Universality Conjecture Research Articles

Crafting the dynamical structure of synchronization by harnessing bosonic multilevel cavity QED

Many-body cavity QED experiments are established platforms to tailor and control the collective responses of ensembles of atoms, interacting through one or more common photonic modes. The rich diversity of dynamical phases they can host, calls for a unified framework. Here we commence this program by showing that a cavity QED simulator assembled from $N$-levels bosonic atoms, can reproduce and extend the possible dynamical responses of collective observables occurring after a quench. Specifically, by initializing the atoms in classical or quantum states, or by leveraging intra-levels quantum correlations, we craft on demand the entire synchronization/desynchronization dynamical crossover of an exchange model for $SU(N)$ spins. We quantitatively predict the onset of different dynamical responses by combining the Liouville-Arnold theorem on classical integrability with an ansatz for reducing the collective evolution to an effective few-body dynamics. Among them, we discover a synchronized chaotic phase induced by quantum correlations and associated to a first order non-equilibrium transition in the Lyapunov exponent of collective atomic dynamics. Our outreach includes extensions to other spin-exchange quantum simulators and a universal conjecture for the dynamical reduction of non-integrable all-to-all interacting systems.

A counter-example to the probabilistic universal graph conjecture via randomized communication complexity

We refute the Probabilistic Universal Graph Conjecture of Harms, Wild, and Zamaraev, which states that a hereditary graph property admits a constant-size probabilistic universal graph if and only if it is stable and has at most factorial speed. Our counter-example follows from the existence of a sequence of n × n Boolean matrices M n , such that their public-coin randomized communication complexity tends to infinity, while the randomized communication complexity of every n × n submatrix of M n is bounded by a universal constant.

Intermittency of gravity wave turbulence on the surface of an infinitely deep fluid: Numerical experiment

The intermittency of gravity wave turbulence on the surface of an infinitely deep fluid has been studied numerically. We estimated the scaling exponent of surface elevation structure-functions directly from the equations of motion, e.g., by numerical solution of the underlying equations of motion, without any additional closure assumptions (e.g., phase randomisation) and reduction to the kinetic equations as in the conventional framework of wave turbulence theory. With our high-resolution numerical methods for accurately modelling the nonlinear water surface, we were able to evaluate the scaling exponents of the structure-function up to relatively high orders (p=15), which has not been previously reported. We investigated the effect of wave steepness and forcing on intermittency and compared our results with other analytical and numerical studies, including results on the intermittency of wave turbulence of a different nature (turbulence of bending waves on a plate and magnetohydrodynamic turbulence). Our results support the conjecture of universality for intermittency phenomena in wave turbulence.

Superuniversality of dimerized SU(N+M) spin chains

We explore the physics of the quantum Hall effect using the Haldane mapping of dimerised $SU(N+M)$ spin chains, the large $N$ expansion and the density matrix renormalization group technique. We show that while the transition is first order for $N+M >2$, the system at zero temperature nevertheless displays a continuously diverging length scale $\xi$ (correlation length). The numerical results for $(M, N) = (1,3), ~ (2, 2),~(1, 5)$ and $(1, 7)$ indicate that $\xi$ is a directly observable physical quantity, namely the spatial width of the edge states. We relate the physical observables of the quantum spin chain to those of the quantum Hall system (and, hence, the $\vartheta$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of super universality which says the dimerised spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of $M$ and $N$. For the cases at hand we show that the singularity structure of the quantum Hall plateau transitions involves a universal function with two scale parameters that may in general depend on $M$ and $N$. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $\xi$ with varying values of $\vartheta \sim \pi$.

Spectrum of the Laplace operator on closed surfaces

AbstractA survey is given of classical and relatively recent results on the distribution of the eigenvalues of the Laplace operator on closed surfaces. For various classes of metrics the dependence of the behaviour of the second term in Weyl’s formula on the geometry of the geodesic flow is considered. Various versions of trace formulae are presented, along with ensuing identities for the spectrum. The case of a compact Riemann surface with the Poincaré metric is considered separately, with the use of Selberg’s formula. A number of results on the stochastic properties of the spectrum in connection with the theory of quantum chaos and the universality conjecture are presented.Bibliography: 51 titles.

