Generalized Random Energy Model II

The formulae for the free energy, when the driving distributions in Generalized Random Energy Model (GREM) are of the form for γ ≥ 1 are derived. The large deviation technique allows the use of different distributions at different levels of the GREM. As an illustration we consider, in detail, a two level GREM with exponential and Gaussian distributions. This simple case itself leads to interesting phenomena.     

Bounds on correlation decay for long-range vector spin glasses

We give upper bounds on the decay of correlation functions for long-rangeSO(N)-symmetric spin-glass models in one and two dimensions using McBryan-Spencer techniques. In doing so we extend recent results of Picco.     

Bounds for free energy derivatives using renormalization group methods

The free energy of a thermodynamic system is known to be a concave function of the temperature. This fact is used, together with some results due to Fisher, to deduce bounds on the internal energy of the two-dimensional Ising model, given reasonably accurate upper and lower bounds on the free energy. These free energy bounds are derived from renormalization group transformations. Unfortunately, the numerical accuracy of the bounds on the internal energy is poor. The reasons for the failure are discussed.     

Temperature Dependence of the Gibbs State in the Random Energy Model

We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida's Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution ^N _{i=1 i i} /N[–1,1] of two spin configurations , under the product of two Gibbs measures, which are taken at temperatures T,T respectively. If TT are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is D _{0 0}+D _{1 1}, with random weights D ₀, D ₁, while for the second one it is ₀. We compute consequently the overlap distribution for the second model whenever T–T0 at different speeds as N.     

Spin Glass Computations and Ruelle’s Probability Cascades

We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure.     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

Metastability and Convergence to Equilibrium for the Random Field Curie–Weiss Model

We study a dynamics for the magnetization of the random field Curie–Weiss model. A metastable behavior is exhibited and asymptotic estimates on the speed of convergence to equilibrium are given. The results are given almost surely and in law with respect to the realizations of the random magnetic fields.     

Local Energy Statistics in Spin Glasses

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. We review some rigorous results confirming the validity of this conjecture. In the context of the SK models, we analyse the limits of the validity of the conjecture for energy levels growing with the volume of the system. In the case of the Generalised Random energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies E _N < β _c N, where β _c is the critical temperature. We also explain the more complex behaviour that sets in at higher energies. Research supported in part by the DFG in the Dutch-German Bilateral Research Group “Mathematics of Random Spatial Models from Physics and Biology” and by the European Science Foundation in the Programme RDSES.     

A Strict Inequality for the Random Triangle Model

The random triangle model on a graph G, is a random graph model where the usual i.i.d. measure is perturbed by a factor q ^t(), where q1 is a constant, and t() is the number of triangles in the random subgraph . Here we consider the case where G is the usual two-dimensional triangular lattice, for which there exists a percolation threshold p _c (q) such that the probability of getting an infinite connected component of retained edges is 0 for p<p _c (q), and 1 for p>p _c (q). It has previously been shown that p _c (q) is a decreasing function of q. Here we strengthen this by showing that p _c (q) is strictly decreasing. This confirms a conjecture by Häggström and Jonasson.     

The Dilute Potts Model on Random Surfaces

We present a new solution of the asymmetric two-matrix model in the large-N limit which only involves a saddle point analysis. The model can be interpreted as Ising in the presence of a magnetic field, on random dynamical lattices with the topology of the sphere (resp. the disk) for closed (resp. open) surfaces; we elaborate on the resulting phase diagram. The method can be equally well applied to a more general (Q+1)-matrix model which represents the dilute Potts model on random dynamical lattices. We discuss in particular duality of boundary conditions for open random surfaces.     

类氦原子体系基态能量的变分法数值研究

选取由指数形式函数的线性组合所构成的试探波函数,利用变分法对类氦原子体系的基态能量进行了数值计算,计算中利用拟牛顿法求非线性方程组的一组实根,计算结果与实验值相当接近．计算表明：当n由1增大到2时,能量的计算值有较大的改进,但由于选取的变分波函数没有考虑到电子关联作用,所以计算只能达到一定的精确度．     

First-Order Formalism for the Quintom Model of Dark Energy

The present paper deals to the quintom model of dark energy. We introduce a first-order formalism, which shows how to relate the potential that specifies the scalar field model to Hubble’s parameter. Reviewing briefly the quintom scenario of dark energy, we present a general procedure to solve the equations of motion for quintom model driven by a couple scalar fields with first-order differential equations.     

