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Learning and computation of equilibria are central problems in game theory, theory of computation, and artificial intelligence. In this work, we introduce proximal regret, a new notion of regret based on proximal operators that lies strictly between external and swap regret. When every player employs a no-proximal-regret algorithm in a general convex game, the empirical distribution of play converges to proximal correlated equilibria (PCE), a refinement of coarse correlated equilibria. Our framework unifies several emerging notions in online learning and game theory-such as gradient equilibrium and semicoarse correlated equilibrium-and introduces new ones. Our main result shows that the classic Online Gradient Descent (GD) algorithm achieves an optimal $O(\sqrt{T})$ bound on proximal regret, revealing that GD, without modification, minimizes a stronger regret notion than external regret. This provides a new explanation for the empirically superior performance of gradient descent in online learning and games. We further extend our analysis to Mirror Descent in the Bregman setting and to Optimistic Gradient Descent, which yields faster convergence in smooth convex games.

While the standard $Λ$CDM model succeeds on large cosmological scales, it faces persistent small-scale challenges, including the core-cusp problem, the diversity of galaxy rotation curves, and the tight correlation between dark matter and baryons observed in the Tully-Fisher relation. To address these issues, we recently proposed an empirical law where the effective dark matter energy density is directly correlated with the baryonic gravitational potential, $ρ_{\rm DM} \propto Φ_b^2$, which reproduces observed rotation curves and resolves the core-cusp and diversity problems. To provide a theoretical foundation for this empirical law, we construct an effective field theory (EFT) introducing massive scalar, vector, and tensor mediators between baryons and a dark sector field $χ$. We demonstrate that aligning the mediator couplings to a specific ratio (4:6:3) with degenerate masses cancels the additional fifth forces acting on baryons up to $\mathcal{O}(v^2)$. We then show that this theoretical framework originates from a 5-dimensional (5D) spacetime. Treating the baryonic source as a 5D null fluid reveals that the three mediators emerge from a single 5D symmetric tensor field. By co

The rotation velocity profiles of galaxies (rotation curves) remain unexpectedly flat at large distances, where visible matter alone should make the rotation velocity decrease with radius. Conventionally, this requires a large amount of unseen dark matter. However, standard dark matter models face persistent small-scale challenges, such as the core-cusp and diversity problems, and struggle to explain the observed correlation between dark matter and baryons. Here, we introduce a simple empirical law for the dark matter distribution, stating that the effective dark matter energy density $ρ_{\rm DM}$ is directly correlated with the baryonic gravitational potential $Φ_b$ with the relation of $ρ_{\rm DM} = μΦ_b^2 / c^4$ in the rest frame of the galaxy. This leads to a Poisson equation for the total gravitational potential $Φ_{\rm tot}$, \[ abla^2Φ_{\rm tot} = 4πG\,ρ_b /c^2 + 4πG\,μ\,Φ_b^2 / c^6. \] Assuming that $μ= K M_b^{-3/2}$ with a parameter $K$, we applied this equation to 91 galaxies from the SPARC database. This baryon-correlated dark matter profile reproduced both the inner rise and outer flat regions of the observed rotation curves, resolving the core-cusp and diversity probl

This paper investigates equilibrium computation and the price of anarchy for Bayesian games, which are the fundamental models of games with incomplete information. In normal-form games with complete information, it is known that efficiently computable no-regret dynamics converge to correlated equilibria, and the price of anarchy for correlated equilibria can be bounded for a broad class of games called smooth games. However, in Bayesian games, as surveyed by Forges (1993), several non-equivalent extensions of correlated equilibria exist, and it remains unclear whether they can be efficiently computed or whether their price of anarchy can be bounded. In this paper, we identify a natural extension of correlated equilibria that can be computed efficiently and is guaranteed to have bounds on the price of anarchy in various games. First, we propose a variant of regret called untruthful swap regret. If each player minimizes it in repeated play of Bayesian games, the empirical distribution of these dynamics is guaranteed to converge to communication equilibria, which is one of the extensions of correlated equilibria proposed by Myerson (1982). We present an efficient algorithm for minimiz

We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix. Further, we study the joint convergence of independent generalized Toeplitz matrices along with other related matrices. The limits depend only on the correlation structure but are universal otherwise, in that they do not depend on the underlying distributions of the entries. In particular, these results provide the joint convergence of asymmetric Hankel matrices. Earlier results in the literature on the joint convergence of random symmetric Toeplitz and symmetric Hankel matrices with real entries follow as special cases.

