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We investigate, by means of extensive Monte Carlo simulations, the magnetic critical behavior of the three-dimensional bimodal random-field Ising model at the strong disorder regime. We present results in favor of the two-exponent scaling scenario, $\bar{\eta}=2\eta$, where $\eta$ and $\bar{\eta}$ are the critical exponents describing the power-law decay of the connected and disconnected correlation functions and we illustrate, using various finite-size measures and properly defined noise to signal ratios, the strong violation of self-averaging of the model in the ordered phase., Comment: 8 pages, 6 figures, to be published in Eur. Phys. J. B

Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p>0, there is a cookie, completely suppressing the branching of any particle located there. Abstract. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is \frac{2}{qn}, where q:=1-p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures “shifted” by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.

The properties of the galaxy distribution at large scales are usually studied using statistics which are assumed to be self-averaging inside a given sample. We present a new analysis able to quantitatively map galaxy large scale structures while testing for the stability of average statistical quantities in different sample regions. We find that the newest samples of the Sloan Digital Sky Survey provide unambiguous evidence that galaxy structures correspond to large amplitude density fluctuations at all scales limited only by sample sizes. The two-point correlations properties are self-averaging up to approximately 30 Mpc/h and are characterized by a fractal dimension D=2.1 +- 0.1. Then at all larger scales probed density fluctuations are too large in amplitude and too extended in space to be self-averaging inside the considered volumes. These inhomogeneities are compatible with a continuation of fractal correlations but incompatible with: (i) a homogeneity scale smaller than 100 Mpc/h, (ii) predictions of standard theoretical models, (iii) mock galaxy catalogs generated from cosmological Nbody simuations., 6 pages with 4 figures. Some changes to match the accepted version. To be published in Europhysics Letters (May 2009)

The Pitman-Yor, or Chinese Restaurant Process, is a stochastic process that generates distributions following a power-law with exponents lower than two, as found in a numerous physical, biological, technological and social systems. We discuss its rich behavior with the tools and viewpoint of statistical mechanics. We show that this process invariably gives rise to a condensation, i.e. a distribution dominated by a finite number of classes. We also evaluate thoroughly the finite-size effects, finding that the lack of stationary state and self-averaging of the process creates realization-dependent cutoffs and behavior of the distributions with no equivalent in other statistical mechanical models., Comment: (5pages, 1 figure)

We provide a systematic treatment of self-averaging identities, whose validity is proven in integral average, for dilute spin glasses. The method is quite general, and as a special case recovers the Ghirlanda—Guerra identities, which are therefore proven, together with their extension, to be valid in dilute spin glasses. We focus on dilute spin glasses, but the results hold in all models enjoying stability with respect to the perturbations we introduce; although such a stability is believed to hold for several models, we do not classify them here.

We study the self-averaging properties of the three dimensional site diluted Heisenberg model. The Harris criterion \cite{critharris} states that disorder is irrelevant since the specific heat critical exponent of the pure model is negative. According with some analytical approaches \cite{harris}, this implies that the susceptibility should be self-averaging at the critical temperature ($R_\chi=0$). We have checked this theoretical prediction for a large range of dilution (including strong dilution) at critically and we have found that the introduction of scaling corrections is crucial in order to obtain self-averageness in this model. Finally we have computed critical exponents and cumulants which compare very well with those of the pure model supporting the Universality predicted by the Harris criterion., Comment: 11 pages, 11 figures, 14 tables. New analysis (scaling corrections in the g2=0 scenario) and new numerical simulations. Title and conclusions change

Kinetic equations are often appropriate to model the energy density of high frequency waves propagating in highly heterogeneous media. The limitations of the kinetic model are quantified by the statistical instability of the wave energy density, i.e., by its sensitivity to changes in the realization of the underlying heterogeneous medium modeled as a random medium. In the simplified Ito-Schrodinger regime of wave propagation, we obtain optimal estimates for the statistical instability of the wave energy density for different configurations of the source terms and the domains over which the energy density is measured. We show that the energy density is asymptotically statistically stable (self-averaging) in many configurations. In the case of highly localized source terms, we obtain an explicit asymptotic expression for the scintillation function in the high frequency limit.

