FKG Inequality Research Articles

A strong FKG inequality for multiple events

A strong FKG inequality for multiple events

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.

On the extension of the FKG inequality to n functions

The 1971 Fortuin–Kasteleyn–Ginibre inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008, one of us (Sahi) conjectured an extended version of this inequality for all n > 2 monotone functions on a distributive lattice. Here, we prove the conjecture for two special cases: for monotone functions on the unit square in Rk whose upper level sets are k-dimensional rectangles and, more significantly, for arbitrary monotone functions on the unit square in R2. The general case for Rk,k>2, remains open.

Sharp phase transition for Cox percolation

We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.

Rank‐optimal assignments in uniform markets

We prove that in a market where agents rank objects independently and uniformly at random, there exists an assignment of objects to agents with a constant average rank (i.e., an average rank independent of the market size). The proof builds on techniques from random graph theory and the FKG inequality (Fortuin et al. (1971)). When the agents' rankings are their private information, no Dominant Strategy Incentive Compatible mechanism can implement the assignment with the smallest average rank; however, we show that there exists a Bayesian Incentive Compatible mechanism that does so. Together with the fact that the average rank under the Random Serial Dictatorship (RSD) mechanism grows infinitely large with the market size, our findings indicate that the average rank under RSD can take a heavy toll compared to the first‐best, and highlight the possibility of using other assignment methods in scenarios where average rank is a relevant objective.

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order sqrt{log N}. The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

Correlation inequalities and monotonicity properties of the Ruelle operator

In this paper, we provide sufficient conditions for the validity of the FKG Inequality, on Thermodynamic Formalism setting, for a class of eigenmeasures of the dual of the Ruelle operator. We use this correlation inequality to study the maximal eigenvalue problem for the Ruelle operator associated to low regular potentials. As an application, we obtain explicit upper bounds for the main eigenvalue (consequently, for the pressure) of the Ruelle operator associated to Ising models with a power law decay interaction energy.

The dixie cup problem and FKG inequality

AbstractLet be the number of purchases required to obtain copies of given items, each purchase choosing at random one of the items. is the expected value of the random variable . The problem of obtaining a formula for is known as the dixie cup problem. The problem is easy for , but difficult for m > 1. Newman and Shepp solve the problem for all m, n. From the formula, they obtain the asymptotics of for each fixed and tending to infinity. Later, Erdös and Rényi obtain the limit law for , for each fixed and tending to infinity. From the limit law, they also derive and improve on the result of Newman and Shepp. The derivation is however incomplete, as they do not address the problem of estimating the tails of the distribution of . In this paper, we provide the estimates. The estimates depend on notions concerning conditional probabilities. In particular, we use the FKG inequality, a correlation inequality which is a fundamental tool in statistical mechanics and probabilistic combinatorics.

Comment on "Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated".

Cator and Van Mieghem [Phys. Rev. E 89, 052802 (2014)PLEEE81539-375510.1103/PhysRevE.89.052802] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian susceptible-infected-susceptible (SIS) and susceptible-infected-removed (SIR) epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the Fortuin, Kasteleyn, and Ginibre (FKG) inequality. In this Comment, we show that although the approach used by the authors applies to the SIS model, it cannot be used for the SIR model as stated in their work. In particular, we observe that monotonicity in the process is crucial for invoking the FKG inequality. Moreover, we provide an example of a simple graph for which the nodal infection in the SIR Markovian model is negatively correlated.

Persistence probabilities and a decorrelation inequality for the Rosenblatt process and Hermite processes

Мы изучаем вероятности персистентности для эрмитовых процессов. В качестве инструмента мы получаем общее неравенство декорреляции для процесса Розенблатта, которое напоминает лемму Слепяна для гауссовских процессов или FKG-неравенство и которое может быть интересно само по себе. Используя это неравенство, мы вычисляем показатель персистентности для процесса Розенблатта. Для общих эрмитовых процессов мы получаем верхнюю и нижнюю границы вероятности персистентности с ожидаемым показателем, но с несогласованными границами.

Correlation inequalities for the Potts model

Correlation inequalities are presented for ferromagnetic Potts models with external field, using the random-cluster representation of Fortuin and Kasteleyn, together with the FKG inequality. These results extend and simplify earlier inequalities of Ganikhodjaev and Razak, and also of Schonmann, and include GKS-type inequalities when the spin-space is taken as the set of $q$th roots of unity.

A note on jointly modeling edges and node attributes of a network

We are interested in modeling networks in which the connectivity among the nodes and node attributes are random variables and interact with each other. We propose a probabilistic model that allows one to formulate jointly a probability distribution for these variables. This model can be described as a combination of a latent space model and a Gaussian graphical model: given the node variables, the edges will follow independent logistic distributions, with the node variables as covariates; given edges, the node variables will be distributed jointly as multivariate Gaussian, with their conditional covariance matrix depending on the graph induced by the edges. We will present some basic properties of this model, including a connection between this model and a dynamical network process involving both edges and node variables, the marginal distribution of the model for edges as a random graph model, its one-edge conditional distributions, the FKG inequality, and the existence of a limiting distribution for the edges in an infinite graph.

