Disorder Chaos Research Articles

Ground-state stability and the nature of the spin glass phase.

We propose an approach toward understanding the spin glass phase at zero and low temperature by studying the stability of a spin glass ground state against perturbations of a single coupling. After reviewing the concepts of flexibility, critical droplet, and related quantities for both finite- and infinite-volume ground states, we study some of their properties and review three models in which these quantities are partially or fully understood. We also review a recent result showing the connection between our approach and that of disorder chaos. We then view four proposed scenarios for the low-temperature spin glass phase-replica symmetry breaking, scaling-droplet, TNT, and chaotic pairs-through the lens of the predictions of each scenario for the lowest-energy large-lengthscale excitations above the ground state. Using a new concept called σ-criticality, which quantifies the sensitivity of ground states to single-bond coupling variations, we show that each of these four pictures can be identified with different critical droplet geometries and energies. We also investigate necessary and sufficient conditions for the existence of multiple incongruent ground states.

Perspectives of the future and global leadership: ideas and concepts

In the article a comparative analysis of the concepts of S. Huntington, F. Fukuyama and I. Wallerstein is carried out in order to identify the theoretical and methodological foundations of the coming future. I. Wallerstein's post-non-classical ideas foreshadowing the crisis of liberalism and the need to transform the foundations of Western democracy, the collapse of capitalism development paradigms, the growing interconnection of populism and authoritarianism, the formation of a new transitional period characterized by systemic disorder (chaos), acute political struggle on the principle “all against all”, lack of peaceful coexistence, stability and legality are most adequately reflected in the modern political process. In general, the future appears as a bundle of scenarios of varying degrees of probability, only some of which, bypassing the zones of uncertainty, can take place. This approach significantly changes the approach to higher education and the teaching of the humanities and social sciences.
 The second part of the article discusses the issues of defining the future by preceding events, highlighting the main and peripheral events that lay the foundation for the future. It is noted that even before the pandemic, the flaws of democracy and the growth of anti-liberal and anti-globalist sentiments in the world became noticeable, leading to national insularity, strengthening xenophobia, contributing to the growth of authoritarian regimes and their image in populist terms. It is emphasized that the majority of contemporary forecasts are alarmist in nature, they are value colored by deep pessimism and apocalyptic sentiments.
 Each country will have to cope with threats on its own, because the world must go through destructive processes in order to establish a new order, confirming Wallerstein's predictions.
 As you can see, we are talking about a serious reduction in the philosophy of education and its systemic paradigms.

A new spin on optimal portfolios and ecological equilibria

We consider the classical problem of optimal portfolio construction with the constraint that no short position is allowed, or equivalently the valid equilibria of multispecies Lotka–Volterra equations with self-regulation in the special case where the interaction matrix is of unit rank, corresponding to species competing for a common resource. We compute the average number of solutions and show that its logarithm grows as N α , where N is the number of assets or species and α ⩽ 2/3 depends on the interaction matrix distribution. We conjecture that the most likely number of solutions is much smaller and related to the typical sparsity m(N) of the solutions, which we compute explicitly. We also find that the solution landscape is similar to that of spin-glasses, i.e. very different configurations are quasi-degenerate. Correspondingly, ‘disorder chaos’ is also present in our problem. We discuss the consequence of such a property for portfolio construction and ecologies, and question the meaning of rational decisions when there is a very large number ‘satisficing’ solutions.

A New Spin on Optimal Portfolios and Ecological Equilibria

We consider the classical problem of optimal portfolio construction with the constraint that no short position is allowed, or equivalently the valid equilibria of multispecies Lotka-Volterra equations with self-regulation in the special case where the interaction matrix is of unit rank, corresponding to species competing for a common resource. We compute the average number of solutions and show that its logarithm grows as $N^\alpha$, where $N$ is the number of assets or species and $\alpha \leq 2/3$ depends on the interaction matrix distribution. We conjecture that the most likely number of solutions is much smaller and related to the typical sparsity $m(N)$ of the solutions, which we compute explicitly. We also find that the solution landscape is similar to that of spin-glasses, i.e. very different configurations are quasi-degenerate. Correspondingly, ``disorder chaos'' is also present in our problem. We discuss the consequence of such a property for portfolio construction and ecologies, and question the meaning of rational decisions when there is a very large number ``satisficing'' solutions.

On Absence of disorder chaos for spin glasses on $\mathbb{Z} ^{d}$

We identify simple mechanisms that prevent a strong form of disorder chaos for edge overlap of the Ising spin glass model on $\mathbb{Z} ^{d}$. This was first shown by Chatterjee in the case of Gaussian couplings. We present three proofs of the theorem for general couplings with continuous distribution based on the presence in the coupling realization of stabilizing features of positive density.

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.