A class of two-parameter Poisson-Dirichlet distributions have non-vanishing coefficient of variation. This phenomenon is also known as non-self averaging stochastic multi-sector endogenous growth model. This model is used to raise questions on the use of means in assessing effectiveness of macroeconomic or other macro-policy effects. The coefficients of variations of the number of total sectors, and of sectors of a given size all remain positive as the model size grows unboundedly.

We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations $\sim r^{4-d}$., Comment: 29 pages

The spectral form factor as well as the two-point correlator of the density of (quasi-)energy levels of individual quantum dynamics are not self-averaging. Only suitable smoothing turns them into useful characteristics of spectra. We present numerical data for a fully chaotic kicked top, employing two types of smoothing: one involves primitives of the spectral correlator, the second a small imaginary part of the quasi-energy. Self-averaging universal (like the CUE average) behavior is found for the smoothed correlator, apart from noise which shrinks like $1\over\sqrt N$ as the dimension $N$ of the quantum Hilbert space grows. There are periodically repeated quasi-energy windows of correlation decay and revival wherein the smoothed correlation remains finite as $N\to\infty$ such that the noise is negligible. In between those windows (where the CUE averaged correlator takes on values of the order ${1\over N^2}$) the noise becomes dominant and self-averaging is lost. We conclude that the noise forbids distinction of CUE and GUE type behavior. Surprisingly, the underlying smoothed generating function does not enjoy any self-averaging outside the range of its variables relevant for determining the two-point correlator (and certain higher-order ones). --- We corroborate our numerical findings for the noise by analytically determining the CUE variance of the smoothed single-matrix correlator.

We establish the averaging property for a queuing process with one server, M(t)/GI/1. It is a new relation between the output flow rate and the input flow rate, crucial in the study of the Poisson Hypothesis. Its implications include the statement that the output flow always possesses more regularity than the input flow., 18 pages, one typo removed

We apply the recently developed critical minimum energy subspace scheme for the investigation of the random-field Ising model. We point out that this method is well suited for the study of this model. The density of states is obtained via the Wang-Landau and broad histogram methods in a unified implementation by employing the N-fold version of the Wang-Landau scheme. The random-fields are obtained from a bimodal distribution ($h_{i}=\pm2$), and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from $L=4$ to $L=32$. Observing the finite-size scaling behavior of the maxima of the specific heats we examine the question of saturation of the specific heat. The lack of self-averaging of this quantity is fully illustrated and it is shown that this property may be related to the question mentioned above., 8 pages, 7 figures, extended version with two new figures, version as accepted for publication to Physical Review E

According to recent progresses in the finite size scaling theory of disordered systems, thermodynamic observables are not self-averaging at critical points when the disorder is relevant in the Harris criterion sense. This lack of self-averageness at criticality is directly related to the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. In this paper, we apply this analysis to disordered Poland-Scheraga models with different loop exponents $c$,corresponding to marginal and relevant disorder. In all cases, we numerically obtain a Gaussian histogram of pseudo-critical temperatures $T_c(i,L)$ with mean $T_c^{av}(L)$ and width $\Delta T_c(L)$. For the marginal case $c=1.5$ corresponding to two-dimensional wetting, both the width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay as $L^{-1/2}$, so the exponent is unchanged ($\nu_{random}=2=\nu_{pure}$) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder $c=1.75$, the width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ decay with the same new exponent $L^{-1/\nu_{random}}$ (where $\nu_{random} \sim 2.7 > 2 > \nu_{pure}$) and there is again no self-averaging at criticality. Finally for the value $c=2.15$, of interest in the context of DNA denaturation, the transition is first-order in the pure case. In the presence of disorder, the width $\Delta T_c(L) \sim L^{-1/2}$ dominates over the shift $[T_c(\infty)-T_c^{av}(L)] \sim L^{-1}$, i.e. there are two correlation length exponents $\nu=2$ and $\tilde \nu=1$ that govern respectively the averaged/typical loop distribution., Comment: 15 pages; 8 figures

This paper is devoted to a detailed study of a p-spins interaction model with external field, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its fluctuations, the thermodynamical limit for the free energy and the definition of an Almeida-Thouless type line. Those results show that the external field dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic field vanishes., Comment: 57 pages

