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Large deviation, Phase transition

On the existence of a glass transition in a Random Energy Model,

, , , J. Phys. A, (2013), 315002,

Abstract

We revisit the Random Energy Model by assuming that the energy of each configuration is given by the sum of Nindependent contributions (local energies'') with finite variances,instead of directly assuming a Gaussian energy distribution with a variance proportional to N. Using the large deviation formalism,we find that the glass transition generically exists when local energies have a smooth distribution. In contrast,if the distribution of the local energies has a Dirac mass at the minimal energy,the glass transition may cease to exist. This property is illustrated on a simple example,in which local energies are equal to zero with a finite probability,and positive otherwise.

Estimation, Phase transition

Critical moment definition and estimation,for finite size observation of log-exponential-power law random variables,

, , , Signal Processing, (2012), 2848-2865,

Abstract

This contribution aims at studying the behaviour of the classical sample moment estimator,$S(n,q)=\sum_k=1^n X_k^q/n$,as a function of the number of available samples $n$,in the case where the random variables $X$ are positive,have finite moments at all orders and are naturally of the form $X=\exp Y$ with the tail of $Y$ behaving like $\exp(-y^\rho)$. This class of laws encompasses and generalizes the classical example of the log-normal law. This form is motivated by a number of applications stemming from modern statistical physics or multifractal analysis. Borrowing heuristic and analytical results from the analysis of the Random Energy Model in statistical physics,a critical moment $q_c(n)$ is defined as the largest statistical order $q$ up to which the sample mean estimator $S(n,q)$ correctly accounts for the ensemble average $\E X^q$,for a given $n$. A practical estimator for the critical moment $q_c(n)$ is then proposed. Its statistical performance are studied analytically and illustrated numerically in the case of i.i.d. samples. A simple modification is proposed to explicitly account for correlation amongst the observed samples. Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series.

Estimation, Multifractal, Phase transition

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

, , , , Physica D, (2011), 1245-1253,

Abstract

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: 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) that multifractal exponents necessarily behave linearly in $q$,for $q > q^*$. Tayloring 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.