Exponent Inequalities Research Articles

New critical exponent inequalities for percolation and the random cluster model

We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: The critical exponent inequalities $\gamma \leq \delta-1$ and $\Delta \leq \gamma +1$ hold for percolation and the random cluster model on any transitive graph. These inequalities are new even in the context of Bernoulli percolation on $\mathbb{Z}^d$, and are saturated in mean-field for Bernoulli percolation and for the random cluster model with $q \in [1,2)$. The volume of a cluster has an exponential tail in the entire subcritical phase of the random cluster model on any transitive graph. This proof also applies to infinite-range models, where the result is new even in the Euclidean setting.

Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension

We consider the one-dimensional model of a spin glass with independent Gaussian-distributed random interactions, which have mean zero and variance 1/|i-j|^{2σ}, between the spins at sites i and j for all i≠j. It is known that, for σ>1, there is no phase transition at any nonzero temperature in this model. We prove rigorously that, for σ>3/2, any translation-covariant Newman-Stein metastate for the ground states (i.e., the frequencies with which distinct ground states are observed in finite-size samples in the limit of infinite size, for given disorder) is trivial and unique. In other words, for given disorder and asymptotically at large sizes, the same ground state, or its global spin flip, is obtained (almost) always. The proof consists of two parts: One is a theorem (based on one by Newman and Stein for short-range two-dimensional models), valid for all σ>1, that establishes triviality under a convergence hypothesis on something similar to the energies of domain walls and the other (based on older results for the one-dimensional model) establishes that the hypothesis is true for σ>3/2. In addition, we derive heuristic scaling arguments and rigorous exponent inequalities which tend to support the validity of the hypothesis under broader conditions. The constructions of various metastates are extended to all values σ>1/2. Triviality of the metastate in bond-diluted power-law models for σ>1 is proved directly.

Uniformly strong consistency and Berry-Esseen bound of frequency polygons for α-mixing samples

ABSTRACTIn this article, the frequency polygon investigated by Scott is studied as a nonparametric estimator for α-mixing samples. By some known exponent and moment inequalities, we obtain the uniformly strong consistency and Berry-Esseen bound of the estimator. The present results relax the relevant conditions used by Carbon et al. Furthermore, the convergence rate of the uniformly asymptotic normality is derived, which is O(n− 1/11) under the given conditions.

Exponent Inequalities in Dynamical Systems

In this Letter, we derive exponent inequalities relating the dynamic exponent z to the steady state exponent Γ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating z and Γ. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems, are discussed.

The complexity of deciding consistency of systems of polynomials in exponent inequalities

Suppose the polynomials P 1…, P k ∈ℤ[ X 1,…, X n , U], h∈ℤ[ X 1,…, X n ] have degrees deg( P i ), deg( h)< d and the absolute value of any coefficient of P i or h is less or equal to 2 M for all 1≤ i≤ k. An algorithm is described which recognizes the consistency in ℝ n of the system of inequalities P 1( X 1,... X n , e h( X 1,…, X n) )≥0,…, P k ( X 1,…, X n , e h(X 1,…,X n )≥0 within time polynomial in M(nkd) n 4 .

Exponent inequalities for the bulk conductivity of a hierarchical model

The bulk conductivityσ *(p) of the bond lattice in ℤ d is considered, where the bonds have conductivity 1 with probabilityp or e≧0 with probability 1-p Various representations of the derivatives ofσ *(p) are developed. These representations are used to analyze the behavior ofσ *(p) for e=0 near the percolation thresholdp c , when the conducting backbone is assumed to have a hierarchical node-link-blob (NLB) structure. This model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds. Exact asymptotics of $$\frac{{d^2 \sigma ^* }}{{dp^2 }}$$ for the NLB model are proven under some technical assumptions. The proof employs a novel technique whereby $$\frac{{d^2 \sigma ^* }}{{dp^2 }}$$ for the NLB model with e=0 andp nearp c is computed using perturbation theory forσ *(p) (for two- and three-component resistor lattices) aroundp=1 with a sequence of e′s converging to 1 as one goes deeper in the hierarchy. These asymptotics establish convexity ofσ *(p) (for the NLB model) nearp c , and that its critical exponentt obeys the inequalities 1≦t≦2 ford=2,3, while 2≦t≦3 ford≧4. The upper boundt=2 ind=3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model.

Convexity and exponent inequalities for conduction near percolation.

The bulk conductivity ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$(p) of the bond lattice in ${\mathit{openZ}}^{\mathit{d}}$ with a fraction p of conducting bonds is analyzed. Assuming a hierarchical node-link-blob (NLB) model of the conducting backbone, it is shown that ${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$(p) (for this model) is convex in p near the percolation threshold ${\mathit{p}}_{\mathit{c}}$, and that its critical exponent t obeys the inequalities 1\ensuremath{\le}t\ensuremath{\le}2 for d=2,3, while 2\ensuremath{\le}t\ensuremath{\le}3 for d\ensuremath{\ge}4. The upper bound t=2 in d=3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion.

Rigorous exponent inequalities for random walks

The exponents for the resistance of a random walk path and the 'random walk on a random walk' problem are related to a number of other exponents for random walks. Some rigorous inequalities for these exponents are then established.