Recent renormalization group results predict non self averaging behaviour at criticality for relevant disorder. However, we find strong self averaging(SA) behaviour in the critical region of a quenched Ising model on an ensemble of small-world networks, despite the relevance of the random bonds at the pure critical point., Comment: 4 pages, revtex, 4 figures; replaced by the published version to appear in Physics Letters A

We consider a large population of asexually reproducing individuals in absence of selective pressure. The population size is maintained constant by the environment. We find out that distances between individuals (time from the last common ancestor) exhibit highly non-trivial properties. In particular their distribution in a single population is random even in the thermodynamical limit, i.e., there is lack of self-averaging. As a result, not only distances are different for different pairs of individuals but also the mean distance of the individuals of a given population is different at different times. All computed quantities are parameters free and only scale linearly with the population size. Results in this paper may have some relevance in the ‘Out of Africa/Multi-regional’ debate about the origin of modern man. In fact, the recovery of mitochondrial DNA from Neandertal fossils in three different loci: Feldhofer (Germany), Mezmaiskaya (Northern Caucaso), Vinjia (Croatia), permitted to compare Neandertal/Neandertal distances with Neandertal/modern and modern/modern ones.

In a previous paper we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit., 7 pages, 4 ps figures

Random walk models in one-dimensional disordered media with an oscillatory input current are investigated theoretically as generic models of the boundary perturbation experiment. It is shown that the complex admittance obtained in the experiment is not self-averaging when the jump rates $w_i$ are random variables with the power-law distribution $\rho(w_i)\sim {w_i}^{\alpha-1} (0 < \alpha \leq 1)$. More precisely, the frequency-dependence of the disorder-averaged admittance $$ disagrees with that of the admittance $\chi$ of any sample. It implies that the Cole-Cole plot of $$ shows a different shape from that of the Cole-Cole plots of $\chi$ of each sample. The condition for absence of self-averaging is investigated with a toy model in terms of the extended central limit theorem. Higher dimensional media are also investigated and it is shown that the complex admittance for two-dimensional or three-dimensional media is also non-self-averaging., Comment: 38 pages, 18 figures, submitted to Phys. Rev. B. Section VII was revised completely with the new results for higher dimensional disordered media

We study the dynamics of the q-state random bond Potts ferromagnet on the square lattice at its critical point by Monte Carlo simulations with single spin-flip dynamics. We concentrate on q=3 and q=24 and find, in both cases, conventional, rather than activated, dynamics. We also look at the distribution of relaxation times among different samples, finding different results for the two q values. For q=3 the relative variance of the relaxation time tau at the critical point is finite. However, for q=24 this appears to diverge in the thermodynamic limit and it is ln(tau) which has a finite relative variance. We speculate that this difference occurs because the transition of the corresponding pure system is second order for q=3 but first order for q=24., 9 pages, 13 figures, final published version

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner transform by the solution of the Liouville equations, and the limit theorem for two-particle motion along the characteristics of the Liouville equations. The results are applied to a mathematical model of the time-reversal experiments for the acoustic waves, and self-averaging properties of the re-transmitted wave are proved.

The Harris-Aharony criterion for a statistical model predicts, that if a specific heat exponent $\alpha \ge 0$, then this model does not exhibit self-averaging. In two-dimensional percolation model the index $\alpha=-{1/2}$. It means that, in accordance with the Harris-Aharony criterion, the model can exhibit self-averaging properties. We study numerically the relative variances $R_{M}$ and $R_{\chi}$ for the probability $M$ of a site belongin to the "infinite" (maximum) cluster and the mean finite cluster size $\chi$. It was shown, that two-dimensional site-bound percolation on the square lattice, where the bonds play the role of impurity and the sites play the role of the statistical ensemble, over which the averaging is performed, exhibits self-averaging properties., Comment: 15 pages, 5 figures

We investigate the transport of a passive tracer in a two-dimensional stratified random medium with flow parallel and perpendicular to the strata. Assuming a Gaussian random flow with a Gaussian correlation function, it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterising the tracer transport, but also to investigate the self-averaging properties of these quantities. We give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time. As it turns out, the sample to sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite times, which implies that the given two-dimensional model is not self-averaging., 14 pages, 4 figures

We develop a generalization of the Thouless-Anderson-Palmer (TAP) mean-field approach of disorder physics. which makes the method applicable to the computation of approximate averages in probabilistic models for real data. In contrast to the conventional TAP approach, where the knowledge of the distribution of couplings between the random variables is required, our method adapts to the concrete set of couplings. We show the significance of the approach in two ways: Our approach reproduces replica symmetric results for a wide class of toy models (assuming a nonglassy phase) with given disorder distributions in the thermodynamic limit. On the other hand, simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional TAP approaches.

