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We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of price fluctuations. The central result of the present study is the remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). In particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management.An important aspect of risk management is the estimation of the correlations between the price movements of different assets. The probability of large losses for a certain portfolio or option book is dominated by correlated moves of its different constituents -for example, a position which is simultaneously long in stocks and short in bonds will be risky because stocks and bonds move in opposite directions in crisis periods. The study of correlation (or covariance) matrices thus has a long history in finance, and is one of the cornerstone of Markowitz's theory of optimal portfolios [1]. However, a reliable empirical determination of a correlation matrix turns out to be difficult: if one considers N assets, the correlation matrix contains N (N − 1)/2 entries, which must be determined from N time series of length T ; if T is not very large compared to N , one should expect that the determination of the covariances is noisy, and therefore that the empirical correlation matrix is to a large extent random, i.e. the structure of the matrix is dominated by measurement noise. If this is the case, one should be very careful when using this correlation matrix in applications. In particular, as we shall show below, the smallest eigenvalues of this matrix are the most sensitive to this 'noise' -on the other hand, it is precisely the eigenvectors corresponding to these smallest eigenvalues which determine, in Markowitz theory, the least risky portfolios [1]. It is thus important to devise methods which allows one to distinguish 'signal' from 'noise', i.e. eigenvectors and eigenvalues of the correlation matrix containing real information (which one would like to include for risk control), from those which are devoid of any useful information, and, as such, unstable in time. From this point of view, it is interesting to compare the properties of an empirical correlation matrix C to a 'null hypothesis' purely random matrix as one could obtain from a finite time series of strictly uncorrelated assets. Deviations from the random matrix case might then suggest the presence of true information. The theory of Random Matrices has a long history in physics since the fifties [2], and many results are known [3]. As shown below, these results are also of genuine interest in a financial context (see also [4]).The empirical correlation matrix C is constructed from the time series of price change...

We consider materials whose mechanical integrity is the result of a jamming process. We argue that such media are generically "fragile": unable to support certain types of incremental loading without plastic rearrangement. Fragility is linked to the marginal stability of force chain networks within the material. Such ideas may be relevant to jammed colloids and poured sand. The crossover from fragile (when particles are rigid) to elastoplastic behavior is explored.

Another set of experiments which basically carry the same information is those of the so-called 'Thermo-Remanent Magnetisation' (TRM) relaxation 13,14 . The system is cooled under a small magnetic field H, which is left from t = 0 (the time of the quench) to t = t w , and then suddenly switched off. The subsequent relaxation of the magnetisation M can be decomposed as 21,14c This requires to perform many independent quenches where the magnetic noise is recorded for different ages tw and then averaged over the different quenches.

Risk control and derivative pricing have become of major concern to financial institutions, and there is a real need for adequate statistical tools to measure and anticipate the amplitude of the potential moves of the financial markets. Summarising theoretical developments in the field, this 2003 second edition has been substantially expanded. Additional chapters now cover stochastic processes, Monte-Carlo methods, Black-Scholes theory, the theory of the yield curve, and Minority Game. There are discussions on aspects of data analysis, financial products, non-linear correlations, and herding, feedback and agent based models. This book has become a classic reference for graduate students and researchers working in econophysics and mathematical finance, and for quantitative analysts working on risk management, derivative pricing and quantitative trading strategies.

We study various models of independent particles hopping between energy 'traps' with a density of energy barriers ρ(E), on a d dimensional lattice or on a fully connected lattice. If ρ(E) decays exponentially, a true dynamical phase transition between a high temperature 'liquid' phase and a low temperature 'aging' phase occurs. More generally, however, one expects that for a large class of ρ(E), 'interrupted' aging effects appear at low enough temperatures, with an ergodic time growing faster than exponentially. The relaxation functions exhibit a characteristic shoulder, which can be fitted as stretched exponentials. A simple way of introducing interactions between the particles leads to a modified model with an effective diffusion constant in energy space, which we discuss in detail.

We reformulate the interpretation of the mean-field glass transition scenario for finite dimensional systems, proposed by Wolynes and collaborators. This allows us to establish clearly a temperature dependent length ξ * above which the mean-field glass transition picture has to be modified. We argue in favor of the mosaic state introduced by Wolynes and collaborators, which leads to the Adam-Gibbs relation between the viscosity and configurational entropy of glass forming liquids. Our argument is a mixture of thermodynamics and kinetics, partly inspired by the Random Energy Model: small clusters of particles are thermodynamically frozen in low energy states, whereas large clusters are kinetically frozen by large activation energies. The relevant relaxation time is that of the smallest 'liquid' clusters. Some physical consequences are discussed.