Dynamical Mean-Field Theory (DMFT) has opened new perspectives for the investigation of strongly correlated electron systems and greatly improved our understanding of correlation effects in models and materials. In contrast to Hartree-Fock-type approximations the mean field of DMFT is dynamical, whereby local quantum fluctuations are fully taken into account. DMFT becomes exact in the limit of high spatial dimensions or coordination number. Using DMFT the dynamics of correlated electron systems can be investigated non-perturbatively at all interaction strengths, electron densities and temperatures. By merging density functional theory with DMFT a powerful method for the calculation of the properties of correlated electron materials has become available, which is applicable to bulk systems and heterostructures, including topological states of matter. The inclusion of non-local correlations into DMFT makes it possible to explore unconventional superconductivity and the critical behavior at thermal or quantum phase transitions. By generalizing DMFT to non-equilibrium states also the real-time dynamics of correlated systems can be investigated. In this brief review the foundations and

Strongly correlated quantum fluids are phases of matter that are intrinsically quantum mechanical, and that do not have a simple description in terms of weakly interacting quasi-particles. Two systems that have recently attracted a great deal of interest are the quark-gluon plasma, a plasma of strongly interacting quarks and gluons produced in relativistic heavy ion collisions, and ultracold atomic Fermi gases, very dilute clouds of atomic gases confined in optical or magnetic traps. These systems differ by more than 20 orders of magnitude in temperature, but they were shown to exhibit very similar hydrodynamic flow. In particular, both fluids exhibit a robustly low shear viscosity to entropy density ratio which is characteristic of quantum fluids described by holographic duality, a mapping from strongly correlated quantum field theories to weakly curved higher dimensional classical gravity. This review explores the connection between these fields, and it also serves as an introduction to the Focus Issue of New Journal of Physics on Strongly Correlated Quantum Fluids: from Ultracold Quantum Gases to QCD Plasmas. The presentation is made accessible to the general physics reader and

Electronic phase behavior in correlated-electron systems is a fundamental problem of condensed matter physics. We argue here that the change in the phase behavior near the surface and interface, i.e., {\em electronic reconstruction}, is the fundamental issue of the correlated-electron surface or interface science. Beyond its importance to basic science, understanding of this behavior is crucial for potential devices exploiting the novel properties of the correlated systems. % We present a general overview of the field, and then illustrate the general concepts by theoretical studies of the model heterostructures comprised of a Mott-insulator and a band-insulator, which show that spin (and orbital) orderings in thin heterostructures are generically different from the bulk and that the interface region, about three-unit-cell wide, is always metallic, demonstrating that {\em electronic reconstruction} generally occurs. % Predictions for photoemission experiments are made to show how the electronic properties change as a function of position, and the magnetic phase diagram is determined as a function of temperature, number of layers, and interaction strength. Future directions for resea

These are introductory lectures to some aspects of the physics of strongly correlated electron systems. I first explain the main reasons for strong correlations in several classes of materials. The basic principles of dynamical mean-field theory (DMFT) are then briefly reviewed. I emphasize the formal analogies with classical mean-field theory and density functional theory, through the construction of free-energy functionals of a local observable. I review the application of DMFT to the Mott transition, and compare to recent spectroscopy and transport experiments. The key role of the quasiparticle coherence scale, and of transfers of spectral weight between low- and intermediate or high energies is emphasized. Above this scale, correlated metals enter an incoherent regime with unusual transport properties. The recent combinations of DMFT with electronic structure methods are also discussed, and illustrated by some applications to transition metal oxides and f-electron materials.

Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here we apply techniques borrowed from this field to quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences $\{ρ_n\}_{n}$ and $\{σ_n\}_n$. We obtain upper bounds on the finite blocklength type II Stein- and Hoeffding errors, which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi and Verstraete [Journal of Mathematical Physics, 53(12), 2012]. We also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a (non-homogeneous) factorization property. Examples of such sequences include Gibbs states of spin chains with translation-invariant finite range interaction, as well as finitely correlated quantum states. We apply our results to find bounds on the capacity of a certain class of classical-quantum channels with memory, which satisfy a so-called channel factorization property- both in the f

Familywise error rate (FWER) has been a cornerstone in simultaneous inference for decades, and the classical Bonferroni method has been one of the most prominent frequentist approaches for controlling FWER. The present article studies the limiting behavior of Bonferroni FWER in a multiple testing problem as the number of hypotheses grows to infinity. We establish that in the equicorrelated normal setup with positive equicorrelation, Bonferroni FWER tends to zero asymptotically. We extend this result for generalized familywise error rates and to arbitrarily correlated setups.