Current theories of heteropolymers are inherently macrpscopic, but are applied to folding proteins which are only mesoscopic. In these theories, one computes the averaged free energy over sequences, always assuming that it is self-averaging -- a property well-established only if a system with quenched disorder is macroscopic. By enumerating the states and energies of compact 18, 27, and 36mers on a simplified lattice model with an ensemble of random sequences, we test the validity of the self-averaging approximation. We find that fluctuations in the free energy between sequences are weak, and that self-averaging is a valid approximation at the length scale of real proteins. These results validate certain sequence design methods which can exponentially speed up computational design and greatly simplify experimental realizations., Comment: 4 pages, 3 figures

We discuss a series of Monte Carlo studies of systems with quenched connectivity disorder - spin models defined on quenched, random lattices generated in simulations of 2D quantum gravity, paying particular attention to non-self-averaging properties.

Using the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm, we simulate the two-dimensional (2D) site-diluted Ising model. Since we can tune the critical point of each random sample automatically with the PCC algorithm, we succeed in studying the sample-dependent ${T}_{c}(L)$ and the sample average of physical quantities at each ${T}_{c}(L)$ systematically. Using the finite-size scaling (FSS) analysis for ${T}_{c}(L),$ we discuss the importance of corrections to FSS both in the strong-dilution and weak-dilution regions. The critical phenomena of the 2D site-diluted Ising model are shown to be controlled by the pure fixed point. The crossover from the percolation fixed point to the pure Ising fixed point with the system size is explicitly demonstrated by the study of the Binder parameter. We also study the distribution of critical temperature ${T}_{c}(L).$ Its variance shows the power-law L dependence, ${L}^{\ensuremath{-}n},$ and the estimate of the exponent n is consistent with the prediction of Aharony and Harris [Phys. Rev. Lett. 77, 3700 (1996)]. Calculating the relative variance of critical magnetization at the sample-dependent ${T}_{c}(L),$ we show that the 2D site-diluted Ising model exhibits weak self-averaging.

We study the distribution of the $n$-th energy level for two different one-dimensional random potentials. This distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function. We first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum. It is demonstrated that, in the limit of a large system ($L\to\infty$), the distribution of the $n$-th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables. In the second part we consider the case of a supersymmetric random Hamiltonian (potential $V(x)=\phi(x)^2+\phi'(x)$). We study first the case of $\phi(x)$ being a white noise with zero mean. It is in particular shown that the ground state energy, which behaves on average like $\exp{-L^{1/3}}$ in agreement with previous work, is not a self averaging quantity in the limit $L\to\infty$ as is seen in the case of diagonal disorder. Then we consider the case when $\phi(x)$ has a non zero mean value., Comment: LaTeX, 33 pages, 9 figures

We derive exact statistical properties of a class of recursive fragmentation processes. We show that introducing a fragmentation probability 0, Comment: 4 pages, revtex

We investigate the non-self-averaging properties of the dynamics of Ising, 4-state Potts and 10-state Potts models in single-cluster Monte Carlo simulations on quenched ensembles of planar, trivalent Phi3 random graphs, which we use as an example of relevant quenched connectivity disorder. We employ a novel application of scaling techniques to the cumulative probability distribution of the autocorrelation times for both the energy and magnetisation in order to discern non-self-averaging. Although the specific results discussed here are for quenched random graphs, the method has quite general applicability., Comment: 9 pages + 11 figures

