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Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in$\mathbb{R}^d$. Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph isself-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.

We find the free-energy in the thermodynamic limit of a one-dimensional XY model associated to a system of N qubits. The coupling among the &sigma, i z is a long range two-body random interaction. The randomness in the couplings is the typical interaction of the Hopfield model with p patterns ( p &lt, N ), where the patterns are p sequences of N independent identically distributed random variables (i.i.d.r.v.), assuming values &plusmn, 1 with probability 1 / 2 . We show also that in the case p &le, &alpha, N , &alpha, &ne, 0 , the free-energy is asymptotically independent from the choice of the patterns, i.e., it is self-averaging.

It is shown that the variance of a perturbation Hamiltonian density vanishes in the infinite-volume limit of the perturbed spin systems with quenched disorder. This is proven in a simpler way and under less assumptions than before. A corollary of this theorem indicates the impossibility of non-spontaneous replica symmetry-breaking in disordered spin systems. The commutativity between the infinite-volume limit and the switched-off limit of a replica symmetry-breaking perturbation implies that the variance of the spin overlap vanishes in the replica symmetric Gibbs state., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1811.04386

International audience; In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In particular, in terms of the renormalized replica Ginzburg–Landau Hamiltonian in dimensions D < 4 an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations is derived. Next, using known fixed-point values for the renormalized coupling parameters the universal curve for such PDF in the dimension D = 3 is obtained. For the specific case of the two-dimensional Ising model, using replica calculations in the renormalization group framework, we derive explicit expressions for the PDF of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energy is Gaussian, and its typical sample-to-sample fluctuations as well as its average value scale with the system size L like ~ Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ-peak in the thermodynamic limit L → ∞.

We consider a recently introduced generalization of the Ising model in which individual spin strength can vary. The model is intended for analysis of ordering in systems comprising agents which, although matching in their binarity (i.e., maintaining the iconic Ising features of `+' or `$-$', `up' or `down', `yes' or `no'), differ in their strength. To investigate the interplay between variable properties of nodes and interactions between them, we study the model on a complex network where both the spin strength and degree distributions are governed by power laws. We show that in the annealed network approximation, thermodynamic functions of the model are self-averaging and we obtain an exact solution for the partition function. This allows us to derive the leading temperature and field dependencies of thermodynamic functions, their critical behavior, and logarithmic corrections at the interface of different phases. We find the delicate interplay of the two power laws leads to new universality classes., Comment: This article belongs to the Special Issue Ising Model: Recent Developments and Exotic Applications

Finite-size effects have been a major and justifiable source of concern for studies of many-body localization, and several works have been dedicated to the subject. In this paper, however, we discuss yet another crucial problem that has received much less attention, that of the lack of self-averaging and the consequent danger of reducing the number of random realizations as the system size increases. By taking this into account and considering ensembles with a large number of samples for all system sizes analyzed, we find that the generalized dimensions of the eigenstates of the disordered Heisenberg spin-1/2 chain close to the transition point to localization are described remarkably well by an exact analytical expression derived for the non-interacting Fibonacci lattice, thus providing an additional tool for studies of many-body localization., Comment: 8 pages, 5 figures

In a disordered system, a quantity is self-averaging when the ratio between its variance for disorder realizations and the square of its mean decreases as the system size increases. Here, we consider a chaotic disordered many-body quantum system and search for a relationship between self-averaging behavior and the properties of the distributions over disorder realizations of various quantities and at different timescales. An exponential distribution, as found for the survival probability at long times, explains its lack of self-averaging, since the mean and the dispersion are equal. Gaussian distributions, however, are obtained for both self-averaging and non-self-averaging quantities. Our studies show also that one can make conclusions about the self-averaging behavior of one quantity based on the distribution of another related quantity. This strategy allows for semianalytical results, and thus circumvents the limitations of numerical scaling analysis, which are restricted to few system sizes., 13 pages, 5 figures

We show that spatial quenched disorder affects polar active matter in ways more complex and far-reaching than believed heretofore. Using simulations of the 2D Vicsek model subjected to random couplings or a disordered scattering field, we find in particular that ergodicity is lost in the ordered phase, the nature of which we show to depend qualitatively on the type of quenched disorder: for random couplings, it remains long-range ordered, but qualitatively different from the pure (disorderless) case. For random scatterers, polar order varies with system size but we find strong non-self-averaging, with sample-to-sample fluctuations dominating asymptotically, which prevents us from elucidating the asymptotic status of order., 6 pages, 5 figures, comments are welcome

