Exponential decay of connectivities in the two-dimensional ising model

We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.On leave from São Paulo University, Brazil.     

On the number of infinite geodesics and ground states in disordered systems

We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to +. As corollaries we show that in any dimensiond2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or +. Proofs employ simple large-deviation estimates and ergodic arguments.     

Maximal Clusters in Non-Critical Percolation and Related Models

We investigate the maximal non-critical cluster in a big box in various percolation-type models. We investigate its typical size, and the fluctuations around this typical size. The limit law of these fluctuations is related to maxima of independent random variables with law described by a single cluster.     

On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point

We consider the two-dimensional stochastic Ising model in finite square with free boundary conditions, at inverse temperature >₀ and zero external field. Using duality and recent results of Ioffe on the Wulff construction close to the critical temperature, we extend some of the results obtained by Martinelli in the low-temperature regime to any temperature below the critical one. In particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of , with a rate constant determined by the surface tension along one of the coordinate axes. We also extend to the same range of temperatures the result due to Shlosman on the equilibrium large deviations of the magnetization with free boundary conditions.     

Percolation of the minority spins in high-dimensional Ising models

We present some new results on the region in the-h plane where the + spins percolate for the nearest neighbor Ising model. In particular, it is shown that in high enough dimensionsd there is percolation of the minority spins at inverse temperatures< ₊ with some ₊>_c, for which ₊/gb_c1/2log(cd),c a constant.On leave from Rutgers University.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Phase transition in an interacting Bose system. An application of the theory of Ventsel' and Freidlin

We consider a system ofN bosons on a complete graph withV vertices interacting through hard core repulsion. With the use of the Ventsel'-Freidlin large-deviation theory of random perturbation of dynamical systems, we calculate the canonical free energy in the thermodynamic limit and prove that the system exhibits a phase transition.On leave from the Mathematical Institute of the Hungarian Academy of Sciences, Budapest, H-1053, Hungary.     

Ising model in a quasiperiodic transverse field,percolation, and contact processes in quasiperiodic environments

Quantum Ising models in a transverse field are related to continuous-time percolation processes whose oriented percolation versions are contact processes. We study such models in the presence of quasiperiodic disorder and prove localization in the ground state, no percolation, and extinction, respectively, for sufficiently large disorder.     

Nonuniversality and continuity of the critical covered volume fraction in continuum percolation

We establish, using mathematically rigorous methods, that the critical covered volume fraction (CVF) for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls. In addition, we show that the critical CVF is a continuous function of the distribution of the radius random variable, in the sense that if a sequence of random variables converges weakly to some random variable, then the critical CVF based on these random variables converges to the critical CVF of the limiting random variable.     

Low-temperature properties of the Ising model in the transverse field and the influence of kinematical effects

We present the low-temperature method for the Ising model in the transverse field based on the approximate boson expression for Pauli-operators. Kinematic effects, which arise through special decoupling of Green's functions, lead to appearance of additional energy levels.     

Assessing,Testing and Estimating the Amount of Fine-Tuning by Means of Active Information

A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes use of in order to reach a certain target. A function f quantifies specificity for each possible outcome x of a search, so that the target of the algorithm is a set of highly specified states, whereas fine-tuning occurs if it is much more likely for the algorithm to reach the target as intended than by chance. The distribution of a random outcome X of the algorithm involves a parameter θ that quantifies how much background information has been infused. A simple choice of this parameter is to use θf in order to exponentially tilt the distribution of the outcome of the search algorithm under the null distribution of no tuning, so that an exponential family of distributions is obtained. Such algorithms are obtained by iterating a Metropolis–Hastings type of Markov chain, which makes it possible to compute their active information under the equilibrium and non-equilibrium of the Markov chain, with or without stopping when the targeted set of fine-tuned states has been reached. Other choices of tuning parameters θ are discussed as well. Nonparametric and parametric estimators of active information and tests of fine-tuning are developed when repeated and independent outcomes of the algorithm are available. The theory is illustrated with examples from cosmology, student learning, reinforcement learning, a Moran type model of population genetics, and evolutionary programming.     

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density ρ=ρ_short+ρ_long into the number density of particles belonging to cycles of finite length (ρ_short) and to infinitely long cycles (ρ_long) in the thermodynamic limit. For this model we prove that when there is Bose condensation, ρ_long is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of ρ_long≠ 0 with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas     

One-, Two-, and Three-Dimensional Ising Model in the Static Fluctuation Approximation

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator _f ^z withquadratic fluctuations is replaced with its mean value [( _f ^z )² ( _f ^z )²]. This means that the true quantum mechanical spectrum of the operator _f ^z is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.     

New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain

In this paper we show how an infinite system of coupled Toda-type nonlinear differential equations derived by one of us can be used efficiently to calculate the time-dependent pair-correlations in the Ising chain in a transverse field. The results are seen to match extremely well long large-time asymptotic expansions newly derived here. For our initial conditions we use new long asymptotic expansions for the equal-time pair correlation functions of the transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising model. Using this one can also study the equal-time wavevector-dependent correlation function of the quantum chain, a.k.a. the q-dependent diagonal susceptibility in the 2d Ising model, in great detail with very little computational effort. Supported in part by the National Science Foundation under grant PHY 07-58139 and by the Australian Research Council under Project ID: LX0989627.     