The Longest Common Subsequence (LCS) Problem asks for the longest sequence of (non-contiguous) matches between two given strings of characters. Using extensive Monte Carlo simulations, we find a finite size scaling law of the form E(L)/N =C + A/(N^1/2 ln N)+... for the mean LCS length of two random strings of size N over S letters. We provide precise estimates of C for S between 2 and 15. We consider also a related Bernoulli Matching model where the different entries of an N times M array are independently occupied with probability 1/S. In that case we find the expression of the limit of L(N,M)/N as N grows to infinity, as a function of r=M/N. This expression provides a very good approximation for the Random String model, which gets more and more accurate as S increases. The question of the ``universality class'' of the LCS problem is also considered. For the Bernoulli Matching model we find very good agreement with recent scaling predictions of Hwa and Lassig for Needleman-Wunsch sequence alignment. We find however that the variance of the LCS length has a different scaling different in the Random String model, suggesting that long-ranged correlations among the matches are relevant in this model. We finally study the ``ground state'' properties of this problem. We find in particular that the number of solutions typically grows exponentially with N, i.e. this system has a residual entropy at T=0. Also the overlap between two LCSs chosen at random is found to be self averaging and to aproach a definite value q(S), 17 pages, 12 figures. Accepted for publication in EPJ B

We present a stochastic evolutionary model obtained through a perturbation of Kauffman's maximally rugged model, which is recovered as a special case. Our main results are: (i) existence of a percolation-like phase transition in the finite phase space case; (ii) existence of non self-averaging effects in the thermodynamic limit. Lack of self-averaging emerges from a fragmentation of the space of all possible evolutions, analogous to that of a geometrically broken object. Thus the model turns out to be exactly solvable in the thermodynamic limit., 22 pages, 1 figure

The distributions $P(X)$ of singular thermodynamic quantities in an ensemble of quenched random samples of linear size $l$ at the critical point $T_c$ are studied by Monte Carlo in two models. Our results confirm predictions of Aharony and Harris based on Renormalization group considerations. For an Ashkin-Teller model with strong but irrelevant bond randomness we find that the relative squared width, $R_X$, of $P(X)$ is weakly self averaging. $R_X\sim l^{\alpha/\nu}$, where $\alpha$ is the specific heat exponent and $\nu$ is the correlation length exponent of the pure model fixed point governing the transition. For the site dilute Ising model on a cubic lattice, known to be governed by a random fixed point, we find that $R_X$ tends to a universal constant independent of the amount of dilution (no self averaging). However this constant is different for canonical and grand canonical disorder. We study the distribution of the pseudo-critical temperatures $T_c(i,l)$ of the ensemble defined as the temperatures of the maximum susceptibility of each sample. We find that its variance scales as $(\delta T_c(l))^2 \sim l^{-2/\nu}$ and NOT as $\sim l^{-d}. We find that $R_\chi$ is reduced by a factor of $\sim 70$ with respect to $R_\chi (T_c)$ by measuring $\chi$ of each sample at $T_c(i,l)$. We analyze correlations between the magnetization at criticality $m_i(T_c,l)$ and the pseudo-critical temperature $T_c(i,l)$ in terms of a sample independent finite size scaling function of a sample dependent reduced temperature $(T-T_c(i,l))/T_c$. This function is found to be universal and to behave similarly to pure systems., Comment: 31 pages, 17 figures, submitted to Phys. Rev. E

Conditions are given under which one may prove that the stochastic dynamics of on-line learning can be described by the deterministic evolution of a finite set of order parameters in the thermodynamic limit. A global constraint on the average magnitude of the increments in the stochastic process is necessary to ensure self-averaging. In the absence of such a constraint, convergence may only be in probability., Comment: 10 pages

