Decay Of Correlations Research Articles

A Vector-Valued Almost Sure Invariance Principle for Random Expanding on Average Cocycles

We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gouëzel’s approach for establishing ASIP (developed in Dragičević and Hafouta in Nonlinearity 34:6773–6798, 2021) and the recent construction of the so-called adapted norms (carried out in Dragičević and Sedro in Quenched limit theorems for expanding on average cocycles. arXiv:2105.00548 , 2021), which in some sense eliminate the non-uniformity of the decay of correlations. For real-valued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of the scaling condition (12), answering a question of Dragičević and Sedro in Quenched limit theorems for expanding on average cocycles. arXiv:2105.00548 , 2021.

Statistical mechanics approach to the holographic renormalization group: Bethe lattice Ising model and p-adic AdS/CFT

Abstract The Bethe lattice Ising model—a classical model of statistical mechanics for the phase transition—provides a novel and intuitive understanding of the prototypical relationship between tensor networks and the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. After analytically formulating a holographic renormalization group for the Bethe lattice model, we demonstrate the underlying mechanism and the exact scaling dimensions for the power-law decay of boundary-spin correlations by introducing the relation between the lattice network and an effective Poincaré metric on a unit disk. We compare the Bethe lattice model in the high-temperature region with a scalar field in AdS2, and then discuss its more direct connection to the p-adic AdS/CFT. In addition, we find that the phase transition in the interior induces a crossover behavior of boundary-spin correlations, depending on the depth of the corresponding correlation path.

Locality Estimates for Complex Time Evolution in 1D

It is a generalized belief that there are no thermal phase transitions in short range 1D quantum systems. However, the only known case for which this is rigorously proven is for the particular case of finite range translationally invariant interactions. The proof was obtained by Araki in his seminal paper of 1969 as a consequence of pioneering locality estimates for the time-evolution operator that allowed him to prove its analyticity on the whole complex plane, when applied to a local observable. However, as for now there is no mathematical proof of the absence of 1D thermal phase transitions if one allows exponential tails in the interactions. In this work we extend Araki’s result to include exponential (or faster) tails. Our main result is the analyticity of the time-evolution operator applied on a local observable on a suitable strip around the real line. As a consequence we obtain that thermal states in 1D exhibit exponential decay of correlations above a threshold temperature that decays to zero with the exponent of the interaction decay, recovering Araki’s result as a particular case. Our result however still leaves open the possibility of 1D thermal short range phase transitions. We conclude with an application of our result to the spectral gap problem for Projected Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to Cirac et al.

Shape and interfacial structure of droplets. Exact results and simulations

We consider the fluctuating interface of a droplet pinned on a flat wall. For such a system we compare results obtained within the exact field theory of phase separation in two dimensions and Monte Carlo (MC) simulations for the Ising model. The interface separating coexisting phases splits and hosts drops whose effect is to produce subleading corrections to the order parameter profile and correlation functions. In this paper we establish the first direct measurement of interface structure effects by means of high-performance MC simulations which successfully confirm the field-theoretical results. Simulations are found to corroborate the theoretical predictions for interface structure effects whose analytical expression has recently been obtained. It is thanks to these higher-order corrections that we are able to correctly resettle a longstanding discrepancy between simulations and theory for the order parameter profile. In addition, our results clearly establish the long-ranged decay of interfacial correlations in the direction parallel to the interface and their spatial confinement within the interfacial region also in the presence of a wall from which the interface is entropically repelled.

Abrupt onset of the capillary-wave spectrum at wall-fluid interfaces.

