The correlation length of commodity markets 2. Theoretical framework

The second of this series of two papers is devoted to a theoretical analysis of spatial interaction between commodity markets. The theoretical framework that we present is referred to as the stochastic spatial arbitrage model (SSAM); it accounts for most of the empirical regularities observed in the first paper. Two basic mechanisms are found to be responsible for spatial inter-market interaction, namely (i) spatial arbitrage and hedging conducted by traders, (ii) spatial correlation between local shocks; the latter is favored by a similar economic and cultural environment. The SSAM includes both effects and offers a wide range of predictions about price volatility, trade, price correlations, price differentials. Statistical tests display a convergent array of evidence in favor of the model. However several predictions cannot be tested by lack of statistical evidence, a circumstance which shows that yet additional “experimental” work is r! equired. Received 17 May 1999 and Received in final form 31 May 1999     

Speculative trading: the price multiplier effect

During a speculative episode the price of an item jumps from an initial level p₁ to a peak level p₂ before more or less returning to level p₁ . The ratio p ₂/p ₁ is referred to as the amplitude A of the peak. This paper shows that for a given market the peak amplitude is a linear function of the logarithm of the price at the beginning of the speculative episode; with p₁ expressed in 1999 euros the relationship takes the form: ; the values of the parameter a turn out to be relatively independent of the market considered: , the values of the parameter b are more market-dependent, but are stable in the course of time for a given market. This relationship suggests that the higher the stakes the more “bullish” the market becomes. Possible mechanisms of this “risk affinity” effect are discussed. Received 29 September 1999     

Determining bottom price-levels after a speculative peak

During a stock market peak the price of a given stock (i) jumps from an initial level p ₁(i) to a peak level p ₂(i) before falling back to a bottom level p ₃(i). The ratios A(i) = p ₂(i)/p ₁(i) and B(i)= p ₃(i)/p ₁(i) are referred to as the peak- and bottom-amplitude respectively. The paper shows that for a sample of stocks there is a linear relationship between A(i) and B(i) of the form: B=0.4A+b. In words, this means that the higher the price of a stock climbs during a bull market the better it resists during the subsequent bear market. That rule, which we call the resilience pattern, also applies to other speculative markets. It provides a useful guiding line for Monte Carlo simulations. Received 9 June 2000     

Magnetic fluids in an external field

Within mean field approximation we investigate the phase diagrams of magnetic fluids in presence of a magnetic field. In a finite field the magnetic phase transition is absent, but instead a line of first order liquid-liquid transitions ending in a critical point occurs for a magnetic interaction, which is sufficiently strong. Varying the magnetic field these critical points extend from the tricritical point at H=0 to a critical endpoint. For a fluid with Ising spins we calculate the critical lines and several tricritical exponents analytically. For Heisenberg fluids we obtain the phase diagrams from a numerical solution of the mean field equations of state. Received 20 March 1998     

Stabilizing co-continuous polymer blend morphologies with ABC block copolymers

We suggest that ABC triblock copolymers provide a convenient and effective route to emulsifying blends of A and C homopolymers into co-continuous morphologies over a wide range of compositions. Direct transitions between disordered tricontinuous phases (“ABC microemulsions”) and spatially-periodic tricontinuous phases (e.g. gyroid or double-diamond cubic phases) should be possible in appropriately formulated alloys. We envision a broad range of potential applications to thermoplastic and thermoset polymeric materials. Received: 4 June 1997 / Revised: 17 September 1997 / Accepted: 13 October 1997     

Critical behavior of a three-state Potts model on a Voronoi lattice

We use the single-histogram technique to study the critical behavior of the three-state Potts model on a (random) Voronoi-Delaunay lattice with size ranging from 250 to 8 000 sites. We consider the effect of an exponential decay of the interactions with the distance, , with a>0, and observe that this system seems to have critical exponents and which are different from the respective exponents of the three-state Potts model on a regular square lattice. However, the ratio remains essentially the same. We find numerical evidences (although not conclusive, due to the small range of system size) that the specific heat on this random system behaves as a power-law for a=0 and as a logarithmic divergence for a=0.5 and a=1.0 Received 5 April 2000     

