Universality of Zipf's Law

We introduce a simple and generic model that reproduces Zipf's law. By regarding the time evolution of the model as a random walk in the logarithmic scale, we explain theoretically why this model reproduces Zipf's law. The explanation shows that the behavior of the model is very robust and universal.Comment: 5 eps files included. To be published in J. Phys. Soc. Jp

Risk bubbles and market instability

We discuss a simple model of correlated assets capturing the feedback effects induced by portfolio investment in the covariance dynamics. This model predicts an instability when the volume of investment exceeds a critical value. Close to the critical point the model exhibits dynamical correlations very similar to those observed in real markets. Maximum likelihood estimates of the model’s parameter for empirical data indeed confirms this conclusion. We show that this picture is confirmed by the empirical analysis for different choices of the time horizon

Dynamic instability in a phenomenological model of correlated assets

We show that financial correlations exhibit a non-trivial dynamic behavior. We introduce a simple phenomenological model of a multi-asset financial market, which takes into account the impact of portfolio investment on price dynamics. This captures the fact that correlations determine the optimal portfolio but are affected by investment based on it. We show that such a feedback on correlations gives rise to an instability when the volume of investment exceeds a critical value. Close to the critical point the model exhibits dynamical correlations very similar to those observed in real markets. Maximum likelihood estimates of the model’s parameter for empirical data indeed confirm this conclusion, thus suggesting that real markets operate close to a dynamically unstable point

Non perturbative renormalization group approach to surface growth

We present a recently introduced real space renormalization group (RG) approach to the study of surface growth. The method permits us to obtain the properties of the KPZ strong coupling fixed point, which is not accessible to standard perturbative field theory approaches. Using this method, and with the aid of small Monte Carlo calculations for systems of linear size 2 and 4, we calculate the roughness exponent in dimensions up to d=8. The results agree with the known numerical values with good accuracy. Furthermore, the method permits us to predict the absence of an upper critical dimension for KPZ contrarily to recent claims. The RG scheme is applied to other growth models in different universality classes and reproduces very well all the observed phenomenology and numerical results. Intended as a sort of finite size scaling method, the new scheme may simplify in some cases from a computational point of view the calculation of scaling exponents of growth processes.Comment: Invited talk presented at the CCP1998 (Granada

Generalized Dielectric Breakdown Model

We propose a generalized version of the Dielectric Breakdown Model (DBM) for generic breakdown processes. It interpolates between the standard DBM and its analog with quenched disorder, as a temperature like parameter is varied. The physics of other well known fractal growth phenomena as Invasion Percolation and the Eden model are also recovered for some particular parameter values. The competition between different growing mechanisms leads to new non-trivial effects and allows us to better describe real growth phenomena. Detailed numerical and theoretical analysis are performed to study the interplay between the elementary mechanisms. In particular, we observe a continuously changing fractal dimension as temperature is varied, and report an evidence of a novel phase transition at zero temperature in absence of an external driving field; the temperature acts as a relevant parameter for the ``self-organized'' invasion percolation fixed point. This permits us to obtain new insight into the connections between self-organization and standard phase transitions.Comment: Submitted to PR

Interacting Individuals Leading to Zipf's Law

We present a general approach to explain the Zipf's law of city distribution. If the simplest interaction (pairwise) is assumed, individuals tend to form cities in agreement with the well-known statisticsComment: 4 pages 2 figure

Financial instability from local market measures

We study the emergence of instabilities in a stylized model of a financial market, when different market actors calculate prices according to different (local) market measures. We derive typical properties for ensembles of large random markets using techniques borrowed from statistical mechanics of disordered systems. We show that, depending on the number of financial instruments available and on the heterogeneity of local measures, the market moves from an arbitrage-free phase to an unstable one, where the complexity of the market - as measured by the diversity of financial instruments - increases, and arbitrage opportunities arise. A sharp transition separates the two phases. Focusing on two different classes of local measures inspired by real markets strategies, we are able to analytically compute the critical lines, corroborating our findings with numerical simulations.Comment: 17 pages, 4 figure

A New Stochastic Strategy for the Minority Game

We present a variant of the Minority Game in which players who where successful in the previous timestep stay with their decision, while the losers change their decision with a probability $p$. Analytical results for different regimes of $p$ and the number of players $N$ are given and connections to existing models are discussed. It is shown that for $p \propto 1/N$ the average loss $\sigma^2$ is of the order of 1 and does not increase with $N$ as for other known strategies.Comment: 4 pages, 3 figure

On the criticality of inferred models

Advanced inference techniques allow one to reconstruct the pattern of interaction from high dimensional data sets. We focus here on the statistical properties of inferred models and argue that inference procedures are likely to yield models which are close to a phase transition. On one side, we show that the reparameterization invariant metrics in the space of probability distributions of these models (the Fisher Information) is directly related to the model's susceptibility. As a result, distinguishable models tend to accumulate close to critical points, where the susceptibility diverges in infinite systems. On the other, this region is the one where the estimate of inferred parameters is most stable. In order to illustrate these points, we discuss inference of interacting point processes with application to financial data and show that sensible choices of observation time-scales naturally yield models which are close to criticality.Comment: 6 pages, 2 figures, version to appear in JSTA