Despite its importance to experiments, numerical simulations, and the development of theoretical models, self-averaging in many-body quantum systems out of equilibrium remains underinvestigated. Usually, in the chaotic regime, self-averaging is taken for granted. The numerical and analytical results presented here force us to rethink these expectations. They demonstrate that self-averaging properties depend on the quantity and also on the time scale considered. We show analytically that the survival probability in chaotic systems is not self-averaging at any time scale, even when evolved under full random matrices. We also analyze the participation ratio, R\'enyi entropies, the spin autocorrelation function from experiments with cold atoms, and the connected spin-spin correlation function from experiments with ion traps. We find that self-averaging holds at short times for the quantities that are local in space, while at long times, self-averaging applies for quantities that are local in time. Various behaviors are revealed at intermediate time scales., Comment: 18 pages, 7 figures (as published)

We discuss the interplay between the degree of dynamical stochasticity, memory persistence and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasi-deterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including mean-field, and the short-range random field Ising model., Introduction strongly revised, changed figures. Accepted for publication as a Rapid Communication in Physical Review E

During my last conversation with Masanao Aoki, he told me that the concept of non-self-averaging in statistical physics, frequently appearing in economic and financial systems, has important consequences to policy implication. Zipf’s law in firms-size distribution is one of such examples. Recent Malevergne, Saichev, and Sornette (MSS) model, simple but realistic, gives a framework of stochastic process including firms’ entry, exit, and growth based on Gibrat’s law of proportionate effect and shows that the Zipf’s law is a robust consequence. By using the MSS model, I would like to discuss about the breakdown of Gibrat’s law and the deviation from Zipf’s law, often observed for the regime of small and medium firms. For the purpose of discussion, I recapitulate the derivation of exact solution for the MSS model with some correction and additional information on the distribution for the age of existing firms. I argue that the breakdown of Gibrat’s law is related to the underlying network of firms, most notably production network, in which firms are mutually correlated among each other leading to the larger volatility in the growth for smaller firms that depend as suppliers on larger customers.

The self-averaging behavior of interacting many-body quantum systems has been mostly studied at equilibrium. The present paper addresses what happens out of equilibrium, as the increase of the strength of on-site disorder takes the system to the localized phase. We consider two local and two nonlocal quantities of great experimental and theoretical interest. In the delocalized phase, self-averaging depends on the observable and on the timescale, but the picture simplifies substantially when localization is reached. In the localized phase, the local observables become self-averaging at all times while the nonlocal quantities are throughout non-self-averaging. These behaviors are explained and scaling analysis is provided using the ℓ-bit model and a toy model., E.J.T.-H. acknowledges funding from VIEP-BUAP (Grant Nos. MEBJ-EXC19-G, LUAGEXC19-G), Mexico. He is also grateful to LNS-BUAP for allowing use of their supercomputing facility. M.S. and L.F.S. were supported by the NSF Grant No. DMR-1603418 and gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University at which some of the research for this paper was performed. F.P.B. thanks the Consejer´ıa de Conocimiento, Investigaci´on y Universidad, Junta de Andaluc´ıa and European Regional Development Fund (ERDF), ref. SOMM17/6105/UGR. Additional computer resources supporting this work were provided by the Universidad de Huelva CEAFMC High Performance Computer located in the Campus Universitario el Carmen and funded by FEDER/MINECO project UNHU15CE-2848. L.F.S. is supported by the NSF Grant No. DMR-1936006. Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. G.D.T. acknowledges the hospitality of MPIPKS Dresden, where part of the work was performed.

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuum space version of scale-free percolation introduced in [DW18]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb R^d$. Each vertex is equipped with a random weight and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the the annealed clustering coefficient of one point, which is a strictly positive quantity., 20 pages

Based on the self-consistent cluster approximation of an effective medium for random walk on a lattice of randomly located traps, the issue of the self-averaging of the diffusion coefficient in the subdispersive mode is examined. It is demonstrated that, in this mode, the diffusion coefficient self-averages slowly according to a power law in the case of three-dimensional space, whereas for the one- and two-dimensional cases, it self-averages poorly, with its relative fluctuations decreasing abnormally slowly, according to a logarithmic law.