Scaling relationship between effective critical exponents throughout the crossover region in thin Ising films

The Monte Carlo (MC) approach is used to check the validity of the scaling relationship for the effective critical exponents in thin Ising films. We investigate this relationship not just in the critical region but throughout the crossover to the expected two-dimensional behavior. Our results indicate that this scaling relationship is very well-fulfilled throughout the entire crossover temperature region, as predicted by a previous renormalization group analysis. The two-dimensional universality class of Ising films is confirmed by means of data collapsing plots for plates with increasing L, up to L=100. The evolution of the maximum value of the effective critical exponents with film thickness is discussed. Received 22 April 1999     

Singularity of the specific heat of two-dimensional random Ising models

The singularity of the specific heat is studied for the dilution (J>J'>0) type and Gaussian type random Ising models using the Pfaffian method numerically. The type of singularity at the paramagnetic-ferromagnetic phase boundary is studied using the standard regression method using data up to system size. It is shown that the logarithmic type singularity is more reliable than the double-logarithmic type and cusp type singularities. The critical temperatures are estimated accurately for both the dilution type and Gaussian type random Ising models. A phase diagram relating strength of the randomness and temperature is also presented. Received: 26 February 1998 / Revised: 15 May 1998 / Accepted: 25 June 1998     

Test of the universality of the critical adsorption profile by neutron reflection

Using neutron reflectivity, we directly and self-consistently obtained all the parameters of the critical adsorption profile of three critical mixtures (n-hexane+perfluorohexane, deuterated methanol+cyclohexane and methanol+deuterated cyclohexane) assuming only a Liu-Fisher form for the profile. The characteristic exponent and the amplitude ratio which can be deduced from the neutron spectra alone are in good agreement with one another, with ellipsometric data and with theoretical values. Owing to auxiliary measurements of coexistence curves and correlation lengths, we could also check the amplitude of the intermediate and distal parts of the profile with good agreements for the methanol cyclohexane mixtures only. The adsorbed amount is much greater in the n-hexane+perfluorohexane mixture where both the power-law amplitude and the characteristic length of the profile are affected. Received 30 May 2001     

Self-organized model for information spread in financial markets

A self-organized model with social percolation process is proposed to describe the propagations of information for different trading ways across a social system and the automatic formation of various groups within market traders. Based on the market structure of this model, some stylized observations of real market can be reproduced, including the slow decay of volatility correlations, and the fat tail distribution of price returns which is found to cross over to an exponential-type asymptotic decay in different dimensional systems. Received 15 March 2000     

High-temperature series analysis of the p-state Potts glass model on d-dimensional hypercubic lattices

We analyze recently extended high-temperature series expansions for the “Edwards-Anderson” spin-glass susceptibility of the p-state Potts glass model on d-dimensional hypercubic lattices for the case of a symmetric bimodal distribution of ferro- and antiferromagnetic nearest-neighbor couplings . In these star-graph expansions up to order 22 in the inverse temperature , the number of Potts states p and the dimension d are kept as free parameters which can take any value. By applying several series analysis techniques to the new series expansions, this enabled us to determine the critical coupling K_c and the critical exponent of the spin-glass susceptibility in a large region of the two-dimensional (p,d)-parameter space. We discuss the thus obtained information with emphasis on the lower and upper critical dimensions of the model and present a careful comparison with previous estimates for special values of p and d. Received: 25 May 1998 / Revised and Accepted: 11 August 1998     

An analytical method to compute an approximate value of the site percolation threshold P<Subscript>c</Subscript>

An analytical method to compute the site percolation threshold is introduced. This method yields an approximate value of larger or equal to the real value. As examples, the computation of is presented for 4 lattices in 2 dimensions: square, triangular, honeycomb and kagome. The results obtained are 0.592 871 6, 0.5, 0.765 069, 0.654 653 7, to be compared with the real values 0.592 746 0, 0.5, 0.697 043, 0.652 703 6. The method is not limited to 2 dimensions. Received 27 July 1999 and Received in final form 29 November 1999     