We formulate a rigorous method for calculating a nonadiabatic (frequency-dependent) exchange-correlation (XC) kernel required for correct description of both equilibrium and nonequilibrium properties of strongly correlated systems within Time-Dependent Density Functional Theory (TDDFT). To do so we use the expression for charge susceptibility provided by Dynamical Mean Field Theory (DMFT) for the effective multi-orbital Hubbard Model. We tested our formalism by applying it to the one-band Hubbard model: our nonadiabatic kernel leads to a significant modification of the excitation spectrum, shifting the peak that appears in adiabatic (simplified) solutions and disclosing a new one, in agreement with the DMFT solution. We also used our method to track the nonequilibrium charge-density response of a multi-orbital perovskite Mott insulator, YTiO3, to a perturbation by a femtosecond (fs) laser pulse. The results were quite different from those provided by the corresponding adiabatic formalism. These initial investigations indicate that electron-electron correlations and nonadiabatic features can significantly affect the spectrum and nonequilibrium properties of strongly correlated syste

This is the text of the 'Theory' opening talk at the 2001 Strongly Correlated Electron Systems conference. It contains opinions about some of the outstanding scientific challenges facing the theory side of the correlated electrons field.

In this third of a series of four articles, we continue the study of the representations of the hamiltonian dynamical transformations of systems of correlated quantized oscillators. By our use of generalized wave function solutions to Schr{ö}dinger's equation (belonging to a rigged Hilbert space), and by considering the algebra of observables as a whole, the presence of Devaney chaos, hyperbolic quasi-invariant measures, complex torus actions, ergodicity and entropy generation associated to the non-invertible decay of Gamow vectors and their associated to Breit-Wigner resonances is shown. A weak (local) form of the second law of thermodynamics is demonstrated through the decay of resonances. Both correlation formation and decorrelation are associated with irreversibility and may be associated with entropy growth, which is due to the dynamical time evolution of resonances. Hilbert space is the manifold of stationary states. There is a fractal structure associated with dynamical time evolution of resonances in the space of generalized states, and the exponential decay of resonances may be identified with quasi-trapping. Equilibrium states may be regarded as strange attractors with re

In this lectures I discuss the electronic liquid crystal (ELC) phases in correlated electronic systems, what these phases are and in what context they arise. I will go over the strongest experimental evidence for these phases in a variety of systems: the two-dimensional electron gas in magnetic fields, the bilayer material Sr$_3$Ru$_2$O$_7$ (also in magnetic fields), and a set of phenomena in the cuprate superconductors (and more recently in the pnictide materials) that can be most simply understood in terms of ELC phases. Finally we will go over the theory of these phases, focusing on effective field theory descriptions and some of the known mechanisms that may give rise to these phases in specific models.

Band theory and BCS theory are arguably the most successful theories of condensed matter physics. Yet, in a number of materials, in particular the high-temperature superconductors and the layered organic superconductors, they fail. In these lecture notes for an international school, I emphasize that even though the low energy properties of a phase of matter are generally emergent and entirely determined by the broken symmetry, there are many differences between a strongly correlated and a weakly correlated state of matter. For example, spin waves are an emergent property for antiferromagnets, but for weak correlations the normal phase of a (Slater) antiferromagnet is metallic, whereas it is insulating (Heisenberg) for strong correlations. As a function of interaction strength, above the antiferromagnetic phase, the crossover between a metal and a local moment paramagnetic insulator is described by the Mott transition, whose mean-field theory down to T=0 is best formulated with dynamical mean-field theory. A similar situation occurs for superconductors: despite similar emergent properties, there are many differences between both kinds of superconductors. Experimental evidence sugges

We investigate thermoelectric properties of correlated quantum dots and molecules, described by a single level Anderson model coupled to conduction electron leads, by using Wilson's numerical renormalization group method. In the Kondo regime, the thermopower, $S(T)$, exhibits two sign changes, at temperatures $T=T_{1}$ and $T=T_{2}>T_{1}$. We find that $T_{2}$ is of order the level width $Γ$ and $T_{1}> T_{p}\approx T_{K}$, where $T_{p}$ is the position of the Kondo induced peak in the thermopower and $T_{K}$ is the Kondo scale. No sign change is found outside the Kondo regime, or, for weak correlations, making a sign change in $S(T)$ a particularly sensitive signature of strong correlations and Kondo physics. For molecules, we investigate the effect of screening by conduction electrons on the thermoelectric transport. We find that a large screening interaction enhances the figure of merit in the Kondo and mixed valence regimes.

We consider the problem of efficient statistical inference for comparing two regression curves estimated from two samples of dependent measurements. Based on a representation of the best pair of linear unbiased estimators in continuous time models as a stochastic integral, an efficient pair of linear unbiased estimators with corresponding optimal designs for finite sample size is constructed. This pair minimises the width of the confidence band for the difference between the estimated curves. We thus extend results readily available in the literature to the case of correlated observations and provide an easily implementable and efficient solution. The advantages of using such pairs of estimators with corresponding optimal designs for the comparison of regression models are illustrated via numerical examples.