Surfaces between 3D solids and fluids exhibit a wide variety of phenomena both at equilibrium, such as roughening transitions, interfacial fluctuations and wetting, and also out-of-equilibrium, such as the surface growth of driven interfaces. These phenomena are described very successfully using lower dimensional (2D) effective models which focus on the physics associated with emergent mesoscopic lengths scales, parallel to the interface, where the 2D-like behaviour is physically transparent. However, the precise conditions under which this dimensional reduction is justifiable have remained unclear. Here we show that, for a wall-fluid interface, a dimensional reduction from 3D-like to 2D-like behaviour - identified via the decay of density correlations - occurs abruptly at a specific value of the contact angle, and indicates the beginning of interfacial-like 2D behaviour and the spontaneous onset of the capillary-wave spectrum. The reduction from 3D to 2D is characterised by the divergence of a correlation length perpendicular to the interface revealing a morphological change in the nature of density correlations. Counter-intuitive effects occur, including that 3D behaviour can persist up to the wetting temperature and also that 2D behaviour can begin when no wetting layer is present and the adsorption is negative.

Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d≥3

For the Bargmann–Fock field on Rd with d≥3, we prove that the critical level ℓc(d) of the percolation model formed by the excursion sets {f≥ℓ} is strictly positive. This implies that for every ℓ sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case ℓ=0), {f=ℓ} contains an unbounded connected component that visits “most” of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behavior of nodal hypersurfaces of these Gaussian fields in Rd for d≥3 is very different from the behavior of nodal lines of their 2-dimensional analogues.

Оценка результатов интерполяции методом взвешенных угловых расстояний среднемесячной температуры воздуха

Article describes the quality analysis of climate maps for the Far Eastern Federal District. Observed station data from 497 meteorological stations are used as meteorological information. Monthly air temperature maps in January and July for two regions are obtained by angular distance weighting interpolation (ADW). The first region is located within the Central Yakut Plain, the Prilenskoe Plateau and the northern slopes of the Aldan Highlands, the absolute heights within the region do not exceed 500 m. The second region is located in Transbaikalia. The relief of the region is represented by plateaus and strongly dissected ridges with a height of more than 2 500 m. Meteorological stations are unevenly distributed over the territory in both regions. Mapping was performed using the free cross-platform geographic information system QGIS. The analysis of the errors carried out for 74 stations, showed that the accuracy of interpolation depends on the number of stations located on the territory and on the land forms. The quality analysis showed that the errors for the winter periods are larger than in the summer period. However, if the data of stations located in areas with high relative elevations are excluded, then the interpolation accuracy decreases sharply due to the underestimation of the features of the spatial temperature correlation between stations. The ADW method requires an understanding of the spatial correlation structure of the station data. The spatial relationships between stations can vary with season. Since the anisotropy of the fields of hydrometeorological characteristics depends on the nature of the relief of the territory and on the season, the errors in the reconstruction of the fields will also differ. The regional variations of the correlation decay distance (CDD) should be taken into account.

Sharpest possible clustering bounds using robust random graph analysis.

Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a robust assessment of complex networks that does not depend on the entire degree distribution, but only on its mean, range, and dispersion: summary statistics that are easy to obtain for most real-world networks. By solving several semi-infinite linear programs, we obtain tight (the sharpest possible) bounds for correlation and clustering measures, for all networks with degree distributions that share the same summary statistics. We identify various extremal random graphs that attain these tight bounds as the graphs with specific three-point degree distributions. We leverage the tight bounds to obtain robust laws that explain how degree-degree correlations and local clustering evolve as a function of node degrees and network size. These robust laws indicate that power-law networks with diverging variance are among the most extreme networks in terms of correlation and clustering, building a further theoretical foundation for the widely reported scale-free network phenomena such as correlation and clustering decay.

On the Strong Forms of the Borel–Cantelli Lemma and Dynamical Systems with Polynomial Decay of Correlations

On the Strong Forms of the Borel–Cantelli Lemma and Dynamical Systems with Polynomial Decay of Correlations

An optimal result on localization in random displacements models

Abstract We study random displacements models with a long-range particle-media interaction potential 𝔲 ⁢ ( r , θ ) = 𝔣 ⁢ ( θ ) ⁢ r - A {\mathfrak{u}(r,\theta)=\mathfrak{f}(\theta)r^{-A}} in polar coordinates, with a smooth function 𝔣 {\mathfrak{f}} which can be sign-indefinite. Spectral and dynamical localization, with an asymptotically exponential decay of eigenfunction correlators, is proved under the optimal condition A > d {A>d} .