In the present Ph.D. dissertation I present results concerning disordered and frustrated models of relevance in statistical mechanics and in combinatorial optimization. As an application of spin glass theory I study the disordered and frustrated Blume-Emery-Griffiths model. The model is treated in its mean-field approximation using replicas. Within the Ansatz of replica-symmetry, I present a complete numerical solution; I also discuss effects of replica symmetry breaking. The stability of the RS solution is studied and the regions of instability inferred. The phase diagram exhibits first and second order transitions. The tricritical point is still present in the frustrated model, in agreement with former work. A version of the BEG model with disordered chemical potential is also studied. The calculations confirm that the disorder decreases the tricritical temperature. Next, I consider the graph partitioning problem, a combinatorial optimization problem, which, from the point of view of statistical mechanics is a spin glass model with the constraint of zero magnetisation. I focus on the statistical properties of low energy solutions generated by "heuristic" algorithms designed to solve such hard combinatorial optimization problems. Several heuristics proposed to solve this problem were implemented. Scaling laws are obtained; in particular, the average cost and its variance grow linearly with the number of vertices of the graphs. As a consequence the cost found by the heuristics is self-averaging. I suggest that this property is quite general, valid for random solutions, quasi-optimal solutions, and probably for the optimum solutions, too. Furthermore a ranking method is proposed and illustrated on an ensemble of graph partitioning problems. This ranking procedure takes into account the quality of the solution as well as the time necessary to find that solution. In the third part of this dissertation I study in detail the zero-temperatures properties of spin glasses on sparse random graphs with fixed connectivity. Spin glasses on these graphs may be considered as a more realistic approximation to real spin glasses as represented by the model of Sherrington and Kirkpatrick. I have designed a new algorithm for finding low energy states. Second, I present a method for deriving the ground state energy from heuristic algorithms, even though they are not guaranteed to find the optimum. Third, I present a numerical test of a conjecture due to Banavar, Sherrington and Sourlas, giving the large volume energy density of the ground states as function of the connectivity. The distribution of the order parameter is found to be non-trivial, and I give evidence for the presence of ultrametricity for all values of the connectivity. These results confirm the expectation that the remarquable properties of the infinite range Sherrington-Kirkpatrick model carry over to more realistic models, as for example the spin glass model on random graphs with finite connectivity studied in the present work., Dans la présente thèse de doctorat je présente des résultats concernant des modèles désordonnés et frustrés venant de la physique statistique et de l'optimisation combinatoire. Comme application de la théorie des verres de spins, j'étudie le modèle de Blume, Emery et Griffiths désordonné et frustré. Ce modèle est traité dans l'approximation de champ moyen dans le cadre de la méthode des répliques A l'aide de l'Ansatz symétrique dans les répliques je présente une solution numérique complète puis je discute des effets de brisure de cette symétrie La stabilité de la solution symétrique a été Rudik et les régions instables identifiées Le diagramme de phase exhibe des transitions de premier et de second ordre. Le point tricritique persiste dans le modèle frustré, Ce qui est en accord avec des travaux antérieurs une version du modèle BEG avec un potentiel chimique désordonné a également été étudiée. les calculs confirment que le point tricritique apparaît à plus basse température quand il y a du désordre. Ensuite je considère le problème de la bipartition d'un graphe. Ce problème correspond du point de vue de la physique statistique h un verre de spins soumis h une contrainte d'aimantation totale nulle. je considère les propriétés statistiques des solutions de faible énergie engendrées par des algorithmes heuristiques. de tels algorithme sont en général conçus pour résoudre des problèmes d'optimisation combinatoire qui sont NP- difficiles. Plusieurs heuristiques ont 60 implémentées pour le problème de la bipartition de graphe. des lois d'échelle ont été obtenues : en particulier la moyenne et la variance du coût obéissent A une loi linéaire en N. Par conséquent le coût obtenu par des heuristiques est une quantité auto-moyennante. je suggère que cette propriété est générale valable aussi pour les solutions aléatoires pour les solutions quasi-optimales et pour les solutions optimales. En outre je propose une procédure pour comparer des algorithmes heuristiques. Cette procédure tient compte de la qualité de la solution aussi bien que du temps de calcul utilisé. Dans la troisième partie de ma thèse j'ai étudié en détail les propriétés h température nulle des verres de spins sur des graphes aléatoires lacunaires avec une coordination fixe. les verres de spins sur de tels graphes peuvent être considérés comme une approximation aux vrais verres de spins qui est plus réaliste que le modèle de Sherrington et Kirkpatrick. J'ai conçu un nouvel algorithme pour trouver les états fondamentaux. Aussi je teste numériquement une conjecture de Banavar, Sherrington et Sourlas qui donne la densité d'énergie du fondamental dans la limite de grande taille en fonction de la coordination. La distribution du paramètre d'ordre se révèle être non triviale et les données présentent une forte indication de la présence d'ultramétricité pour toutes les valeur de la coordination. Ces résultats confirment que les propriétés particulières des verres de spin, déduites an niveau de l'approximation de champ moyen dans le cadre du modèle de Sherrington et Kirkpatrick, sont aussi présentes pour des modèles plus réalistes comme les verres de spins sur des graphes aléatoires lacunaires avec une coordination fixe.

