Random walks with imperfect trapping in the decoupled-ring approximation

We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical simulations reveal this solution, which is exact in the limit of perfect traps, to be remarkably robust with respect to a significant lowering of the trapping probability. We demonstrate that for randomly distributed traps, the long-time asymptotics of our result recovers the known stretched exponential decay. We also study an anisotropic three-dimensional version of our model. We discuss possible applications of some of our findings to the decay of excitons in semiconducting organic polymer materials, and emphasize the crucial influence of the spatial trap distribution on the kinetics. Received 23 July 2001 / Received in final form 14 May 2002 Published online 13 August 2002     

Experimental modeling of physical laws

A physical law is represented by the probability distribution of a measured variable. The probability density is described by measured data using an estimator whose kernel is the instrument scattering function. The experimental information and data redundancy are defined in terms of information entropy. The model cost function, comprised of data redundancy and estimation error, is minimized by the creation-annihilation process. Received 24 February 2000 and Received in final form 21 February 2001     

A simple model of DNA denaturation and mutually avoiding walks statistics

Recently Garel, Monthus and Orland [Europhys. Lett. 55, 132 (2001)] considered a model of DNA denaturation in which excluded volume effects within each strand are neglected, while mutual avoidance is included. Using an approximate scheme they found a first order denaturation. We show that a first order transition for this model follows from exact results for the statistics of two mutually avoiding random walks, whose reunion exponent is c > 2, both in two and three dimensions. Analytical estimates of c due to the interactions with other denaturated loops, as well as numerical calculations, indicate that the transition is even sharper than in models where excluded volume effects are fully incorporated. The probability distribution of distances between homologous base pairs decays as a power law at the transition. Received 8 July 2002 / Received in final form 25 July 2002 Published online 17 September 2002     

Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion

A generalised random walk scheme for random walks in an arbitrary external potential field is investigated. From this concept which accounts for the symmetry breaking of homogeneity through the external field, a generalised master equation is constructed. For long-tailed transfer distance or waiting time distributions we show that this generalised master equation is the genesis of apparently different fractional Fokker-Planck equations discussed in literature. On this basis, we introduce a generalisation of the Kramers-Moyal expansion for broad jump length distributions that combines multiples of both ordinary and fractional spatial derivatives. However, it is shown that the nature of the drift term is not changed through the existence of anomalous transport statistics, and thus to first order, an external potential Φ(x) feeds back on the probability density function W through the classical term ∝/ x (x)W(x, t), i.e., even for Lévy flights, there exists a linear infinitesimal generator that accounts for the response to an external field. Received 30 June 2000 and Received in final form 12 November 2000     

Value-at-risk prediction using context modeling

In financial market risk measurement, Value-at-Risk (VaR) techniques have proven to be a very useful and popular tool. Unfortunately, most VaR estimation models suffer from major drawbacks: the lognormal (Gaussian) modeling of the returns does not take into account the observed fat tail distribution and the non-stationarity of the financial instruments severely limits the efficiency of the VaR predictions. In this paper, we present a new approach to VaR estimation which is based on ideas from the field of information theory and lossless data compression. More specifically, the technique of context modeling is applied to estimate the VaR by conditioning the probability density function on the present context. Tree-structured vector quantization is applied to partition the multi-dimensional state space of both macroeconomic and microeconomic priors into an increasing but limited number of context classes. Each class can be interpreted as a state of aggregation with its own statistical and dynamic behavior, or as a random walk with its own drift and step size. Results on the US S&P500 index, obtained using several evaluation methods, show the strong potential of this approach and prove that it can be applied successfully for, amongst other useful applications, VaR and volatility prediction. The October 1997 crash is indicated in time. Received 2 September 2000 and Received in final form 12 October 2000     

Nonextensive statistics and incomplete information

We comment on some open questions and theoretical peculiarities in Tsallis nonextensive statistical mechanics. It is shown that the theoretical basis of the successful Tsallis' generalized exponential distribution shows some worrying properties with the conventional normalization and the escort probability. These theoretical difficulties may be avoided by introducing an so called incomplete normalization allowing to deduce Tsallis' generalized distribution in a more convincing and consistent way. Received 29 October 2001 and Received in final form 30 January 2002     