Disorder chaos in some diluted spin glass models

We prove disorder chaos at zero temperature for three types of diluted models with large connectivity parameter: $K$-spin antiferromagnetic Ising model for even $K\geq2$, $K$-spin spin glass model for even $K\geq2$, and random $K$-sat model for all $K\geq2$. We show that modifying even a small proportion of clauses results in near maximizers of the original and modified Hamiltonians being nearly orthogonal to each other with high probability. We use a standard technique of approximating diluted models by appropriate fully connected models and then apply disorder chaos results in this setting, which include both previously known results as well as new examples motivated by the random $K$-sat model.

Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound

The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [18]. The central argument therein relied on a very dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It is believed that this bound and its higher dimensional generalization are highly related to the conjectures of temperature chaos and ultrametricity in the SK model, but a complete investigation remains elusive. Motivated by Bovier-Klimovsky [2] and Auffinger-Chen [3], the aim of this paper is to present a novel approach to analyzing the Parisi functional and the two-dimensional GT bound in the mixed $p$-spin models in terms of optimal stochastic control problems. We compute the directional derivative of the Parisi functional and derive equivalent criteria for the Parisi measure. We demonstrate how our approach provides a simple and efficient control for the GT bound that yields several new results on Talagrand's positivity of the overlap [20,Section 14.12] and disorder chaos in Chatterjee [5] and Chen [6]. In particular, we provide some examples of the models containing odd $p$-spin interactions.

The geometry of the Gibbs measure of pure spherical spin glasses

We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,\beta}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,\beta}$ and prove that, for an appropriate centering $c_{N,\beta}$, $NF_{N,\beta}-c_{N,\beta}$ is a tight sequence. We establish the absence of temperature chaos and analyze the transition rate to disorder chaos of the Gibbs measure and ground state. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical `bands' centered at deep minima, playing the role of so-called `pure states'. For the pure models, the latter makes precise the so-called picture of `many valleys separated by high mountains' and significant parts of the TAP analysis from the physics literature.

On the energy landscape of the mixed even p-spin model

We investigate the energy landscape of the mixed even $p$-spin model with Ising spin configurations. We show that for any given energy level between zero and the maximal energy, with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level. Furthermore, their magnetizations and overlaps are concentrated around some fixed constants. In particular, at the level of maximal energy, we prove that the Hamiltonian exhibits exponentially many orthogonal peaks. This improves the results of Chatterjee and Ding-Eldan-Zhai, where the former established a logarithmic size of the number of the orthogonal peaks, while the latter proved a polynomial size. Our second main result obtains disorder chaos at zero temperature and at any external field. As a byproduct, this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a Gaussian limit law when the external field is present.

Some examples of quenched self-averaging in models with Gaussian disorder

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.

Parisi Formula, Disorder Chaos and Fluctuation for the Ground State Energy in the Spherical Mixed p-Spin Models

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

Universality of Chaos and Ultrametricity in Mixed p‐Spin Models

AbstractWe prove disorder universality of chaos phenomena and ultrametricity in the mixed p‐spin model under mild moment assumptions on the environment. This establishes the longstanding belief among physicists that the solution of mean‐field models with Gaussian disorder also holds for different environments. Our results extend to the mixed p‐spin model as well as to different spin glass models. These include universality of quenched disorder chaos in the Edwards‐Anderson (EA) model and quenched concentration for the magnetization in both EA and mixed p‐spin models under non‐Gaussian environments. In addition, we show quenched self‐averaging for the overlap in the random field Ising model under small perturbation of the external field.© 2015 Wiley Periodicals, Inc.

Disorder Chaos in the Spherical Mean-Field Model

We consider the problem of disorder chaos in the spherical mean-field model. It is concerned about the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra's replica symmetry breaking scheme, we establish this at the level of the free energy as well as the Gibbs measure irrespective the presence or absence of the external field.

Assessment of Geographic Space Arrangement Based on the Amount of Information Contained In the Map Image

AbstractIn this paper an attempt was made to determine whether the amount of information, necessary to record a unit area of the geographic space, mapped on the large-scale map, may define the state of the arrangement of this space. The aim of this study was to make a comprehensive assessment of the geographical space in terms of its reasonable development. Intuition suggests that for two areas with similar functions - a picture of orderly and reasonably developed land and a disordered and chaotically developed land - one can record more concisely the image of the first than the latter. The work is an attempt to formulate an indicator that could characterize a state of the general arrangement of the geographical space and could serve as an objective assessment of such arrangement. The proposed statement was put to practical verification.At the introduction the numerical amount of information was specified, necessary to record unit area of large-scale cadastral maps, containing only the boundaries of the ownership. In the task the concept of the amount of information proposed by C. E. Shannon was used. For a typical map at scale 1:1000, to record the area of one square decimetre (i. e. one hectare in the terrain) 531 bits of information are necessary. For the area of fragmented fields of southeast Poland this number increased to 4980 bits. Basing on previous work of the author, quoted for comparison, the amount of information, necessary to record the identical unit of a typical city in the same coding method, is 23360 bits of information. In this case extensive infrastructure of the city recorded on the source layers has to be included.Later in the article, the amount of information necessary to record the same unit area of geographical space for the state of extreme disorder (chaos) and the state of the theoretical ideal arrangement was determined. The real determined values were much closer to the ideal in the ordered state than to the chaos. Subsequently, the amount of information was considered necessary for recording sites with different ways of building. Basing on this experience a clear relationship between the amount of information that is necessary to record unit area of geographical space and the degree of rationality of its development could not be established. Thus, the amount of information, required to record of the unit area of large-scale map image, cannot be unequivocal indicator of geographical arrangement and consequently also the image map, because the large-scale map accurately reflects the shape of the geographical space. The growth of the indicator can show the advanced infrastructure of the selected area - on the basis of comparable data.The indicator of the amount of information can have other wide application - for planning of memory means required to store the geographic space and to determine the transmission time of the recorded data by telecommunication links. The search for a universal indicator of geographical space arrangement, characterizing its organization and rational development - must remain the subject of further research.

