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arXiv2018

Numerical assessment of the percolation threshold using complement networks

Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold $p_c$ and its model-predicted value $π_c$. Here we show the existence of an empirical linear relation between $p_c$ and $π_c$ across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of $π_c$. We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, $p_c$, and that of its complement, $\bar{p}_c$.

physics.soc-ph↗

arXiv2019

Prototype-based classifiers in the presence of concept drift: A modelling framework

We present a modelling framework for the investigation of prototype-based classifiers in non-stationary environments. Specifically, we study Learning Vector Quantization (LVQ) systems trained from a stream of high-dimensional, clustered data.We consider standard winner-takes-all updates known as LVQ1. Statistical properties of the input data change on the time scale defined by the training process. We apply analytical methods borrowed from statistical physics which have been used earlier for the exact description of learning in stationary environments. The suggested framework facilitates the computation of learning curves in the presence of virtual and real concept drift. Here we focus on timedependent class bias in the training data. First results demonstrate that, while basic LVQ algorithms are suitable for the training in non-stationary environments, weight decay as an explicit mechanism of forgetting does not improve the performance under the considered drift processes.

cs.LG↗

arXiv2013

Optimal scales in weighted networks

The analysis of networks characterized by links with heterogeneous intensity or weight suffers from two long-standing problems of arbitrariness. On one hand, the definitions of topological properties introduced for binary graphs can be generalized in non-unique ways to weighted networks. On the other hand, even when a definition is given, there is no natural choice of the (optimal) scale of link intensities (e.g. the money unit in economic networks). Here we show that these two seemingly independent problems can be regarded as intimately related, and propose a common solution to both. Using a formalism that we recently proposed in order to map a weighted network to an ensemble of binary graphs, we introduce an information-theoretic approach leading to the least biased generalization of binary properties to weighted networks, and at the same time fixing the optimal scale of link intensities. We illustrate our method on various social and economic networks.

physics.data-an↗

arXiv2018

What can spin glass theory and analogies tell us about ferroic glasses?

As well as several different kinds of periodically ordered ferroic phases, there are now recognized several different examples of ferroic glassiness, although not always described as such and in material fields of study that have mostly been developed separately. In this chapter an attempt is made to indicate common conceptual origins and features, observed or anticipated. Throughout, this aim is pursued through the use of simple models, in an attempt to determine probable fundamental origins within a larger picture of greater complication, and analogies between systems in different areas, both experimental and theoretical, in the light of significant progress in spin glass understanding.

cond-mat.mtrl-sci↗

arXiv2025

Dynamics of Pseudoentanglement

The dynamics of quantum entanglement plays a central role in explaining the emergence of thermal equilibrium in isolated many-body systems. However, entanglement is notoriously hard to measure. Recent works have introduced a notion of pseudoentanglement describing ensembles of many-body states that, while only weakly entangled, cannot be efficiently distinguished from states with much higher entanglement, such as random states in the Hilbert space. This prompts the question: how much entanglement is truly necessary to achieve thermal equilibrium in quantum systems? In this work we address this question by introducing random circuit models of quantum dynamics that, at late times, equilibrate to pseudoentangled ensembles -- a phenomenon we name ensemble pseudothermalization. These models replicate all the efficiently observable predictions of thermal equilibrium, while generating only a small (and tunable) amount of entanglement. We examine (i) how a pseudoentangled ensemble on a small subsystem spreads to the whole system as a function of time, and (ii) how a pseudoentangled ensemble can be generated from an initial product state. We map the above problems onto a family of classical Markov chains on subsets of the computational basis. The mixing times of such Markov chains are related to the time scales at which the states produced from the dynamics become indistinguishable from Haar-random states at the level of each statistical moment, or number of copies. Based on a combination of rigorous bounds and conjectures supported by numerics, we argue that each Markov chain's relaxation time and mixing time have different asymptotic behavior in the limit of large system size. This is a necessary condition for a cutoff phenomenon: an abrupt dynamical transition to equilibrium. We thus conjecture that our random circuits give rise to asymptotically sharp distinguishability transitions.