Second-order Markov random fields for independent sets on the infinite Cayley tree

Recently, there has been significant interest in understanding the properties of Markov random fields (M.r.f.) defined on the independent sets of sparse graphs. When these M.r.f. are restricted to pairwise interactions (i.e., hardcore model), much progress has been made. However, considerably less is known in the presence of higher-order interactions, which arise, for example, in the analysis of independent sets with special properties and the study of resource-constrained communication networks. In this paper, we further our understanding of such models by analyzing M.r.f. with second-order interactions on the independent sets of the infinite Cayley tree. We prove that the associated Gibbsian specification satisfies the celebrated FKG inequality whenever the local potentials defining the Hamiltonian satisfy a log-convexity condition. Under this condition, we give necessary and sufficient conditions for the existence of a unique infinite-volume Gibbs measure in terms of an explicit system of equations, prove the existence of a phase transition and give explicit bounds on the associated critical activity, which we prove to exhibit a certain robustness. For potentials which are small perturbations of those coinciding to the hardcore model at the critical activity, we characterize whether the resulting specification has a unique infinite-volume Gibbs measure in terms of whether these perturbations satisfy an explicit linear inequality. Our analysis reveals an interesting nonmonotonicity with regards to biasing toward excluded nodes with no included neighbors.

On the correlation of increasing families

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand [19] established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory.In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight.A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof.

Maximal Rank Correlation

Based on the notion of maximal correlation, Kimeldorf, May and Sampson (1980) introduce a measure of correlation between two random variables, called the "concordant monotone correlation" (CMC). We revisit, generalize and prove new properties of this measure of correlation. It is shown that CMC captures various types of correlation detected in measures of rank correlation like the Kendall tau correlation. We show that the CMC satisfies the data processing and tensorization properties (that make ordinary maximal correlation applicable to problems in information theory). Furthermore, CMC is shown to be intimately related to the FKG inequality. Furthermore, a combinatorical application of CMC is given for which we do not know of another method to derive its result. Finally, we study the problem of the complexity of the computation of the CMC, which is a non-convex optimization problem with local maximas. We give a simple but exponential-time algorithm that is guaranteed to output the exact value of the generalized CMC.

Graphical Representations for Ising and Potts Models in General External Fields

This work is concerned with the theory of Graphical Representation for the Ising and Potts Models over general lattices with non-translation invariant external field. We explicitly describe in terms of the Random Cluster Representation the distribution function and, consequently, the expected value of a single spin for the Ising and $q$-states Potts Models with general external fields. We also consider the Gibbs States for the Edwards-Sokal Representation of the Potts Model with non-translation invariant magnetic field and prove a version of the FKG Inequality for the so called General Random Cluster Model (GRC Model) with free and wired boundary conditions in the non-translation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the quasilocality of the Gibbs Measures for the GRC Model with such general magnetic fields. As a final application of the theory developed, we show the uniqueness of the Gibbs Measures for the Ferromagnetic Ising Model with a positive power law decay magnetic field, as conjectured in [8].

Crossover distributions at the edge of the rarefaction fan

We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_-<\rho_+$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834 (2010) 523-542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman-Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.

Gibbs states over the cone of discrete measures

We construct Gibbs perturbations of the Gamma process on Rd, which may be used in applications to model systems of densely distributed particles. First we propose a definition of Gibbs measures over the cone of discrete Radon measures on Rd and then analyze conditions for their existence. Our approach works also for general Lévy processes instead of Gamma measures. To this end, we need only the assumption that the first two moments of the involved Lévy intensity measures are finite. Also uniform moment estimates for the Gibbs distributions are obtained, which are essential for the construction of related diffusions. Moreover, we prove a Mecke type characterization for the Gamma measures on the cone and an FKG inequality for them.

Power law Pólya’s urn and fractional Brownian motion

We introduce a natural family of random walks \(S_n\) on \(\mathbb{Z }\) that scale to fractional Brownian motion. The increments \(X_n := S_n - S_{n-1} \in \{\pm 1\}\) have the property that given \(\{ X_k : k < n \}\), the conditional law of \(X_n\) is that of \(X_{n - k_n}\), where \(k_n\) is sampled independently from a fixed law \(\mu \) on the positive integers. When \(\mu \) has a roughly power law decay (precisely, when \(\mu \) lies in the domain of attraction of an \(\alpha \)-stable subordinator, for \(0<\alpha <1/2\)) the walks scale to fractional Brownian motion with Hurst parameter \(\alpha + 1/2\). The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense we describe, they are the natural “fractional” analogues of simple random walk on \(\mathbb{Z }\).

A q-analogue of the FKG inequality and some applications

Let L be a finite distributive lattice and µ: L → ℝ+ a log-supermodular function. For functions k: L → ℝ+ let $E_\mu (k;q)^{\underline{\underline {def}} } \sum\limits_{x \in L} {k(x)\mu (x)q^{rank(x)} \in \mathbb{R}^ + [q]} .$ We prove for any pair g,h: L → ℝ+ of monotonely increasing functions, that $E_\mu (g;q) \cdot E_\mu (h;q) \ll E_\mu (1;q) \cdot E_\mu (gh;q),$ where “≪” denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to q=1. The polynomial FKG inequality has applications to f-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of Schubert varieties, and to correlation-type inequalities for a class of power series weighted by Young tableaux. This class contains series involving Plancherel measure for the symmetric groups and its poissonization.