Affinity-based geometric discord and quantum speed limits of its creation and decay

In this article, we define a faithful quantifier of bipartite quantum correlation, namely geometric version of quantum discord using affinity based metric. It is shown that the newly-minted measure resolves the local ancilla problem of Hilbert-Schmidt measures. Exploiting the notion of affinity-based discord, we derive Margolus-Levitin (ML) and Mandelstamm-Tamm (MT) bounds for the quantum speed limit time for the creation and decay of quantum correlation. The dynamical study suggests that the affinity measure is a better resource compared to entanglement. Finally, we study the role of quantum correlation on quantum speed limit.

Upper bounds on the one-arm exponent for dependent percolation models

We prove upper bounds on the one-arm exponent $$\eta _1$$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson–Voronoi and Poisson–Boolean percolation. More precisely, in dimension $$d=2$$ we prove that $$\eta _1 \le 1/3$$ for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann–Fock field), and in $$d \ge 3$$ we prove $$\eta _1 \le d/3$$ for finite-range fields, both discrete and continuous, and $$\eta _1 \le d-2$$ for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.

On the Exponent Governing the Correlation Decay of the hbox {Airy}_1 Process

We study the decay of the covariance of the hbox {Airy}_1 process, {{mathcal {A}}}_1, a stationary stochastic process on mathbb {R} that arises as a universal scaling limit in the Kardar–Parisi–Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that {{,textrm{Cov},}}({{mathcal {A}}}_1(0),mathcal{A}_1(u))= e^{-(frac{4}{3}+o(1))u^3} as urightarrow infty . The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of mathcal{A}_1 to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the hbox {Airy}_1 process together with the FKG inequality.

Quenched Linear Response for Smooth Expanding on Average Cocycles

We establish an abstract quenched linear response result for random dynamical systems, which we then apply to the case of smooth expanding on average cocycles on the unit circle. In sharp contrast to the existing results in the literature, we deal with the class of random dynamics that does not necessarily exhibit uniform decay of correlations. Our techniques rely on the infinite-dimensional ergodic theory and in particular, on the study of the top Oseledets space of a parametrized transfer operator cocycle.

Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials

Consider the subspace {{{mathscr {W}}}_{n}} of L^2({{mathbb {C}}},dA) consisting of all weighted polynomials W(z)=P(z)cdot e^{-frac{1}{2}nQ(z)}, where P(z) is a holomorphic polynomial of degree at most n-1, Q(z)=Q(z,{bar{z}}) is a fixed, real-valued function called the “external potential”, and dA=tfrac{1}{2pi i}, d{bar{z}}wedge dz is normalized Lebesgue measure in the complex plane {{mathbb {C}}}. We study large n asymptotics for the reproducing kernel K_n(z,w) of {{mathscr {W}}}_n; this depends crucially on the position of the points z and w relative to the droplet S, i.e., the support of Frostman’s equilibrium measure in external potential Q. We mainly focus on the case when both z and w are in or near the component U of hat{{{mathbb {C}}}}setminus S containing infty , leaving aside such cases which are at this point well-understood. For the Ginibre kernel, corresponding to Q=|z|^2, we find an asymptotic formula after examination of classical work due to G. Szegő. Properly interpreted, the formula turns out to generalize to a large class of potentials Q(z); this is what we call “Szegő type asymptotics”. Our derivation in the general case uses the theory of approximate full-plane orthogonal polynomials instigated by Hedenmalm and Wennman, but with nontrivial additions, notably a technique involving “tail-kernel approximation” and summing by parts. In the off-diagonal case zne w when both z and w are on the boundary {partial }U, we obtain that up to unimportant factors (cocycles) the correlations obey the asymptotic Kn(z,w)∼2πnΔQ(z)14ΔQ(w)14S(z,w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n(z,w)\\sim \\sqrt{2\\pi n}\\,\\Delta Q(z)^{\\frac{1}{4}}\\,\\Delta Q(w)^{\\frac{1}{4}}\\,S(z,w) \\end{aligned}$$\\end{document}where S(z, w) is the Szegő kernel, i.e., the reproducing kernel for the Hardy space H^2_0(U) of analytic functions on U vanishing at infinity, equipped with the norm of L^2({partial }U,|dz|). Among other things, this gives a rigorous description of the slow decay of correlations at the boundary, which was predicted by Forrester and Jancovici in 1996, in the context of elliptic Ginibre ensembles.