The distributions $P(X)$ of singular thermodynamic quantities, on an ensemble of $d$-dimensional quenched random samples of linear size $L$ near a critical point, are analyzed using the renormalization group. For $L$ much larger than the correlation length $\ensuremath{\xi}$, we recover strong self-averaging (SA): $P(X)$ approaches a Gaussian with relative squared width ${R}_{X}\ensuremath{\sim}(L/\ensuremath{\xi}{)}^{\ensuremath{-}d}$. For $L\ensuremath{\ll}\ensuremath{\xi}$ we show weak SA ( ${R}_{X}$ decays with a small power of $L$) or no SA [ $P(X)$ approaches a non-Gaussian, with universal $L$-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.

We study the distribution of (1+1)-dimensional self-affine lines resulting from the zero-temperature directed polymer in a random medium or from critical directed percolation problems. The related amplitude ratios depend on whether the finite size scaling statistics is made over finite segments of the infinite objects or over the finite objects of diverging size. As a by-product we obtain the correction to scaling exponents for the finite size scaling of the distribution functions.

We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random patterns, $M$, as a function of the system size $N$ satisfies $\limsup M(N)/N=0$. In this case the rate function (or free energy as a function of the overlap parameters) is independent of the disorder for almost all realization of the patterns and given by an explicit variational formula., 31pp; Plain-TeX, hardcopy available on request from bovier@iaas-berlin.d400.de

We investigate self-averaging properties in the transport of particles through random media. We show rigorously that in the subdiffusive anomalous regime transport coefficients are not self--averaging quantities. These quantities are exactly calculated in the case of directed random walks. In the case of general symmetric random walks a perturbative analysis around the Effective Medium Approximation (EMA) is performed., 4 pages, RevTeX , No figures, submitted to Physical Review E (Rapid Communication)

The self averaging properties of conductance $g$ are explored in random resistor networks with a broad distribution of bond strengths $P(g)\simg^{\mu-1}$. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of size $L$ and distribution tail parameter $\mu$. For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit $\mu$ --> 0. A {\it disorder length} $\xi_D$ is identified beyond which the system is effectively homogeneous. This length diverges as $\xi_D \sim |\mu|^{-\nu}$ ($\nu$ is the regular percolation correlation length exponent) as $\mu$-->0. This suggest that exactly the same critical behavior can be induced by geometrical disorder and bu strong bond disorder with the bond occupation probability $p$$\mu$. Only lattices at the percolation threshold have renormalized probability distribution in a {\it Levy-like} basin. At the threshold the disorder length diverges at a vritical tail strength $\mu_c$ as $|\mu-\mu_c|^{-z}$, with $z=3.2\pm 0.1$, a new exponent. Critical path analysis is used in a generalized form to give form to give the macroscopic conductance for lattice above $p_c$., Comment: 16 pages plain TeX file, 6 figures available upon request.IBC-1603-012

The purpose of this thesis is the study of the asymptotic behaviour of the Hopfield model, a model simulating the neural phenomenon of associative memory., L'objet de cette thèse est l'étude asymptotique du modèle de Hopfield dont le but est de simuler le phénomène neuronal de mémoire associative. Après une brève introduction au calcul neuronal et une description générale de la modélisation mathématique de la mémoire associative, nous définissons le modèle étudié dans le cadre d'une dynamique d'évolution déterministe, respectivement séquentielle (modèle de Hopfield) ou parallèle (modèle de Little). Nous étudions alors la stabilité asymptotique de $p$ images originales, au sens presque sûr pour l'espace de probabilité associé aux variables aléatoires modélisant ces images, ainsi que l'attraction de certaines configurations, en une seule étape de la dynamique, si $p$ est de l'ordre $N/\log N$ ($N$ la taille du réseau). La fonction énergie ayant notamment pour minima locaux les images originales et tous les autres points fixes de l'application associée à la dynamique, il est intéressant d'en connaître les fluctuations sur l'espace des configurations. Après avoir rappelé les résultats de Newman, relatifs à l'existence de barrières énergétiques, nous montrons que asymptotiquement et presque sûrement, sous certaines hypothèses sur $p$, les images combinées, combinaisons d'un nombre fine ou de toutes les images combinées, ne peuvent être des minima plus profonds que les images originales elles-mêmes. En ces points, nous calculons la limite presque sûre du hamiltonien normalisé. Au chapitre suivant, nous décrivons la dynamique stochastique de Glauber qui nous conduit à définir les mesures de Gibbs pour la limite thermodynamique de ce systèmes. Nous étudions alors, dans un dernier chapitre, le comportement asymptotique de l'énergie libre: pour toute température, cette variable aléatoire converge presque sûrement vers une constante, si $p/N$ converge vers 0, et vérifie la propriété d'être auto-moyennée, si $p$ est inférieur au proportionnel à $N$. En conclusion, nous terminons en évoquant quelques problèmes ouverts et des extensions possibles du modèle de Hopfield.