quant-ph↗

arXiv2019

The structure of state transition graphs in hysteresis models with return point memory: I. General Theory

We consider the athermal quasi-static dynamics (AQS) of disordered systems driven by a slowly varying external field. Our interest is in an automaton description (AQS-A) that represents the AQS dynamics in terms of the graph of state transitions triggered by the driving field. A particular feature of these systems is return point memory (RPM), a tendency for the system to return to the same microstate upon cycling the field. It is known that the existence of three conditions, (1) a partial order on the set of configuration; (2) a no-passing property; and (3) an adiabatic response to monotonous driving fields, implies RPM. When periodically driven, such systems settle into a cyclic response after a transient of at most one period. However conditions (1)-(3) are only sufficient but not necessary. In fact, we show that the AQS dynamics naturally provides a more selective partial order which, due to its connection to hysteresis loops, is a natural choice for establishing RPM. This enables us to consider AQS-A exhibiting RPM without necessarily possessing the no-passing property. We call such automata $\ell$AQS-A and work out the structure of their state transition graphs. Our central finding is that RPM constrains the {\em intra-loop} structure of hysteresis loops, namely its hierarchical organization into sub loops, but not the {\em inter-loop} structure. We prove that the topology of the intra-loop structure has a natural representation in terms of an ordered tree and that the corresponding state transition graph is planar. On the other hand, the RPM property does not significantly restrict inter-loop transitions. A system exhibiting RPM and subject to periodic forcing can thus undergo a large number of transient cycles before settling into a periodic response. Such systems can even exhibit subharmonic response.

cond-mat.dis-nn↗

arXiv2012

Orbital polarization and magnetization for independent particles in disordered media

Formulas for the contribution of the conduction electrons to the polarization and magnetization are derived for disordered systems and within a one-particle framework. These results generalize known formulas for Bloch electrons and the presented proofs considerably simplify and strengthen prior justifications. The new formulas show that orbital polarization and magnetization are of geometric nature. This leads to quantization for a periodically driven Piezo effect as well as the derivative of the magnetization w.r.t. the chemical potential. It is also shown how the latter is connected to boundary currents in Chern insulators. The main technical tools in the proofs are an adaption of Nenciu's super-adiabatic theory to C$^*$-dynamical systems and Bellissard's Ito derivatives w.r.t. the magnetic field.

math-ph↗

arXiv2017

Generalized stealthy hyperuniform processes : maximal rigidity and the bounded holes conjecture

We study translation invariant stochastic processes on $\mathbb{R}^d$ or $\mathbb{Z}^d$ whose diffraction spectrum or structure function $S(k)$, i.e. the Fourier transform of the truncated total pair correlation function, vanishes on an open set $U$ in the wave space. A key family of such processes are stealthy hyperuniform point processes, for which the origin $k=0$ is in $U$; these are of much current physical interest. We show that all such processes exhibit the following remarkable maximal rigidity : namely, the configuration outside a bounded region determines, with probability 1, the exact value (or the exact locations of the points) of the process inside the region. In particular, such processes are completely determined by their tail. In the 1D discrete setting (i.e. $\mathbb{Z}$-valued processes on $\mathbb{Z}$), this can also be seen as a consequence of a recent theorem of Borichev, Sodin and Weiss; in higher dimensions or in the continuum, such a phenomenon seems novel. For stealthy hyperuniform point processes, we prove the Zhang-Stillinger-Torquato conjecture that such processes have bounded holes (empty regions), with a universal bound that depends inversely on the size of $U$.

math.PR↗

arXiv2018

A Relation between Disorder Chaos and Incongruent States in Spin Glasses on ${\mathbb Z}^d$