Analyticity for Classical Gasses via Recursion

We give a new criterion for a classical gas with a repulsive pair potential to exhibit uniqueness of the infinite volume Gibbs measure and analyticity of the pressure. Our improvement on the bound for analyticity is by a factor $$e^2$$ over the classical cluster expansion approach and a factor e over the known limit of cluster expansion convergence. The criterion is based on a contractive property of a recursive computation of the density of a point process. The key ingredients in our proofs include an integral identity for the density of a Gibbs point process and an adaptation of the algorithmic correlation decay method from theoretical computer science. We also deduce from our results an improved bound for analyticity of the pressure as a function of the density.

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our coupling method only requires that the expanding map satisfies three assumptions: (H1) Chernov’s one-step expansion at q-scale; (H2) dynamically Hölder continuity of log Jacobian (on each branch); (H3) eventually covering over a magnet interval, which are much weaker than the standard assumptions for uniformly expanding maps. We further conclude the existence of an absolutely continuous invariant probability measure and establish the regularity of its density function. Moreover, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem) with respect to a large class of unbounded observables that we call “dynamically Hölder series". Our approach is particularly powerful for the piecewise expanding maps which do not satisfy the “big image property" and the maps that have inverse Jacobian of low regularity. As few is known on the statistical properties of such maps in the literature, we demonstrate our results for a specific class of piecewise expanding maps of this kind.

An optimal transportation approach to the decay of correlations for non-uniformly expanding maps – CORRIGENDUM

An optimal transportation approach to the decay of correlations for non-uniformly expanding maps – CORRIGENDUM

Berezinskii-Kosterlitz-Thouless phases in ultrathin PbTiO3/SrTiO3 superlattices

We study the emergence of Berezinskii-Kosterlitz-Thouless (BKT) phases in ${({\mathrm{PbTiO}}_{3})}_{3}/{({\mathrm{SrTiO}}_{3})}_{3}$ superlattices by means of second-principles simulations. Between a tensile epitaxial strain of $\ensuremath{\epsilon}=0.25--1\phantom{\rule{0.16em}{0ex}}%$, the local dipole moments within the superlattices are confined to the film-plane, and thus the polarization can be effectively considered as two-dimensional. The analysis of the decay of the dipole-dipole correlation with the distance, together with the study of the density of defects and its distribution as a function of temperature, supports the existence of a BKT phase in a range of temperature mediating the ordered ferroelectric (stable at low $T$), and the disordered paraelectric phase that appears beyond a critical temperature ${T}_{\mathrm{BKT}}$. This BKT phase is characterized by quasi-long-range order (whose signature is a power-law decay of the correlations with the distance), and the emergence of tightly bound vortex-antivortex pairs whose density is determined by a thermal activation process. The proposed ${\mathrm{PbTiO}}_{3}/{\mathrm{SrTiO}}_{3}$ superlattice model and the imposed mechanical boundary conditions are both experimentally feasible, making it susceptible for an experimental observation of these new topological phases in ferroelectric materials.

Equilibrium states for non-transitive random open and closed dynamical systems

AbstractWe prove a random Ruelle–Perron–Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides the existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.