The Ising spin glass in finite dimensions: A perturbative study of the free energy

Replica field theory is used to study the n  -dependent free energy of the Ising spin glass in a first order perturbative treatment. Large sample-to-sample deviations of the free energy from its quenched average prove to be Gaussian, independently of the special structure of the order parameter. The free energy difference between the replica symmetric and (infinite level) replica symmetry broken phases is studied in details: the line n(T)$n(T)$ where it is zero coincides with the Almeida–Thouless line for d>8$d>8$. The dimensional domain 6<d<8$6<d<8$ is more complicated, and several scenarios are possible.     

Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

We consider an open one dimensional lattice gas on sites i=1,..., N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N. The probability of microscopic configurations corresponding to some other profile (x), x=i/N, has the asymptotic form exp[–N ({})]; is the large deviation functional. In contrast to equilibrium systems, for which _eq({}) is just the integral of the appropriately normalized local free energy density, the we find here for the nonequilibrium system is a nonlocal function of . This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of in general SNS, where the long range correlations have been observed experimentally.     

Interface, Surface Tension and Reentrant Pinning Transition in the 2D Ising Model

We develop a new way to look at the high-temperature representation of the Ising model up to the critical temperature and obtain a number of interesting consequences. In the two-dimensional case, it is possible to use these tools to prove results on phase-separation lines in the whole phase-coexistence regime, by way of a duality transformation. We illustrate the power of these techniques by studying an Ising model with a boundary magnetic field, in which a reentrant pinning transition takes place; more precisely we show that the typical configurations of the model can be described, at the macroscopic level, by interfaces which are solutions of the corresponding thermodynamic variational problem; this variational problem is solved explicitly. There exist values of the boundary magnetic field and temperatures 0<T ₁<T ₂<T _c such that the interface is not pinned for T<T ₁ or T>T ₂, but is pinned for T ₁<T<T ₂; we can also find values of the boundary magnetic field and temperatures 0<T ₁<T ₂<T ₃<T _c such that for T<T ₁ or T ₂<T<T ₃ the interface is pinned, while for T ₁<T<T ₂ or T>T ₃ it is not pinned. An important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results. The techniques used in this work are robust and can be used in a variety of different situations. Received: 28 January 1998 / Accepted: 22 December 1998     

Negative-temperature states and large-scale,long-lived vortices in two-dimensional turbulence

We study Onsager's theory of large, coherent vortices in turbulent flows in the approximation of the point-vortex model for two-dimensional Euler hydrodynamics. In the limit of a large number of point vortices with the energy perpair of vortices held fixed, we prove that the entropy defined from the microcanonical distribution as a function of the (pair-specific) energy has its maximum at a finite value and thereafter decreases, yielding the negative-temperature states predicted by Onsager. We furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy, and, under regularity assumptions, obeys the Joyce-Montgomery mean-field equation. We also prove that, under appropriate conditions, the vorticity distribution is the same as that for the canonical distribution, a form of equivalence of ensembles. We establish a large-fluctuation theory for the microcanonical distributions, which is based on a level-3 large-deviations theory for exchangeable distributions. We discuss some implications of that property for the ergodicity requirements to justify Onsager's theory, and also the theoretical foundations of a recent extension to continuous vorticity fields by R. Robert and J. Miller. Although the theory of two-dimensional vortices is of primary interest, our proofs actually apply to a very general class of mean-field models with long-range interactions in arbitrary dimensions.     

Hysteresis of the Magnetic Particle in a Dipolar Ising Model

Zero-temperature Monte Carlo simulations are used to investigate the hysteresis of a magnetic particle in a dipolar Ising model.The magnetic particle is described in a systemm of permanent dipoles,and the dipoles are located in a cubic lattice site.The effects of the shape and the size of the particle on the hysteresis loop at zero temperature are obtained.For strong exchange interactions,the shapes of magnetic hysteresis loops approach rectangle.For weak exchange interactions,the effects of the size and the shape of the particle on the loops are more remarkable than those of strong exchange interactions case.The slope of the hysteresis loop decreases with the increase of the ratio of the semi major axis to the semi minor axis of the ellipsoidal magnetic particle,and there is an increase of the slope of the hysteresis with the decrease of the size of the magnetic particle.The effects of the shape and size of the particle on the coercive force at zero temperature are also investigated.     

伊辛模型中量子的热纠缠和保真度

研究了伊辛模型中两个粒子在均匀和非均匀磁场中的热纠缠以及利用它作为量子信息传输的通道的传输保真度.计算出纠缠度的度量Concurrence,以及在不同种情况下呈现出来的纠缠度的表现形为.与均匀磁场相反,我们发现在非均匀磁场中传输的保真度能够得到增强,同时我们发现保真度还与耦合系数和温度有关.我们通过图形清楚地表示它们的性质,从图形中我们得出,第一:在其它条件相同的情况下,无论磁场方向是相同还是相反,它们的纠缠度都是相同的;但是当磁场方向相反时,平均保真度比均匀磁场具有更大的值.第二:为了提高纠缠和平均保真度我们可以通过选择适当的磁场强度、耦合系数和降低温度来实现.