Universal thermodynamic relations with constant corrections for rotating AdS black holes

In Goon and Penco (2020) [17], a universal relation between corrections to entropy and extremality was proposed. The relation was also found to exactly hold for the four-dimensional charged AdS black hole. In this paper, we extend the study to the rotating BTZ and Kerr-AdS black holes when a constant correction to General Relativity is considered for the first time. The entropy and extremality bound are calculated, and they have a closely dependent behavior with the coupling parameter of the constant correction. We confirm the universal relation for the rotating AdS black holes. Furthermore, taking into consideration of the shift of the angular momentum, we confirm one more new universal relation for the rotating cases. In particular, as a more general argument, we propose a novel universal extremal relation, which gives a universal conjecture relation between the shifted thermodynamic quantities for arbitrary black hole backgrounds. We believe that these universal relations will shed new light on the region of the quantum gravity.

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969).

Tourism destinations: A universality conjecture based on network science

The structural characteristics of a tourism destination are of crucial importance since they influence its dynamic behavior. Many studies have shown that destinations have apparently similar topologies. The question posited in this paper is to what extent does this similarity hold, and whether these topologies can be considered as a universal trait. This study reanalyzes available data to test this conjecture. In particular, several quantities representing the topological structures of the destination are calculated; further, we obtain size-invariant and scalable similarity scores. The results indicate that destinations hold structurally similar, and arguably universal, characteristics. This finding is important as it suggests that there are some very basic (and consistent) strategies destination managers can develop when designing plans and actions.

Cusp universality for random matrices, II: The real symmetric case

We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the accurate local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752]. We also present a PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.

Circle problem and the spectrum of the Laplace operator on closed 2-manifolds

In this survey the circle problem is treated in the broad sense, as the problem of the asymptotic properties of the quantity , the remainder term in the circle problem. A survey of recent results in this direction is presented. The main focus is on the behaviour of on short intervals. Several conjectures on the local behaviour of which lead to a solution of the circle problem are presented. A strong universality conjecture is stated which links the behaviour of with the behaviour of the second term in Weyl’s formula for the Laplace operator on a closed Riemannian 2-manifold with integrable geodesic flow. Bibliography: 43 titles.

On random walk on growing graphs

Nous considérons un modèle de marche aléatoire sur un graphe dynamique. Pour une suite de graphes finis croissant vers un graphe limite infini, nous montrons une borne supérieure pour la probabilité de transition. Cela donne un critère de transience pour la marche simple, pour des graphes à croissante lente, à partir du volume et de la constante de Cheeger de chaque graphe. Pour des cas plus particuliers, nous montrons une borne inférieure du même ordre et déduisons un critère de récurrence (dans un sens faible). Nous répondons aussi à la question de la récurrence directement, en lien avec une conjecture d’universalité de (Electron. J. Probab. 19 (2014) Article ID 106). Nous répondons aussi négativement à une question reliée de (Ann. Probab. 39 (2011) 1161–1203, Problem 1.8), à propos du « plongement inhomogène ».

Semiclassical calculation of spectral correlation functions of chaotic systems

We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality.

Towards sharp Bohnenblust–Hille constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.

Energy solutions of KPZ are unique

The Kardar–Parisi–Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the stationary KPZ equation on the real line by showing that itsenergy solutions, as introduced by Gonçalves and Jara in 2010 and refined by Gubinelli and Jara, are unique. This is the first time that a singular stochastic PDE can be tackled using probabilistic methods, and the combination of the convergence results of the first work and many follow-up papers with our uniqueness proof establishes the weak KPZ universality conjecture for a wide class of models. Our proof builds on an observation of Funaki and Quastel from 2015, and a remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole–Hopf solution, but it involves an additional driftt/12t/12.

Coarsening in one dimension: invariant and asymptotic states

We study a coarsening process of one-dimensional cell complexes. We show that if cell boundaries move with velocities proportional to the difference in size of neighboring cells, then the average cell size grows at a prescribed exponential rate and the Poisson distribution is precisely invariant for the distribution of the whole process, rescaled in space by its average growth rate. We present numerical evidence toward the following universality conjecture: starting from any finite mean stationary renewal process, the system when rescaled by e −2t converges to a Poisson point process. For a limited case, this makes precise what has been observed previously in experiments and simulations, and lays the foundation for a theory of universal asymptotic states of dynamical cell complexes.