Based on of the self-consistent cluster approximation of an effective medium for random walk on a lattice with random traps, the kinetics of the self-averaging of the spectral relaxation of the partial populations is analyzed. It is demonstrated that the decrease of the fluctuations in the partial populations in the dispersive mode at long times occurs very slowly, according to a power law. With increasing degree of disorder (fraction of deep traps), a transition from self-averaging to non-self-averaging takes place in spaces of arbitrary dimension.

The stochastic dynamics of a quantum system driven by $N$ statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with increasing $N$ the system approaches a deterministic limit, indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite-dimensional quantum system which, according to an asymptotic analysis, decreases at least as $1/N$. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite $N$ the distance (according to the Frobenius norm) between the averaged evolution unitary operator generated by the Hamiltonian $H$ and the unitary evolution operator generated by the averaged Hamiltonian $\ensuremath{\langle}H\ensuremath{\rangle}$ scales as $1/N$. Numerical simulations enlarge this result to a broader class of noncommuting protocols.

On the basis of a self-consistent cluster effective-medium approximation for random trapping transport, we study the problem of self-averaging of the diffusion coefficient in a nonstationary formulation. In the long-time domain, we investigate different cases that correspond to the increasing degree of disorder. In the regular and subregular cases the diffusion coefficient is found to be a self-averaging quantity-its relative fluctuations (relative standard deviation) decay in time in a power-law fashion. In the subdispersive case the diffusion coefficient is self-averaging in three dimensions (3D) and weakly self-averaging in two dimensions (2D) and one dimension (1D), when its relative fluctuations decay anomalously slowly logarithmically. In the dispersive case, the diffusion coefficient is self-averaging in 3D, weakly self-averaging in 2D, and non-self-averaging in 1D. When non-self-averaging, its fluctuations remain of the same order as, or larger than, its average value. In the irreversible case, the diffusion coefficient is non-self-averaging in any dimension. In general, with the decreasing dimension and/or increasing disorder, the self-averaging worsens and eventually disappears. In the cases of weak self-averaging and, especially, non-self-averaging, the reliable reproducible experimental measurements are highly problematic. In all the cases under consideration, asymptotics with prefactors are obtained beyond the scaling laws. Transition between all cases is analyzed as the disorder increases.

The combination of field tunable bandgap, topological edge states, and valleys in the band structure, makes insulating bilayer graphene a unique localized system, where the scaling laws of dimensionless conductance g remain largely unexplored. Here we show that the relative fluctuations in ln g with the varying chemical potential, in strongly insulating bilayer graphene (BLG) decay nearly logarithmically for channel length up to L/${\xi}$ ${\approx}$ 20, where ${\xi}$ is the localization length. This 'marginal' self averaging, and the corresponding dependence of on L, suggest that transport in strongly gapped BLG occurs along strictly one-dimensional channels, where ${\xi}$ ${\approx}$ 0.5${\pm}$0.1 ${\mu}$m was found to be much longer than that expected from the bulk bandgap. Our experiment reveals a nontrivial localization mechanism in gapped BLG, governed by transport along robust edge modes., Comment: This document is the Author's version of a submitted work that was subsequently accepted for publication in Physical Review Letters

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in Artificial Intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mechanics perspective these systems display the same (random) Gibbs measure of bi-partite spin-glasses, whose rigorous treatment is notoriously difficult. In this work, beyond providing a brief review on RBMs from both the learning and the retrieval perspectives, we aim to contribute to their analytical investigation, by considering two distinct realizations of their weights (i.e., Boolean and Gaussian) and studying the properties of their related free energies. More precisely, focusing on a RBM characterized by digital couplings, we first extend the Pastur-Shcherbina-Tirozzi method (originally developed for the Hopfield model) to prove the self-averaging property for the free energy, over its quenched expectation, in the infinite volume limit, then we explicitly calculate its simplest approximation, namely its annealed bound. Next, focusing on a RBM characterized by analogical weights, we extend Guerra's interpolating scheme to obtain a control of the quenched free-energy under the assumption of replica symmetry: we get self-consistencies for the order parameters (in full agreement with the existing Literature) as well as the critical line for ergodicity breaking that turns out to be the same obtained in AGS theory. As we discuss, this analogy stems from the slow-noise universality. Finally, glancing beyond replica symmetry, we analyze the fluctuations of the overlaps for an estimate of the (slow) noise affecting the retrieval of the signal, and by a stability analysis we recover the Aizenman-Contucci identities typical of glassy systems., Comment: 21 pages, 1 figure