Disorder chaos in the Sherrington–Kirkpatrick model with external field

We consider a spin system obtained by coupling two distinct Sherrington-Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of external field, this statement was first confirmed by Chatterjee [Disorder chaos and multiple valleys in spin glasses (2009) Preprint]. In the present paper, using Guerra's replica symmetry breaking bound, we prove that the SK model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to Guerra's bound and the Parisi measure.

Evolution of immunity: no development without risk

Signal transduction by cell surface receptors in the context of heterogeneous and variable cellular environments plays a pivotal role in regulating many biological processes, including development, activation, and homeostasis of the immune system. In some receptors, extracellular ligand-binding and intracellular signaling domains are located on the same protein chain (single-chain receptors), while in the so-called multichain immune recognition receptors (MIRRs), recognition and signaling functions are separated between different protein chains. Why did nature separate recognition and signaling functions for MIRRs, thereby increasing the risk of malfunction and potential attack by pathogens? The risk is real: in order to escape the immune response, viruses are able to disrupt functional coupling between recognition and signaling aspects of MIRR machinery. Intrinsic disorder of intracellular signal-generating regions of MIRRs adds further intrigue to the story. Why did nature select protein disorder for MIRRs to translate recognition of distinct antigens into appropriate activation signals that would induce specific functional outcomes? Here, I suggest that nature takes the risks associated with intrareceptor separation of functions as well as with the chaos and indeterminacy of protein disorder in exchange for providing diversity and variability of signal transduction. Not only does this phenomenon serve as the molecular basis for the development and evolution of the immune and other complex biological systems, but it fits closely to Darwinian evolutionary biology.

Cells diversify transmembrane signaling through the controlled chaos of protein disorder.

Cell surface receptors function to transduce signals across the cell membrane leading to a variety of biologic responses. Structurally, these integral proteins can be classified into two main families, depending on whether extracellular ligand-binding and intracellular signaling domains are located on the same protein chain (single-chain receptors, SRs) or on separate subunits (multichain receptors, MRs). Since most MRs are immune receptors, they are all commonly referred to as multi-chain immune recognition receptors (MIRRs). Recent studies reveal that, in contrast to well-structured signaling domains of SRs, those of MIRRs represent intrinsically disordered regions, the regions that lack a well-defined three-dimensional structure under physiological conditions. Why did nature separate recognition and signaling functions of MIRRs? Why for MIRRs did nature select to provide highly specific signaling through the chaos of protein disorder? What mechanisms could control this chaos in the process of transmembrane signal transduction to provide the specificity and diversity of the immune response? Here, I summarize recent findings that may not only shed light on these and other questions but also add significantly to our understanding of receptor signaling, a fundamental process that plays a critical role in health and disease.

Domain walls and chaos in the disordered SOS model

Domain walls, optimal droplets and disorder chaos at zero temperature are studied numericallyfor the solid-on-solid model on a random substrate. It is shown that the ensemble ofrandom curves represented by the domain walls obeys Schramm’s left passage formula withκ = 4 whereas theirfractal dimension is ds = 1.25, and therefore their behavior cannot be described as showing ‘Schramm– (orstochastic) Loewner evolution’ (SLE). Optimal droplets with a lateral size betweenL and2L have the same fractal dimension as domain walls but an energy that saturates at a value oforder for such that arbitrarily large excitations exist which cost only a small amount of energy.Finally it is demonstrated that the sensitivity of the ground state to small changes of orderδ in the disorder is subtle: beyond a crossover length scaleLδ∼δ−1 the correlations of the perturbed ground state with the unperturbed ground state, rescaledusing the roughness, are suppressed and approach zero logarithmically.

Temperature and Disorder Chaos in Three-Dimensional Ising Spin Glasses

We study the effects of small temperature as well as disorder perturbations on the equilibrium state of three-dimensional Ising spin glasses via an alternate scaling ansatz. By using Monte Carlo simulations, we show that temperature and disorder perturbations yield chaotic changes in the equilibrium state and that temperature chaos is considerably harder to observe than disorder chaos.