind=2). For two dimensions, it is also established that the infinite-time invaded region is essentially independent of initial conditions.National Science Foundation Postdoctoral Research Fellows. Work supported in part by the National Science Foundation under Grant No. PHY-82-03669John S. Guggenheim Memorial Fellow. Work supported in part by the National Science Foundation under Grant No. MCS-80-19384     

An intermediate phase with slow decay of correlations in one dimensional 1/|x−y|2 percolation,Ising and Potts models

We rigorously establish the existence of an intermediate ordered phase in one-dimensional 1/|x–y|² percolation, Ising and Potts models. The Ising model truncated two-point function has a power law decay exponent which ranges from its low (and high) temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary. Similar results are obtained for percolation and Potts models.Alfred P. Sloan Research Fellow. Research supported in part by NSF Grants No. PHY-8706420 and PHY-8645122Research supported in part by NSF Grant No. DMS-8514834 and AFOSR Contract F49620-86-C0130 at the Arizona Center for Math. Sciences     

Scaling relations for 2D-percolation

We prove that the relations 2D-percolation hold for the usual critical exponents for 2D-percolation, provided the exponents andv exist. Even without the last assumption various relations (inequalities) are obtained for the singular behavior near the critical point of the correlation length, the percolation probability, and the average cluster size. We show that in our models the above critical exponents have the same value for approach ofp to the critical probability from above and from below.Research supported by the NSF through grants to Cornell University and to the Institute for Mathematics and its Applications, Minneapolis     

Duality relation for 32-vertex model on the triangular lattice

The duality relation is derived for a vertex model on the triangular lattice. Vertex configurations are limited to the 32 that have an odd number of incoming arrows, and vertex energies are invariant to rotations ofp/3 and reversal of all arrows. Special cases of the model include the triangular Ising model and Baxter's three-spin model, for each of which the duality relation gives the critical temperature.Research supported in part by NSF Grant No. 33535X.     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

On the universality of geometrical and transport exponents of rigidity percolation

We develop a three-parameter position-space renormalization group method and investigate the universality of geometrical and transport exponents of rigidity (vector) percolation in two dimensions. To do this, we study site-bond percolation in which sites and bonds are randomly and independently occupied with probabilitiess andb, respectively. The global flow diagram of the renormalization transformation is obtained which shows that thegeometrical exponents of the rigid clusters in both site and bond percolation belong to the same universality class, and possibly that of random (scalar) percolation. However, if we use the same renormalization transformation to calculate the critical exponents of the elastic moduli of the system in bond and site percolation, we find them to be very different (although the corresponding values of the correlation length exponent are the same). This indicates that the critical exponent of the elastic moduli of rigidity percolation may not be universal, which is consistent with some of the recent numerical simulations.     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

A three-dimensional solid-liquid two-phase turbulence model with the effect of vegetation in non-orthogonal curvilinear coordinates

A three-dimensional k-ε-Ap solid-liquid two-phase two-fluid model with the effect of vegetation is solved numerically with a finite-volume method on an adaptive grid to study water-sediment movements and bed evolution in vegetated channels. The additional drag force and additional turbulence generation due to vegetation are added to the relevant control equations for simulating the interaction between vegetation and flow. The flow structure and the bed-topography changes in a 60° partly vegetated channel be...     

Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems IV

We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method ofinfrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.Research partially supported by US National Science Foundation under Grant MPS-75-11864Research partially supported by Canadian National Research Council under Grant A4015Research partially supported by US National Science Foundation under Grant MCS-75-21684-A01     

Stark effect in multielectron systems: Non-existence of bound states

We prove non-existence of bound states for a class ofN-body systems in homogeneous electric fields. This class includes atoms and Born—Oppenheimer molecules. This result in conjunction with a stability result of [HS] implies existence of resonances for such systems.Research partially supported by NSERC under Grant No. A7901 and NSF under Grant No. DMS8507040     

A mean field spin glass with short-range interactions

We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.Research supported by the NSF under Grant No. DMR-8314625Research supported by the DOE under Grant No. DE-AC02-83ER13044Research supported by the NSF under Grant No. DMR-8503544Research supported by the NSF under Grant No. DMR-8319301     

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.Research supported in part by NSF Vigre Grant DMS-0091946Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158Acknowledgement The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.     

Cluster and percolation inequalities for lattice systems with interactions

For a lattice gas with attractive potentials of finite range we use the inequalities of Fortuin, Kasteleyn, and Ginibre (FKG) to obtain fairly accurate upper and lower bounds on the equilibrium probabilityp(K) of finding the set of sitesK occupied and the adjacent sites unoccupied, i.e., on the probabilities of finding specified clusters. The probability that a given site, say the origin, is empty or belongs to a cluster of at mostl particles is shown to be a nonincreasing function of the fugacityz and the reciprocal temperature=(T) ^–1; hence the percolation probability is a nondecreasing function ofz and. If the forces are not entirely attractive, or if the ensemble is restricted by forbidding clusters larger than a certain size, the FKG inequalities no longer apply, but useful upper and lower bounds onp(K) can still be obtained if the density of the system and the size of the clusterK are not too large. They are obtained from a generalization of the Kirkwood-Salsburg equation, derived by regarding the system as a mixture of different types of cluster, whose only interaction is that they cannot overlap or touch.Research supported in part by AFOSR Grant #2430B.     

Decay of Solutions of the Wave Equation in the Kerr Geometry

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L ^∞ _loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant #RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30. An erratum to this article is available at .     

Continuous percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process

The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability p. At p = 0.5, the GAP becomes the random growth model and leads to the minority product rule at p = 1. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with 0.5 ≤ p ≤ 1 are always continuous and their critical exponents depend on p. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter p in addition.     

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.Research supported in part by NSF Grant No. MCS-76-05857Research supported in part by NSF Grant No. MCS-74-07313-A02     

Thermal conductivity of liquid R507 refrigerant

Thermal conductivity of ozone-safe liquid refrigerant R507 was studied by the method of high-frequency thermal waves within the temperature range of 297.95 … 332.55 K and pressures from the saturation line up to 3.7 MPa. The estimated errors of temperature, pressure, and thermal conductivity measurements are 0.1 K, 3 kPa, and 1.5 %, correspondingly. Thermal conductivity of liquid R507 was calculated on the saturation line. Approximation dependences for thermal conductivity were derived for the whole range of studied temperatures and pressures and on the saturation line. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 07-08-00295-a).     

Wall shear stress in an upward slug flow in a vertical tube

This paper deals with wall shear stress in an upward gas-liquid slug flow inside a vertical tube. Local characteristics were measured by the electrodiffusion method. The method of conditional averaging over realization ensemble was used, and this allows distinguishing large-scale structures on the background of turbulent pulsation of liquid. While averaging, each slug velocity measured by a double probe of electric conductivity was taken into account. Averaged distributions of shear stress over the wall under a gas slug were obtained for different mode parameters. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 04-01-00328) and the Russian Science Support Foundation (Grant “The Best Post-Graduate of RAS-2006”).