Tracking tracer particles in heterogeneous environments plays an important role in unraveling the material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here, we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceeds trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants with the aid of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities do not depend on the initial condition nor on the thermal histories but depend crucially on the disorder realization., Comment: 12 pages, 7 figures

International audience; Diffusion in natural and engineered media is quantified in terms of stochastic models for the heterogeneity-induced fluctuations of particle motion. However, fundamental properties such as ergodicity and self-averaging and their dependence on the disorder distribution are often not known. Here, we investigate these questions for diffusion in quenched disordered media characterized by spatially varying retardation properties, which account for particle retention due to physical or chemical interactions with the medium. We link self-averaging and ergodicity to the disorder sampling efficiency Rn, which quantifies the number of disorder realizations a noise ensemble may sample in a single disorder realization. Diffusion for disorder scenarios characterized by a finite mean transition time is ergodic and self-averaging for any dimension. The strength of the sample to sample fluctuations decreases with increasing spatial dimension. For an infinite mean transition time, particle motion is weakly ergodicity breaking in any dimension because single particles cannot sample the heterogeneity spectrum in finite time. However, even though the noise ensemble is not representative of the single-particle time statistics, subdiffusive motion in q≥2 dimensions is self-averaging, which means that the noise ensemble in a single realization samples a representative part of the heterogeneity spectrum.

We introduce and study a generalization of martingales with the following self-averaging property: at each time, the conditional expectation of future random variables given the past, is a weighted average of all the random variables comprising the past. We assume only that more recent random variables are weighted no less than older random variables. We investigate conditions under which important properties satisfied by martingales, such as maximal inequalities and convergence, are present in an appropriate form.

Bulk properties of equilibrium liquids are a manifestation of intermolecular forces. Here, we show how these forces imprint on dynamical fluctuations in the Lyapunov exponents for simple fluids with and without attractive forces. While the bulk of the spectrum is strongly self-averaging, the first Lyapunov exponent self-averages only weakly and at a rate that depends on the length scale of the intermolecular forces; short-range repulsive forces quantitatively dominate longer range attractive forces, which act as a weak perturbation that slows the convergence to the thermodynamic limit. Regardless of intermolecular forces, the fluctuations in the Kolmogorov-Sinai entropy rate diverge, as one expects for an extensive quantity, and the spontaneous fluctuations of these dynamical observables obey fluctuation-dissipation like relationships. Together, these results are a representation of the van der Waals picture of fluids and another lens through which we can view the liquid state., Comment: 5 pages, 4 figures

In this paper we give an elementary approach to several results of Chatterjee in arXiv:0907.3381 and arXiv:1404.7178, as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards-Anderson type models with Gaussian disorder. The proof extends to systems at different temperatures and covers a number of other models, such as the mixed $p$-spin model, Sherrington-Kirkpatrick model with multi-dimensional spins and diluted $p$-spin model. Next, we adapt the same idea to prove quenched self-averaging of the bond magnetization for one system and use it to show quenched self-averaging of the site overlap for random field models with positively correlated spins. Finally, we show self-averaging for certain modifications of the random field itself., 17 pages

The study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings, provided that the system is large enough. Careful examination suggests that this idea might have a flaw, when the correlation length becomes of the order of the size of the system. We find that certain bounded quantities are not self-averaging when the correlation length becomes of the order of the size of the system. This suggests that the lack of self-averaging, expressed in terms of properly chosen signal-to-noise ratios, may serve to identify phase boundaries. This is demonstrated by using such signal-to-noise ratios to identify the boundary of the ferromagnetic phase of the random field Ising system and compare the findings with more traditional measures.

[1] We study dispersion in heterogeneous porous media for solutes evolving from point-like and extended source distributions in d=2 and d=3 spatial dimensions. The impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field. We focus here on the sample-to-sample fluctuations of the dispersion coefficients about their ensemble mean. For finite source sizes, the definition of dispersion coefficients in single realizations is not unique. We consider dispersion measures that describe the extension of the solute distribution, as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution. While the ensemble averages of these dispersion quantities may be identical, their fluctuation behavior is found to be different. Using a perturbation approach in the fluctuations of the random flow field, we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations. Their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution. We find that the dispersion variance decreases toward zero with time in d=3 spatial dimensions, while in d=2 it converges toward a finite long time value that is independent of the source dimensions.