The DLCA-RLCA transition arising in 2D-aggregation: simulations and mean field theory

A sticking probability model based on the average cluster lifetime is employed for deducing a kernel capable to describe the kinetics of computer simulated irreversible aggregation processes in two dimensions. The deduced kernel describes not only the time evolution of the cluster size distribution for diffusion limited aggregation (DLCA) and reaction limited aggregation (RLCA) but also for the entire transition region between both regimes. The model predicts a crossover to diffusion limited cluster aggregation for all sticking probabilities at long aggregation times. The time needed for reaching the DLCA limit increases for decreasing sticking probability. Received 16 April 2001 and Received in final form 24 May 2001     

Relationships between a microscopic parameter and the stochastic equations for interface's evolution of two growth models

The relationship between a microscopic parameter p, that is related to the probability of choosing a mechanism of deposition, and the stochastic equation for the interface's evolution is studied for two different models. It is found that in one model, that is similar to ballistic deposition, the corresponding stochastic equation can be represented by a Kardar-Parisi-Zhang (KPZ) equation where both λ and ν depend on p in the following way: ν(p) = νp and λ(p) = λp ^3/2. Furthermore, in the other studied model, which is similar to random deposition with relaxation, the stochastic equation can be represented by an Edwards-Wilkinson (EW) equation where ν depends on p according to ν(p) = νp ². It is expected that these results will help to find a framework for the development of stochastic equations starting from microscopic details of growth models. Received 26 August 2002 / Received in final form 20 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: ealbano@inifta.unlp.edu.ar     

Election results and the Sznajd model on Barabasi network

The network of Barabasi and Albert, a preferential growth model where a new node is linked to the old ones with a probability proportional to their connectivity, is applied to Brazilian election results. The application of the Sznajd rule, that only agreeing pairs of people can convince their neighbours, gives a vote distribution in good agreement with reality Received 19 September 2001 and Received in final form 2 November 2001     

Universal crossing probabilities and incipient spanning clusters in directed percolation

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the system is strongly anisotropic, the shape dependence in space-time enters through the effective aspect ratio r _eff = ct/L ^z, where c is a non-universal constant and z the anisotropy exponent. A particular attention is paid to the influence of the initial state on the universal behaviour of the crossing probability. Using anisotropic finite-size scaling and generalizing a simple argument given by Aizenman for isotropic percolation, we also obtain the behaviour of the probability to find n incipient spanning clusters on a finite system at time t. The numerical results are in good agreement with the conjecture. Received 10 February 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: turban@lpm.u-nancy.fr RID="b" ID="b"UMR CNRS 7556     

Beliefs and stochastic modelling of interest rate scenario risk

We present a framework that allows for a systematic assessment of risk given a specific model and belief on the market. Within this framework the time evolution of risk is modeled in a twofold way. On the one hand, risk is modeled by the time discrete and nonlinear garch(1,1) process, which allows for a (time-)local understanding of its level, together with a short term forecast. On the other hand, via a diffusion approximation, the time evolution of the probability density of risk is modeled by a Fokker-Planck equation. Then, as a final step, using Bayes theorem, beliefs are conditioned on the stationary probability density function as obtained from the Fokker-Planck equation. We believe this to be a highly rigorous framework to integrate subjective judgments of future market behavior and underlying models. In order to demonstrate the approach, we apply it to risk assessment of empirical interest rate scenario methodologies, i.e. the application of Principal Component Analysis to the the dynamics of bonds. Received 1st August 2000     

Localization of polymers in a finite medium with fixed random obstacles

In this paper we investigate the conformation statistics of a Gaussian chain embedded in a medium of finite size, in the presence of quenched random obstacles. The similarities and differences between the case of random obstacles and the case of a Gaussian random potential are elucidated. The connection with the density of states of electrons in a metal with random repulsive impurities of finite range is discussed. We also interpret the results obtained in some previous numerical simulations. Received 14 August 2001     

