Correlation inequalities for non-purely-ferromagnetic systems

Correlation inequalities are proven for spin systems with non-purely-ferromagnetic interactions possessing a certain symmetry. These inequalities generalize well-known inequalities of Griffiths, Ginibre, Lebowitz, Schrader, Messager-Miracle-Sole, and Percus.     

GHS and other inequalities

We use a transformation due to Percus to give a simple derivation of the Griffiths, Hurst, and Sherman, and some other new inequalities, for Ising ferromagnets with pair interactions. The proof makes use of the Griffiths, Kelly, and Sherman and the Fortuin, Kasteleyn, and Ginibre inequalities.Work supported in part by USAFOSR-73-2430.     

Inequalities for higher order Ising spins and for continuum fluids

The recently derived Fortuin, Kasteleyn and Ginibre (FKG) inequalities for lattice gasses are investigated for higher order Ising spin systems and multi-component lattice gasses. Conditions are given for the validity of the FKG inequalities for higher order spin systems with Hamiltonians of the form used recently as models for various physical systems, e.g.He ³–He ⁴ mixtures. We also investigate various inequalities for binary lattice gases and show how these can be carried over to continuum systems.Supported in part by U.S.A.F.O.S.R. # F 44620-71-C-0013.N.S.F. Graduate Trainee.     

n--> infinity limit of O(n) ferromagnetic models on graphs

Thirty years ago, H. E. Stanley showed that an O(n) spin model on a lattice tends to a spherical model as n-->infinity. This means that at any temperature the corresponding free energies coincide. This fundamental result is no longer valid on more general discrete structures lacking in translation invariance, i.e., on graphs. However, only the singular parts of the free energies determine the critical behavior of the two statistical models. Here we show that for ferromagnetic models such singular parts still coincide even on graphs in the thermodynamic limit. This implies that the critical exponents of O(n) models on graphs for n-->infinity tend to the spherical ones and depend only on the graph spectral dimension.     

Equivalence of Bose-Einstein condensation and symmetry breaking

Based on a classic paper by Ginibre [Commun. Math. Phys. 8, 26 (1968)] it is shown that, whenever Bogoliubov's approximation, that is, the replacement of a(0) and a*(0) by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |a(0)|(2)/V and a*(0)a(0)/V in symmetry breaking fields, irrespective of the existence or absence of Bose-Einstein condensation. Because the former was proved by Ginibre to hold for absolutely integrable superstable pair interactions, the latter is equally valid in this case. Apart from Ginibre's work, our proof uses only a simple convexity inequality due to Griffiths.     

Percolation on Infinite Graphs and Isoperimetric Inequalities

We consider the Bernoulli bond percolation process (with parameter p) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay when p is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for p arbitrarily close to one.     

On the positivity of correlations in nonequilibrium spin systems

We consider Ising spin systems, equivalently lattice gases evolving under discrete- or continuous-time Markov processes, i.e., stochastic cellular automata or interacting particle systems. We show that for certain spin-flip probabilities or rates and suitable initial states the expectation values of products of spin variables taken at equal or different times are nonnegative; they satisfy the same inequalities as the equal-time correlations of ferromagnetic systems in equilibrium (first Griffiths inequality). Extensions of FKG inequalities to time-displaced correlations are also discussed.     

Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model

It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin–Kasteleyn–Ginibre property in this model and the Ghirlanda–Guerra identities in artificial models in a path integral representation based on the Lie–Trotter–Suzuki formula enable us to extend Chatterjee’s proof for the random field Ising model to the quantum model.     

General formulation of Griffiths' inequalities

We present a general framework in which Griffiths inequalities on the correlations of ferromagnetic spin systems appear as natural consequences of general assumptions. We give a method for the construction of a large class of models satisfying the basic assumptions. Special cases include the Ising model with arbitrary spins, and the plane rotator model. The general theory extends in a straightforward way to the non-commutative (quantum) case, but non-commutative examples satisfying all the assumptions are lacking at the moment.     

Dimensional implications of dynamical data on manifolds to empirical KL analysis

We explore the approximation of attracting manifolds of complex systems using dimension reducing methods. Complex systems having high-dimensional dynamics typically are initially analyzed by exploring techniques to reduce the dimension. Linear techniques, such as Galerkin projection methods, and nonlinear techniques, such as center manifold reduction are just some of the examples used to approximate the manifolds on which the attractors lie. In general, if the manifold is not highly curved, then both linear and nonlinear methods approximate the surface well. However, if the manifold curvature changes significantly with respect to parametric variations, then linear techniques may fail to give an accurate model of the manifold. This may not be a surprise in itself, but it is a fact so often overlooked or misunderstood when utilizing the popular KL method, that we offer this explicit study of the effects and consequences. Here we show that certain dimensions defined by linear methods are highly sensitive when modeled in situations where the attracting manifolds have large parametric curvature. Specifically, we show how manifold curvature mediates the dimension when using a linear basis set as a model. We punctuate our results with the definition of what we call, a “curvature induced parameter,” $d_{CI}$. Both finite- and infinite-dimensional models are used to illustrate the theory.     