We derive lower bounds for the variance of the difference of energies between incongruent ground states, i.e., states with edge overlaps strictly less than one, of the Edwards-Anderson model on ${\mathbb Z}^d$. The bounds highlight a relation between the existence of incongruent ground states and the absence of edge disorder chaos. In particular, it suggests that the presence of disorder chaos is necessary for the variance to be of order less than the volume. In addition, a relation is established between the scale of disorder chaos and the size of critical droplets. The results imply a long-conjectured relation between the droplet theory of Fisher and Huse and the absence of incongruence.

math.PR↗

arXiv2012

Complex Random Energy Model: Zeros and Fluctuations

The partition function of the random energy model at inverse temperature $β$ is a sum of random exponentials $Z_N(β)=\sum_{k=1}^N \exp(β\sqrt{n} X_k)$, where $X_1,X_2,...$ are independent real standard normal random variables (= random energies), and $n=\log N$. We study the large $N$ limit of the partition function viewed as an analytic function of the complex variable $β$. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $β$, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.

math.PR↗

arXiv2013

Low-temperature specific heat of graphite and CeSb2: Validation of a quasi-adiabatic continuous method

We present the application of a fast quasi-adiabatic continuous method to the measurement of specific heat at 4He temperatures, which can be used for the study of a wide range of materials. The technique can be performed in the same configuration used for the relaxation method, as the typical time constants between calorimetric cell and thermal sink at 4.2 K are chosen to be of the order of tau~30 s. The accuracy in the absolute values have been tested by comparing them to relaxation-method results obtained in the same samples (performed in situ using the same set-up), with a deviation between the absolute values < 3% in the whole temperature range. This new version of the continuous calorimetric method at low temperatures allows us to completely characterize and measure a sample within a few hours with a high density of data points, whereas when employing other methods we typically need a few days. An exhaustive study has been performed for reproducibility to be tested. In the present work, we have applied this method to two different substances: CeSb2, which exhibits three magnetic transitions at 15.5 K, 11.7 K and 9.5 K, and graphite, both highly-oriented pyrolytic graphite (HOPG) and natural crystals. Our results on these graphites are discussed in comparison with previous published data on different kinds of graphite samples.

cond-mat.mtrl-sci↗

arXiv2018

Thermal coupling of silicon oscillators in cryogen-free dilution refrigerators

Silicon double paddle osillators (DPO) have been successfully used for measuring the elastic properties of amorphous films down to 10 mK (see for example \cite{Fefferman16,Liu14}). Until now, our group has used a wet dilution refrigerator for the lowest temperature measurements. We present measurements carried out on a Bluefors cryogen-free dilution refrigerator that demonstrate an extreme sensitivity of the thermal coupling of the DPO to its environment. These measurements show that it is necessary to enclose the DPO in a shield at the mixing chamber (MXC) temperature. Any gaps in the shield limit its effectiveness, even if there is no line-of-sight path to the DPO. In the absence of a cryogenic hermetic shield surrounding the DPO, turning off the pulse tube while maintaining the MXC and still temperatures leads to heating of the DPO. This demonstrates that any heating of the sample due to pulse tube vibrations is a less important effect.

physics.ins-det↗

arXiv2012

Region graph partition function expansion and approximate free energy landscapes: Theory and some numerical results

Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph method, are theoretical approaches for treating the complicated short-loop-induced local correlations. For graphical models represented by non-redundant or redundant region graphs, approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion. Several free energy functionals are obtained, each of which use a set of probability distribution functions or functionals as order parameters. These probability distribution function/functionals are required to satisfy the region graph belief-propagation equation or the region graph survey-propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges. As a simple application of the general theory, we perform region graph belief-propagation simulations on the square-lattice ferromagnetic Ising model and the Edwards-Anderson model. Considerable improvements over the conventional Bethe-Peierls approximation are achieved. Collective domains of different sizes in the disordered and frustrated square lattice are identified by the message-passing procedure. Such collective domains and the frustrations among them are responsible for the low-temperature glass-like dynamical behaviors of the system.