Deconfinement transition: a three dimensional model

We present a three–dimensional model for quark matter with a density dependent quark–quark (confining) potential, which allows to describe a sort of deconfinement transition as the system evolves from a low density assembly of bound structures to a high density free Fermi gas of quarks. We consider different confining potentials, some of which successfully utilized in hadron spectroscopy. We find that a proper treatment of the many–body correlations induced by the medium is essential to disentangle the different nature of the two (hadronic and deconfined) phases of the system. For this purpose the ground state energy per particle and the pair correlation function are investigated. Received: 10 June 1998 / Revised version: 24 September 1998     

Adsorption of a polyampholyte on a charged spherical particle

We develop a scaling theory for a single polyampholyte chain adsorbed on a charged spherical particle in a theta-solvent. Adsorption of a polyampholyte molecule is due to its polarization in the electrostatic field of the particle. For large particles with sizes exceeding the thickness of the adsorbed layer, the conformations of the chain are similar to the one found for polyampholyte adsorption on charged planar surface. However, an adsorbed polyampholyte chain forms a self-similar flower-like structure near the particles with sizes smaller than its Gaussian size. These self-similar structures result from the balance of the polarization energy of loops and the excluded volume interactions between monomers. The structure of an adsorbed polyampholyte in the flower-like conformation is similar to that of a neutral star polymer. Received 3 March 2000 and Received in final form 5 July 2000     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

Dynamics of heuristic optimization algorithms on random graphs

In this paper, the dynamics of heuristic algorithms for constructing small vertex covers (or independent sets) of finite-connectivity random graphs is analysed. In every algorithmic step, a vertex is chosen with respect to its vertex degree. This vertex, and some environment of it, is covered and removed from the graph. This graph reduction process can be described as a Markovian dynamics in the space of random graphs of arbitrary degree distribution. We discuss some solvable cases, including algorithms already analysed using different techniques, and develop approximation schemes for more complicated cases. The approximations are corroborated by numerical simulations. Received 14 March 2002 Published online 31 July 2002     

Fluctuations of the quark densities in nuclei

We study the static scalar susceptibility of the nuclear medium, i.e., the change of the quark condensate for a small modification of the quark mass. In the linear sigma model it is linked to the in-medium sigma propagator and its magnitude increases due to the mixing with the softer modes of the nucleon-hole excitations. We show that the pseudoscalar susceptibility, which is large in the vacuum, owing to the smallness of the pion mass, follows the density evolution of the quark condensate and thus decreases. At normal nuclear matter density the two susceptibilities become much closer, a partial chiral symmetry restoration effect as they become equal when the full restoration is achieved. Received: 20 July 2002 / Accepted: 14 September 2002 / Published online: 21 January 2003 RID="a" ID="a"e-mail: chanfray@ipnl.in2p3.fr Communicated by A. Molinari     

Optimal investment horizons

In stochastic finance, one traditionally considers the return as a competitive measure of an asset, i.e., the profit generated by that asset after some fixed time span Δt, say one week or one year. This measures how well (or how bad) the asset performs over that given period of time. It has been established that the distribution of returns exhibits “fat tails” indicating that large returns occur more frequently than what is expected from standard Gaussian stochastic processes [1-3]. Instead of estimating this “fat tail” distribution of returns, we propose here an alternative approach, which is outlined by addressing the following question: What is the smallest time interval needed for an asset to cross a fixed return level of say 10%? For a particular asset, we refer to this time as the investment horizon and the corresponding distribution as the investment horizon distribution. This latter distribution complements that of returns and provides new and possibly crucial information for portfolio design and risk-management, as well as for pricing of more exotic options. By considering historical financial data, exemplified by the Dow Jones Industrial Average, we obtain a novel set of probability distributions for the investment horizons which can be used to estimate the optimal investment horizon for a stock or a future contract. Received 20 February 2002 Published online 25 June 2002