The purpose of this thesis is the study of the asymptotic behaviour of the Hopfield model, a model simulating the neural phenomenon of associative memory.; L'objet de cette thèse est l'étude asymptotique du modèle de Hopfield dont le but est de simuler le phénomène neuronal de mémoire associative. Après une brève introduction au calcul neuronal et une description générale de la modélisation mathématique de la mémoire associative, nous définissons le modèle étudié dans le cadre d'une dynamique d'évolution déterministe, respectivement séquentielle (modèle de Hopfield) ou parallèle (modèle de Little). Nous étudions alors la stabilité asymptotique de $p$ images originales, au sens presque sûr pour l'espace de probabilité associé aux variables aléatoires modélisant ces images, ainsi que l'attraction de certaines configurations, en une seule étape de la dynamique, si $p$ est de l'ordre $N/\log N$ ($N$ la taille du réseau). La fonction énergie ayant notamment pour minima locaux les images originales et tous les autres points fixes de l'application associée à la dynamique, il est intéressant d'en connaître les fluctuations sur l'espace des configurations. Après avoir rappelé les résultats de Newman, relatifs à l'existence de barrières énergétiques, nous montrons que asymptotiquement et presque sûrement, sous certaines hypothèses sur $p$, les images combinées, combinaisons d'un nombre fine ou de toutes les images combinées, ne peuvent être des minima plus profonds que les images originales elles-mêmes. En ces points, nous calculons la limite presque sûre du hamiltonien normalisé. Au chapitre suivant, nous décrivons la dynamique stochastique de Glauber qui nous conduit à définir les mesures de Gibbs pour la limite thermodynamique de ce systèmes. Nous étudions alors, dans un dernier chapitre, le comportement asymptotique de l'énergie libre: pour toute température, cette variable aléatoire converge presque sûrement vers une constante, si $p/N$ converge vers 0, et vérifie la propriété d'être auto-moyennée, si $p$ est inférieur au proportionnel à $N$. En conclusion, nous terminons en évoquant quelques problèmes ouverts et des extensions possibles du modèle de Hopfield.

We consider the Hopfield model with M(N) = αN patterns, where N is the number of neurons. We show that if α is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work [BGPl]. The key new ingredient is a self averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.

The statistics of equally weighted random paths (ideal polymer) is studied in $2$ and $3$ dimensional percolating clusters. This is equivalent to diffusion in the presence of a trapping environment. The number of $N$ step walks follows a log-normal distribution with a variance growing asymptotically faster than the mean which leads to a weak non self-averaging behaviour. Critical exponents associated with the scaling of the two-points correlation function do not obey standard scaling laws., 11 pages, Revtex 3.0, 5 figures available upon request, PURD-TH-93-02. Email:giacometti@padova.infn.it

We consider fluctuations in both the shattering and normal (nonshattering) fragmentation processes. The continuum shattering process is shown to lack the self-averaging property: the rms fluctuation in the total number of the finite-size fragements is larger than the mean value. The discrete shattering process exhibits strong fluctuations at the intermediate times when the total number of fragments is much greater than 1 but smaller than the number of the monomers contained in the primary particle. For the normal process, the fluctuations are small and self-averaging exists.