cond-mat.dis-nn↗

arXiv2013

Complex Network Structure of Flocks in the Standard Vicsek Model

In flocking models, the collective motion of self-driven individuals leads to the formation of complex spatiotemporal patterns. The Standard Vicsek Model (SVM) considers individuals that tend to adopt the direction of movement of their neighbors under the influence of noise. By performing an extensive complex network characterization of the structure of SVM flocks, we show that flocks are highly clustered, assortative, and non-hierarchical networks with short-tailed degree distributions. Moreover, we also find that the SVM dynamics leads to the formation of complex structures with an effective dimension higher than that of the space where the actual displacements take place. Furthermore, we show that these structures are capable of sustaining mean-field-like orientationally ordered states when the displacements are suppressed, thus suggesting a linkage between the onset of order and the enhanced dimensionality of SVM flocks.

cond-mat.stat-mech↗

arXiv2013

Topology trivialization and large deviations for the minimum in the simplest random optimization

Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N-1)- dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the "trust region subproblem" or "constraint least square problem". When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number N_{tot} of critical (stationary) points in the cost function landscape. In the first regime N_{tot} remains of the order $N$ and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor.

cond-mat.dis-nn↗

arXiv2014

Anomalies in conductance and localization length of disordered ladders

We discuss the conditions under which an anomaly occurs in conductance and localization length of Anderson model on a lattice. Using the ladder hamiltonian and analytical calculation of average conductance we find the set of resonance conditions which complements the $π$-coupling rule for anomalies. We identify those anomalies that might vanish due to the symmetry of the lattice or the distribution of the disorder. In terms of the dispersion relation it is known from strictly one-dimensional model that the lowest order (i.e., the most strong) anomalies satisfy the equation $E(k)=E(3k)$. We show that the anomalies of the generalized model studied here are also the solutions of the same equation with modified dispersion relation.

cond-mat.dis-nn↗

arXiv2014

Survival of classical and quantum particles in the presence of traps

We present a detailed comparison of the motion of a classical and of a quantum particle in the presence of trapping sites, within the framework of continuous-time classical and quantum random walk. The main emphasis is on the qualitative differences in the temporal behavior of the survival probabilities of both kinds of particles. As a general rule, static traps are far less efficient to absorb quantum particles than classical ones. Several lattice geometries are successively considered: an infinite chain with a single trap, a finite ring with a single trap, a finite ring with several traps, and an infinite chain and a higher-dimensional lattice with a random distribution of traps with a given density. For the latter disordered systems, the classical and the quantum survival probabilities obey a stretched exponential asymptotic decay, albeit with different exponents. These results confirm earlier predictions, and the corresponding amplitudes are evaluated. In the one-dimensional geometry of the infinite chain, we obtain a full analytical prediction for the amplitude of the quantum problem, including its dependence on the trap density and strength.

cond-mat.stat-mech↗

arXiv2014

Free-energy bounds for hierarchical spin models

In this paper we study two non-mean-field spin models built on a hierarchical lattice: The hierarchical Edward-Anderson model (HEA) of a spin glass, and Dyson's hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the existence of the thermodynamic limit of the free energy and the replica-symmetry-breaking (RSB) free-energy bounds previously derived for the Sherrington-Kirkpatrick model of a spin glass. These RSB mean-field bounds are exact only if the order-parameter fluctuations (OPF) vanish: Given that such fluctuations are not negligible in non-mean-field models, we develop a novel strategy to tackle part of OPF in hierarchical models. The method is based on absorbing part of OPF of a block of spins into an effective Hamiltonian of the underlying spin blocks. We illustrate this method for DHM and show that, compared to the mean-field bound for the free energy, it provides a tighter non-mean-field bound, with a critical temperature closer to the exact one. To extend this method to the HEA model, a suitable generalization of Griffith's correlation inequalities for Ising ferromagnets is needed: Since correlation inequalities for spin glasses are still an open topic, we leave the extension of this method to hierarchical spin glasses as a future perspective.

math-ph↗