Gaussian Inequality Research Articles

On Boundary Correlations in Planar Ashkin–Teller Models

Abstract We generalize the switching lemma of Griffiths, Hurst, and Sherman to the random current representation of the Ashkin–Teller model. We then use it together with properties of two-dimensional topology to derive linear relations for multipoint boundary spin correlations and bulk order–disorder correlations in planar models. We also show that the same linear relations are satisfied by products of Pfaffians. As a result, a clear picture arises in the noninteracting case of two independent Ising models where multipoint correlation functions are given by Pfaffians and determinants of their respective two-point functions. This gives a unified treatment of both the classical Pfaffian identities and recent total positivity inequalities for boundary spin correlations in the planar Ising model. We also derive the Simon and Gaussian inequalities for general Ashkin–Teller models with negative four-body coupling constants.

Sharp upper and lower bounds for maximum likelihood solutions to random Gaussian bilateral inequality systems

This paper focuses on finding a solution maximizing the joint probability of satisfaction of a given set of (independent) Gaussian bilateral inequalities. A specially structured reformulation of this nonconvex optimization problem is proposed, in which all nonconvexities are embedded in a set of 2-variable functions composing the objective. From this, it is shown how a polynomial-time solvable convex relaxation can be derived. Extensive computational experiments are also reported, and compared to previously existing results, showing that the approach typically yields feasible solutions and upper bounds within much sharper confidence intervals.

Global probability maximization for a Gaussian bilateral inequality in polynomial time

The present paper investigates Gaussian bilateral inequalities in view of solving related probability maximization problems. Since the function f representing the probability of satisfaction of a given Gaussian bilateral inequality is not concave everywhere, we first state and prove a necessary and sufficient condition for negative semi-definiteness of the Hessian. Then, the (nonconvex) problem of globally maximizing f over a given polyhedron in $$\mathbb {R}^{n}$$ is adressed, and shown to be polynomial-time solvable, thus yielding a new-comer to the (short) list of nonconvex global optimization problems which can be solved exactly in polynomial time. Application to computing upper bounds to the maximum joint probability of satisfaction of a set of m independent Gaussian bilateral inequalities is discussed and computational results are reported.

Gaussian inequalities for Wulff shapes

It is shown that there are two Gaussian inequalities for Wulff shapes corresponding to Schuster and Weberndorfer’s results for the Lebesgue measure. Barthe’s mean width inequality for continuous isotropic measures and its dual inequality are special cases of these new inequalities. Moreover, two new Gaussian inequalities related to the LYZ ellipsoid are obtained.

A Gaussian Inequality for Expected Absolute Products

We prove the inequality that \({\mathbb{E}}|X_{1}X_{2}\cdots X_{n}|\leq \sqrt{\mathrm{per}(\varSigma )}\), for any centered Gaussian random variables X1,…,Xn with the covariance matrix Σ, followed by several applications and examples. We also discuss a conjecture on the lower bound of the expectation.

Thinning, Entropy, and the Law of Thin Numbers

Renyi's "thinning" operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables. The properties of thinning are investigated in an information-theoretic context, especially in connection with information-theoretic inequalities related to Poisson approximation results. The classical Binomial-to-Poisson convergence (sometimes referred to as the "law of small numbers" is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is provided for this limit, and nonasymptotic bounds are also established. This development parallels, in part, the development of Gaussian inequalities leading to the information-theoretic version of the central limit theorem. In particular, a "thinning Markov chain" is introduced, and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck process in connection to the entropy power inequality.

Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments

Generalizing to higher dimensions the classical gambler’s ruin estimates, we give pointwise estimates for the transition kernel corresponding to a spatially inhomogeneous random walk on the half-space. Our results hold under some strong but natural assumptions of symmetry, boundedness of the increments, and ellipticity. Among the most important steps in our proof are: discrete variants of the boundary Harnack estimate, as proven by Bauman, Bass and Burdzy, and Fabes et al., based on comparison arguments and potential-theoretical tools; the existence of a positive $\tilde{L}$-harmonic function globally defined in the half-space; and some Gaussian inequalities obtained by a treatment inspired by Varopoulos.

Instantaneous global spatial interaction? Exploring the Gaussian inequality, distance and Internet pings in a global network

The Internet has been publicly portrayed as a new technological horizon yielding instantaneous interaction to a point where geography no longer matters. This research aims to dispel this impression by applying a dynamic form of trip modelling to investigate pings in a global computer network compiled by the Stanford Linear Accelerator Centre (SLAC) from 1998 to 2004. Internet flows have been predicted to have the same mathematical operators as trips to a supermarket, since they are both periodic and constrained by a distance metric. Both actual and virtual trips are part of a spectrum of origin–destination pairs in the time–space convergence of trip time-lines. Internet interaction is very near to the convergence of these time-lines (at a very small time scale in milliseconds, but with interactions over thousands of kilometres). There is a lag effect and this is formalised by the derivation of Gaussian and gravity inequalities between the time taken (Δt) and the partitioning of distance (Δx). This inequality seems to be robust for a regression of Δt to Δx in the SLAC data set for each year (1998 to 2004). There is a constant ‘forbidden zone’ in the interaction, underpinned by the fact that pings do not travel faster than the speed of light. Superimposed upon this zone is the network capacity where a linear regression of Δt to Δx is a proxy summarising global Internet connectivity for that year. The results suggest that there has been a substantial improvement in connectivity over the period with R 2 increasing steadily from 0.39 to 0.65 from less Gaussian spreading of the ping latencies. Further, the regression line shifts towards the inequality boundary from 1998 to 2004, where the increased slope shows a greater proportional rise in local connectivity over global connectivity. A conclusion is that national geography still does matter in spatial interaction modelling of the Internet.

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Correlation inequalities and the Kosterlitz-Thouless transition for anisotropic rotators

I give a proof of the Kosterlitz-Thouless transition for sufficiently anisotropic (JzJ x −1 =JzJ y −1 <2q−(JKT)−J) two-dimensionalN-component rotators (N⩾ 3). The method is based on Wells' inequality and is related to mean field Gaussian inequalities.

Fluctuation susceptibility relations for classical spin systems

We prove identities between integrated Ursell functions and derivatives of the pressure in the thermodynamic limit, for multicomponent classical spin systems which obey the Lee-Yang theorem and some form of Gaussian domination, when the susceptibility is finite (T>Tc). Following Refs. 3 and 4, we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function. Fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants, taken in zero external field. High-order cumulants are bounded in terms of the susceptibility using Gaussian and Simon's inequalities for short-range interactions.

Zeros of the partition function and Gaussian inequalities for the plane rotator model

The partition function for ferromagnetic plane rotators in a complex external field μ, with ¦Im μ¦ ⩽ ¦Re μ ¦, is bounded below in modulus by its value at μ=0. The proof is based on complex combinations of duplicated variables which are positive definite on a subgroup of the configuration group. In the isotropic situation (and μ=0), the associated “Gaussian inequalities” imply that all truncated correlation functions decay at least as the two-point function.

The Gaussian inequality for multicomponent rotators

The Gaussian inequality is proven for multicomponent rotators with negative correlations between two spin components. In the case of one-component systems, the Gaussian inequality is shown to be a consequence of the Lebowitz inequality. For multicomponent models, the Gaussian inequality implies that the decay rate of the truncated correlation (or Schwinger) functions is dominated by that of the two-point function. Applied to field theory these inequalities give information on the absence of bound states in the lambda ( phi /sub 1 //sup 2/+ phi /sub 2//sup 2/)/sup 2/ model.