Poisson point processes,cascades, and random coverings ofR n

The generalized random energy model (GREM) is formulated in terms of hierarchies of Poisson point processes. This allows one to relate the high-temperature region with a random covering ofR ⁿ .     

Variational Bounds for the Generalized Random Energy Model

We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds,” and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand’s concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.     

A random covering interpretation for the phase transition of the random energy model

The random energy model is related to a random covering of the real line. The phase transition is interpreted as the passage from a regime where a family of random intervals covers the line (high temperature) to a noncovering regime (low temperature).     

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.     

Ground State Energy of the Low Density Hubbard Model

We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the low density limit the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by 8π a ? _u ? _d , where ? _u(d) denotes the density of the spin-up (down) particles, and a is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.     

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.     

磁场中量子点内弱耦合磁极化子基态束缚能的温度依赖性

用线性组合算符法、LLP变换法和量子统计的方法,研究了温度对磁场中抛物量子点内弱耦合磁极化子的影响,得到了磁极化子基态能量和基态束缚能量与量子点的受限强度、回旋频率和温度的依赖关系。数值计算结果表明,磁极化子的基态束缚能量 E_b 随回旋频率ω_c的增加而增大,随温度参量γ的增加而减小, E_b 随γ的增加而减小的幅度,不仅与γ的取值范围有关,而且还与ω_c有显著关系, E_b 显著变化的γ范围随ω_c的增加而减小。     

低能带电粒子核反应中电子屏蔽效应的温度相关性

为了揭示低能带电粒子核反应中电子屏蔽效应与温度之间的关系, 在德国鲁尔大学实验室的100kV加速器上系统测量了T=200\textcelsius时元素周期表中第三、四族以及镧系元素氘化靶中d(d,p)t反应的电子屏蔽效应. 由于氘在介质中的溶度(介质对氘的吸附能力)随温度升高而迅速下降, 该温度下金属表面不能形成氘化物, 导致金属性增强, 因而观测到了比常温下更显著的电子屏蔽效应. 这一测量结果可以用德拜模型来解释. 为了进一步验证德拜模型, 还测量了不同温度下氘化Co和Pt靶中d(d,p)t反应的电子屏蔽效应, 得到了电子屏蔽效应和温度 相关性的曲线. 实验结果与德拜模型的预言相符.     

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.     

Linearization effect in multifractal analysis: Insights from the Random Energy Model

The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show the following: (i) the existence of a critical order q^∗ beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation; (ii) multifractal exponents necessarily behave linearly in q, for q>q^∗. Tailoring the analysis conducted for the Random Energy Model to that of Compound Poisson motion, we provide explicative and quantitative predictions for the values of q^∗ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte Carlo simulations, conducted over a large number of large sample size realizations of compound Poisson motion, comfort and extend these analyses.     

随机级联模型的连续标度行为

将传统的阶乘矩统计方法推广到分割数或相空间元胞大小连续变化的情形,发现以往元胞大小分立变化的随机级联α模型存在＂链条效应＂,没有连续的标度率.考虑到实验过程及实际测量情况,对模型进行了切合实际的改进,让相空间的大小和位置随机起伏.这样得到的分形体,在相空间范围的起伏足够大时,有较好的标度不变性.能达到这一要求的相空间范围起伏的临界值,对模型参数α不敏感.用Monte Carlo模拟证明了,相空间范围的起伏对模型的反常标度指数影响不大.     

Fluctuations in Derrida's random energy and generalized random energy models

The fluctuations of the finite-size corrections to the free energy per site of the random energy model (REM) and the generalized random energy model (GREM) are investigated. Almost sure behavior for the corrections of order (logN)/N is given. We also prove convergence in distribution for the corrections of order 1/N.     

On the Distribution of Overlaps in the Sherrington–Kirkpatrick Spin Glass Model

This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington–Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.     

Path-Integral Approach to the Statistical Physics of One-Dimensional Random Systems

A path-integral method is extended and developed to investigate the statistical physics of one-dimensional random systems. Evaluation of the one-particle partition function and density matrix is simplified to finding a solution for a second-order ordinary differential equation. This makes it possible to obtain analytic solutions or conduct accurate numerical calculations for the random systems. With this approach, an analytical solution for the Gaussian model is obtained and the statistical physics of the Frisch–Lloyd model is studied.     

Stochastic Symmetry-Breaking in a Gaussian Hopfield Model

We study a two-pattern Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables.     

On Epistemics in Expected Free Energy for Linear Gaussian State Space Models

Active Inference (AIF) is a framework that can be used both to describe information processing in naturally intelligent systems, such as the human brain, and to design synthetic intelligent systems (agents). In this paper we show that Expected Free Energy (EFE) minimisation, a core feature of the framework, does not lead to purposeful explorative behaviour in linear Gaussian dynamical systems. We provide a simple proof that, due to the specific construction used for the EFE, the terms responsible for the exploratory (epistemic) drive become constant in the case of linear Gaussian systems. This renders AIF equivalent to KL control. From a theoretical point of view this is an interesting result since it is generally assumed that EFE minimisation will always introduce an exploratory drive in AIF agents. While the full EFE objective does not lead to exploration in linear Gaussian dynamical systems, the principles of its construction can still be used to design objectives that include an epistemic drive. We provide an in-depth analysis of the mechanics behind the epistemic drive of AIF agents and show how to design objectives for linear Gaussian dynamical systems that do include an epistemic drive. Concretely, we show that focusing solely on epistemics and dispensing with goal-directed terms leads to a form of maximum entropy exploration that is heavily dependent on the type of control signals driving the system. Additive controls do not permit such exploration. From a practical point of view this is an important result since linear Gaussian dynamical systems with additive controls are an extensively used model class, encompassing for instance Linear Quadratic Gaussian controllers. On the other hand, linear Gaussian dynamical systems driven by multiplicative controls such as switching transition matrices do permit an exploratory drive.     

Ferrimagnetic Properties of Bond Dilution Mixed Blume-Capel Model with Random Single-Ion Anisotropy

We study the ferrimagnetic properties of spin 1/2 and spin-1 systems by means of the effective field theory. The system is considered in the framework of bond dilution mixed Blume-Capel mode/ （BCM） with random single-ion anisotropy. The investigation of phase diagrams and magnetization curves indicates the existence of induced magnetic ordering and single or multi-compensation points. Special emphasis is placed on the influence of bond dilution and random single-ion anisotropy on normal or induced magnetic ordering states and single or multi-compensation points. Normal magnetic ordering states take on new phase diagrams with increasing randomness （bond and anisotropy）, while anisotropy induced magnetic ordering states are always occurrence no matter whether concentration of anisotropy is large or small. Existence and disappearance of compensation points rely strongly on bond dilution and random singleion anisotropy. Some results have not been revealed in previous papers and predicted by Néel theory of ferrimagnetism.     

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.     

Computer Simulations for Some One-Dimensional Models of Random Walks in Fluctuating Random Environment

We report some results of computer simulations for two models of random walks in random environment (rwre) on the one-dimensional lattice for fixed space–time configuration of the environment (“quenched rwre”): a “Markov model” with Markov dependence in time, and a “quasi stationary” model with long range space–time correlations. We compare with the corresponding results for a model with i.i.d. (in space time) environment. In the range of times available to us the quenched distributions of the random walk displacement are far from gaussian, but as the behavior is similar for all three models one cannot exclude asymptotic gaussianity, which is proved for the model with i.i.d. environment. We also report results on the random drift and on some time correlations which show a clear power decay