Synchronization in power-law networks

We consider realistic power-law graphs, for which the power-law holds only for a certain range of degrees. We show that synchronizability of such networks depends on the expected average and expected maximum degree. In particular, we find that networks with realistic power-law graphs are less synchronizable than classical random networks. Finally, we consider hybrid graphs, which consist of two parts: a global graph and a local graph. We show that hybrid networks, for which the number of global edges is proportional to the number of total edges, almost surely synchronize.     

Some Circumstances Where Extra Updates Can Delay Mixing

Peres and Winkler proved a ‘censoring' inequality for Glauber dynamics on monotone spins systems such as the Ising model. Specifically, if, starting from a constant-spin configuration, the spins are updated at some sequence of sites, then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution. We show by means of simple counterexamples that the analogous statements fail for Glauber dynamics on proper colorings of a graph, and for lazy transpositions on permutations, answering two questions of Peres. It is not known whether the censoring property holds in other natural settings such as the Potts model.     

On One Example and One Counterexample in Counting Rational Points on Graph Hypersurfaces

In this paper, we present a concrete counterexample to the conjecture of Kontsevich about the polynomial countability of graph hypersurfaces. In contrast to this, we show that the “wheel with spokes” graphs WS _n are polynomially countable.     

Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume–Emery–Griffiths Models

Renormalization group based on the Migdal–Kadanoff bond removal approach is often considered a simple and valuable tool to understand the critical behavior of complicated statistical mechanical models. In presence of quenched disorder, however, predictions obtained with the Migdal–Kadanoff bond removal approach quite often fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In an attempt to overcome this limitation we analyze the behavior of Ising and Blume–Emery–Griffiths models on more structured hierarchical lattices. We find that, apart from some exceptions, the failure is not limited to Midgal–Kadanoff cells but originates right from the hierarchization of Bravais lattices on small cells, and shows up also when in-cell loops are considered.     

General monogamy property of global quantum discord and the application

We provide a family of general monogamy inequalities for global quantum discord (GQD), which can be considered as an extension of the usual discord monogamy inequality. It can be shown that those inequalities are satisfied under the similar condition for the holding of usual monogamy relation. We find that there is an intrinsic connection among them. Furthermore, we present a different type of monogamy inequality and prove that it holds under the condition that the bipartite GQDs do not increase when tracing out some subsystems. We also study the residual GQD based on the second type of monogamy inequality. As applications of those quantities, we investigate the GQDs and residual GQD in characterizing the quantum phase transition in the transverse field Ising model.     

Duality Between Spin Networks and the 2D Ising Model

The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories that couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.     

Existence and uniqueness of equilibrium states for some spin and continuum systems

The one to one correspondence between the existence of a unique equilibrium state and the differentiability of the free energy density with respect to the external field previously shown for Ising ferromagnetis is extendend to higher valued spin systems as well as to continuum systems satisfying the Fortuin, Kasteleyn and Ginibre inequalities. In particular this is shown to hold for a mixture ofA –B particles in which there is no interaction between like particles and a repulsion between unlike particles. Where the derivative of the free energy is discontinuous there are at least two equilibrium states.Supported in part by Air Force Grant n. 732430.     

On the ferromagnetic Ising model in noninteger spatial dimension

Using the necessary and sufficient conditions in terms of the high-field series coefficients that the Yang-Lee theorem holds, we prove rigorously by counterexample that it cannot be extended to general noninteger dimension, when such models are defined by the natural analytic continuation.Work supported in part by the US DOE.     

Renormalization group study of random quantum magnets

We have developed a very efficient numerical algorithm of the strong disorder renormalization group method to study the critical behaviour of the random transverse field Ising model, which is a prototype of random quantum magnets. With this algorithm we can renormalize an N-site cluster within a time NlogN, independently of the topology of the graph, and we went up to N ～ 4 × 10(6). We have studied regular lattices with dimension D ≤ 4 as well as Erd?s-Rényi random graphs, which are infinite dimensional objects. In all cases the quantum critical behaviour is found to be controlled by an infinite disorder fixed point, in which disorder plays a dominant role over quantum fluctuations. As a consequence the renormalization procedure as well as the obtained critical properties are asymptotically exact for large systems. We have also studied Griffiths singularities in the paramagnetic and ferromagnetic phases and generalized the numerical algorithm for other random quantum systems.     

Macroscopic realizations of quantum logics

Summary Two-particle quantum systems with spin can be simulated by classical automata described by graphs. These graphs are associated with nondistributive property lattices of these quantum systems. We emphasize that to non-local properties of a quantum system being in a certain eigenstate of the permutation operator there correspond merely some additional vertices in the graph which have nothing nonlocal in their nature. This leads to the possibility of violating Bell's inequalities in classical systems described by graphs (see Section 6) without violating relativity theory.The subjective interpretation of quantum mechanics of von Neumann, London, and Bauer can be connected with the Boolean nature of mind grasping the non-Boolean nature of the world, which results in the projection postulate: wave packet reduction. A simple example is given for a two-particle system with spin.