J=0fixed pole andD-term form factor in deeply virtual Compton scattering

S.~Brodsky, F.~J.~Llanes-Estrada, and A.~Szczepaniak emphasized the importance of the $J=0$ fixed pole manifestation in real and (deeply) virtual Compton scattering measurements and argued that the $J=0$ fixed pole is universal, {\it i.e.}, independent on the photon virtualities \cite{Brodsky:2008qu}. In this paper we review the $J=0$ fixed pole issue in deeply virtual Compton scattering. We employ the dispersive approach to derive the sum rule that connects the $J=0$ fixed pole contribution and the subtraction constant, called the $D$-term form factor for deeply virtual Compton scattering. We show that in the Bjorken limit the $J=0$ fixed pole universality hypothesis is equivalent to the conjecture that the $D$-term form factor is given by the inverse moment sum rule for the Compton form factor. This implies that the $D$-term is an inherent part of corresponding generalized parton distribution (GPD). Any supplementary $D$-term added to a GPD results in an additional $J=0$ fixed pole contribution and implies the violation of the universality hypothesis. We argue that there exists no theoretical proof for the $J=0$ fixed pole universality conjecture.

Exact results for corner contributions to the entanglement entropy and Rényi entropies of free bosons and fermions in 3d

In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Rényi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient σ for the EE and σn for the Rényi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for σ and σn for all n=2,3,… . The results for σ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that σ/CT=π2/24 in all 3d CFTs, where CT is the central charge. For the Rényi entropies, the ratios σn/CT do not indicate similar universality. However, in the limit n→∞, the asymptotic values satisfy a simple relationship and equal 1/(4π2) times the asymptotic values of the free energy of free scalars/fermions on the n-covered 3-sphere.

Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices

The paper contains two main parts: in the first part, we analyze the general case of \({p \geq 2}\) matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a \({(p+1) \times (p+1)}\) matrix valued solution of a Riemann–Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields, which include the Bessel\({_{\nu}}\) universality for 1-level and the Meijer-G universality for 2-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a \({(p+1) \times (p+1)}\) origin parametrix in terms of Meijer G-functions. The solution of the full Riemann–Hilbert problem is derived rigorously only for p = 3 but the general framework of the proof can be extended to the Cauchy chain of arbitrary length p.

Critical behavior for scalar nonlinear waves

In the long wave regime, nonlinear waves may undergo a phase transition from a smooth behavior to a fast oscillatory behavior. In this study, we consider this phenomenon, which is commonly known as dispersive shock, in the light of Dubrovin’s universality conjecture (Dubrovin, 2006; Dubrovin and Elaeva, 2012) and we argue that the transition can be described by a special solution of a model universal partial differential equation. This universal solution is constructed using the string equation. We provide a classification of universality classes and an explicit description of the transition with special functions, thereby extending Dubrovin’s universality conjecture to a wider class of equations. In particular, we show that the Benjamin–Ono equation belongs to a novel universality class with respect to those known previously, and we compute its string equation exactly. We describe our results using the language of statistical mechanics, where we show that dispersive shocks share many of the features of the tricritical point in statistical systems, and we also build a dictionary of the relations between nonlinear waves and statistical mechanics.

Closed string Ramond–Ramond proposed higher derivative interactions on fermionic amplitudes in IIB

The complete form of the amplitude of one closed string Ramond–Ramond (RR), two fermionic strings and one scalar field in IIB superstring theory has been computed in detail. Deriving 〈VCVψ¯VψVϕ〉 by using suitable gauge fixing, we discover some new vertices and their higher derivative corrections. We investigate both infinite gauge and scalar u-channel poles of this amplitude. In particular, by using the fact that the kinetic term of fermion fields has no correction, employing Born–Infeld action, the Wess–Zumino terms and their higher derivative corrections, we discover all infinite t,s-channel fermion poles. The couplings between one RR and two fermions and all their infinite higher derivative corrections have been explored. In order to look for all infinite (s+t+u)-channel scalar/gauge poles for p+2=n, p=n cases, we obtain the couplings between two fermions–two scalars and two fermions, one scalar and one gauge field as well as all their infinite higher derivative corrections in type IIB. Specifically we make various comments based on arXiv:1205.5079 in favor of universality conjecture for all order higher derivative corrections (with or without low energy expansion) and the relation of open/closed string that is responsible for all superstring scattering amplitudes in IIA, IIB.