Extended Quark Potential Model from Random Phase Approximation

The quark potential model is extended to include the sea quark excitation using the random phase approx-imation. The effective quark interaction preserves the important QCD properties - chiral symmetry and confinementsimultaneously. A primary qualitative analysis shows that the π meson as a well-known typical Goldstone boson andthe other mesons made up of valence qq quark pair such as the ρ meson can also be described in this extended quarkpotential model.     

Bounds for effective parameters of multicomponent media by analytic continuation

Recently D. Bergman introduced a method for obtaining bounds for the effective dielectric constant (or conductivity) of a two-component medium. This method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters. We extend the method to multicomponent media using techniques of several complex variables.     

On the Gibbsian Nature of the Random Field Kac Model Under Block-Averaging

We consider the Kac–Ising model in an arbitrary configuration of local magnetic fields = , in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormalized' infinite volume measures obtained by block averaging any of the Gibbs-measures corresponding to fixed , with block-length small enough compared to the range of the Kac-interaction. We show that these measures are Gibbs measures for the same renormalized interaction potential. This potential depends locally on the field configuration and decays exponentially, uniformly in , for which we give explicit bounds. The construction of the potential is based on a high temperature-type cluster expansion.     

Solution Space Coupling in the Random K-Satisfiability Problem

The random K-satisfiability(K-SAT)problem is very difcult when the clause density is close to the satisfiability threshold.In this paper we study this problem from the perspective of solution space coupling.We divide a given difcult random K-SAT formula into two easy sub-formulas and let the two corresponding solution spaces to interact with each other through a coupling field x.We investigate the statistical mechanical property of this coupled system by mean field theory and computer simulations.The coupled system has an ergodicity-breaking(clustering)transition at certain critical value x d of the coupling field.At this transition point,the mean overlap value between the solutions of the two solution spaces is very close to 1.The mean energy density of the coupled system at its clustering transition point is less than the mean energy density of the original K-SAT problem at the temperature-induced clustering transition point.The implications of this work for designing new heuristic K-SAT solvers are discussed.     

Exactly Solvable Model of Reactions on a Random Catalytic Chain

In this paper we study a catalytically-activated A+A0 reaction taking place on a one-dimensional regular lattice which is brought in contact with a reservoir of A particles. The A particles have a hard-core and undergo continuous exchanges with the reservoir, adsorbing onto the lattice or desorbing back to the reservoir. Some lattice sites possess special, catalytic properties, which induce an immediate reaction between two neighboring A particles as soon as at least one of them lands onto a catalytic site. We consider three situations for the spatial placement of the catalytic sites: regular, annealed random, and quenched random. For all these cases we derive exact results for the partition function, and the disorder-averaged pressure per lattice site. We also present exact asymptotic results for the particles' mean density and the system's compressibility. The model studied here furnishes another example of a 1D Ising-type system with random multisite interactions which admits an exact solution.     

Symmetry Relations for the Six-Vertex Model

The exact solution of the asymmetric six-vertex model is cast in an algebraically simple form in which the extraction of physical quantities is transparent. This is used to derive a symmetry relation corresponding to the exchange of spatial x and y axes. As an application we study the field-induced phase transition of a two-dimensional analogue of spin ice, a frustrated Ising magnet.     

热核表面能的Thomas－Fermi统计模型理论

用Ｓｅｙｌｅｒ－Ｂｌａｎｃｈａｒｄ动量相关非定域相互作用的含温Ｔｈｏｍａｓ－Ｆｅｒｍｉ统计理论，对半无穷大核物质模型计算了核物质表面能系数σ（Ｔ，δ）随温度Ｔ和不对称度δ的变化，发现在低温Ｔ≤５ＭｅＶ和不对称度δ≤０．２时，可以近似写成σ（Ｔ，δ）＝σ０（Ｔ）［１＋Ｋ（Ｔ）δ２］，其中σ０（Ｔ）和Ｋ（Ｔ）可以拟合成温度Ｔ的二次函数．     

A New Method for the Atomic Ground-State Energy in the Screened Coulomb Potential

The new method proposed recently by Friedberg,Lee and Zhao is applied to the derivation of the atomic ground-state energy with the inclusion of the screening effect.The present results are compared with those obtained in the pure Coulomb potential and by the variational approach.The overall good results are obtained with this new method.``