Specific-heat study of the triangular-lattice antiferromagnet RbCuCl

We report on the magnetic phase diagram of the distorted triangular-lattice antiferromagnet RbCuCl₃ for a magnetic field applied parallel to the basal plane (). High-resolution measurements of the specific heat and of the magnetocaloric effect have been performed in magnetic fields up to 14 T. The high-field specific-heat data reveal the existence of an intermediate phase between the paramagnetic and the frustrated antiferromagnetic phase. Received 18 October 1999     

Quantum fluctuations in a scale-free network-connected Ising system

We study the effect of quantum fluctuations in an Ising spin system on a scale-free network of degree exponent γ>5 using a quantum Monte Carlo simulation technique. In our model, one can adjust the magnitude of the magnetic field perpendicular to the Ising spin direction and can therefore control the strength of quantum fluctuations for each spin. Our numerical analysis shows that quantum fluctuations reduce the transition temperature T_c of the ferromagnetic-paramagnetic phase transition. However, the phase transition belongs to the same mean-field type universality class both with and without the quantum fluctuations. We also study the role of hubs by turning on the quantum fluctuations exclusively at the nodes with the most links. When only a small number of hub spins fluctuate quantum mechanically, T_c decreases with increasing magnetic field until it saturates at high fields. This effect becomes stronger as the number of hub spins increases. In contrast, quantum fluctuations at the same number of “non-hub” spins do not affect T_c. This implies that the hubs play an important role in maintaining order in the whole network.     

Two-point correlation function in systems with van der Waals type interaction

The behavior of the bulk two-point correlation function G(;T| d ) in d-dimensional system with van der Waals type interactions is investigated and its consequences on the finite-size scaling properties of the susceptibility in such finite systems with periodic boundary conditions is discussed within mean-spherical model which is an example of Ornstein and Zernike type theory. The interaction is supposed to decay at large distances r as r ^{- (d + σ)}, with 2 < d < 4, 2 < σ < 4 and d + σ≤6. It is shown that G(;T| d ) decays as r ^{- (d - 2)} for 1 ≪r≪ξ, exponentially for ξ≪r≪r ^*, where r ^* = (σ - 2)ξlnξ, and again in a power law as r ^{- (d + σ)} for r≫r ^*. The analytical form of the leading-order scaling function of G(;T| d ) in any of these regimes is derived. Received 28 May 2001     

Phase diagram of the 3D bimodal random-field Ising model

The one-parametric Wang-Landau (WL) method is implemented together with an extrapolation scheme to yield approximations of the two-dimensional (exchange-energy, field-energy) density of states (DOS) of the 3D bimodal random-field Ising model (RFIM). The present approach generalizes our earlier WL implementations, by handling the final stage of the WL process as an entropic sampling scheme, appropriate for the recording of the required two-parametric histograms. We test the accuracy of the proposed extrapolation scheme and then apply it to study the size-shift behavior of the phase diagram of the 3D bimodal RFIM. We present a finite-size converging approach and a well-behaved sequence of estimates for the critical disorder strength. Their asymptotic shift-behavior yields the critical disorder strength and the associated correlation length's exponent, in agreement with previous estimates from ground-state studies of the model.     

Self-organizing Ising model of financial markets

We study a dynamical Ising-like model of agents' opinions (buy or sell) with learning, in which the coupling coefficients are re-assessed continuously in time according to how past external news (time-varying magnetic field) have explained realized market returns. By combining herding, the impact of external news and private information, we find that the stylized facts of financial markets are reproduced only when agents misattribute the success of news to predict return to herding effects, thereby providing positive feedbacks leading to the model functioning close to the Ising critical point.     

Boundary critical behaviour of two-dimensional random Potts models

Using extensive Monte Carlo simulations, transfer matrix techniques and conformal invariance, ferromagnetic random q-state Potts models for are studied in the vicinity of the critical temperature. In particular the surface and bulk magnetization exponents and are found monotonically increasing with q. At the critical temperature, different moments (n) of the magnetization profiles are calculated which are all found to accurately follow predictions of conformal invariance. The critical correlation functions show multifractal behaviour, the decay exponents of the different moments both in the volume and at the surface, are n-dependent. Received 4 June 1999