A realistic, minimal “middle layer” for neural networks

To demonstrate some dynamical consequences of making the formal characteristic of neuronal elements and their connectivities more realistic than those of modern neural computer algorithms, we study two multiplicatively coupled logistic maps, each with the quadratic response characteristics of neurons with autoreceptor mechanisms, which iteratively exchange their outputs. The dynamic phenomenology of this two-neuron system, though noninvertible, demonstrates characteristics of both automorphisms of the real line and diffeomorphisms of the plane, with smooth parametric transitions between them. We portray a unified parameter space containing the negative, positive, and mixed-coupled regimes, categorize the phase space structures using characteristic exponent inequalities, explore representative parametric paths, demonstrate generic bifurcation sequences, and report number theoretic ordering in parameter space. We conjecture that most, if not all, “middle-layer-like” brain systems, in contrast to primary sensory and motor information transport systems, are dominated by intrinsic dynamics of the sort demonstrated here. External input serves as a perturbation of these already ongoing complex systems. The intrinsic instabilities of these middle-layer dynamics return the unpredictability required by theoretical studies of real brain function to models of neural networks. We demonstrate a variety of parameter-dependent intrinsically ordered attractor sequences and number theoretic series which emerge from the global dynamics and exemplify the rich dynamical “machine” language potential for specific coding by brain-like information transport and learning algorithms.

On the upper critical dimensions of random spin systems

A set of critical exponent inequalities is proved for a large class of classical random spin systems. The inequalities imply rigorous (and probably the optimal) lower bounds for the upper critical dimensions, i.e.,d u≥4 for regular and random ferromagnets,d u≥6 for spin glasses and random field systems.

Rigorous upper and lower bounds on the non-linear susceptibility of the Mattis model

For a class of quenched random-site models (including the Mattis model), I prove rigorous upper and lower bounds for the non-linear susceptibility χ 2 in terms of the spontaneous magnetization M and susceptibility χ of the corresponding non-random ferromagnetic spin model. These bounds imply the critical exponent inequalities γ 2⩽ max (0,(2− d 2 )γ), γ′ 2⩽ max(0,(2− d 2 )γ′, γ′−2β ) for lattice dimension d⩾2.

Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 −1 /δ)≥2 −η, dv(1 − 1/δ)≥γ, anddμ> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.

On limit theorems for the bivariate (magnetization, energy) variable at the critical point

The limiting (magnetization, energy) bivariate variable is studied for Ising ferromagnets at the critical point. The factorization property of the limiting bivariate moment generating function is shown to be intimately connected to critical point exponent inequalities and to the behaviour of the scaling limit near and at the critical point. The validity of this can be deduced from the study of the second and the fourth magnetization cumulants at zero external field. The limiting bivariate variable is exactly calculated at the critical point for the Curie-Weiss model (MF) and for the edge of a two-dimensional Ising ferromagnet wrapped on a cylinder. It is shown that the mean field case leads to a non-Gaussian limiting distribution in contradistinction with the particular Ising model we consider for which we obtain a product of two Gaussian probability distributions.

Hyperscaling inequalities for percolation

A set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved. One of the consequences of the inequalities is the lower bounddC≧6 for the upper critical dimension. The proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as Ising ferromagnets.

Correlation length and its critical exponent for percolation processes

Some critical exponent inequalities are given involving the correlation length of site percolation processes on ℤd. In particular, it is shown thatv⩾2/d, which implies that the critical exponentv cannot take its mean-field value for the three-dimensional percolation processes.

Resistance noise in nonlinear resistor networks.

Cohn s theorem is extended to the case of circuit elements with the nonlinear I-V characteristic V rI'. This simplifies the study of resistance noise in nonlinear resistor networks. Exact exponent inequalities are derived. Fractal and percolating structures are considered. The infinite number of exponents necessary to characterize completely the electrical properties of linear and nonlinear percolating networks are calculated within the Migdal-Kadanoff approximation.

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP ∞(p) is discontinuous atp c , then the critical exponentγ (defined by the divergence of expected cluster size, ∑nP n (p) ∼ (P c −P)−γ asp ↑p c ) must satisfyγ ≥ 2. (2)γ orγ′ (defined analogously toγ, but asp ↓p c ) and δ [P n (p c ) ∼ (n −1−1/δ) asn → ∞ ] must satisfyγ,γ′ ≥ 2(1 − 1/δ). These inequalities forγ improve the previously known boundγ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P x has no discontinuity atp c .

Bulk transport properties and exponent inequalities for random resistor and flow networks

The properties of random resistor and flow networks are studied as a function of the density,p, of bonds which permit transport. It is shown that percolation is sufficient for bulk transport, in the sense that the conductivity and flow capacity are bounded away from zero wheneverp exceeds an appropriately defined percolation threshold. Relations between the transport coefficients and quantities in ordinary percolation are also derived. Assuming critical scaling, these relations imply upper and lower bounds on the conductivity and flow exponents in terms of percolation exponents. The conductivity exponent upper bound so derived saturates in mean field theory.

A diagrammatic representation of relations between critical exponents

It is shown that if exponent estimates are represented by lines on a graph then a particularly simple representation of exponent inequalities and scaling relations results. The exponents alpha ', beta , gamma ', delta are considered and the three-state Potts model is used to illustrate the technique.

Exponent Inequalities at the Roughening Transition

The exponents describing the divergence of various measures of the width of an interface between two phases in an Ising ferromagnet at the roughening transition are shown to satisfy a set of rigorous inequalities.