Random pinning model with finite range correlations: Disorder relevant regime

The purpose of this paper is to show how one can extend some results on disorder relevance obtained for the random pinning model with i.i.d disorder to the model with finite range correlated disorder. In a previous work, the annealed critical curve of the latter model was computed, and equality of quenched and annealed critical points, as well as exponents, was proved under some conditions on the return exponent of the interarrival times. Here we complete this work by looking at the disorder relevant regime, where annealed and quenched critical points differ. All these results show that the Harris criterion, which was proved to be correct in the i.i.d case, remains valid in our setup. We strongly use Markov renewal constructions that were introduced in the solving of the annealed model.     

Hierarchical pinning models, quadratic maps and quenched disorder

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by Derrida et al. (J Stat Phys 66:1189–1213, 1992), which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R _n } _{n=1,2, ...}, which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known logistic map. The large-n limit of the sequence of random variables 2^?n log R _n , a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α ? (0, 1), related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R ₀ is larger than a certain threshold value, and it is zero otherwise. It was conjectured in Derrida et al. (J Stat Phys 66:1189–1213, 1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 < α < 1 (respectively, α < 1/2 or α = 1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if α ≥ 1/2, but not if α <  1/2. Our main result is a proof of these conjectures for the case α ≠ 1/2. We emphasize that for α >  1/2 we find the correct scaling form (for weak disorder) of the critical point shift.     

Sharp critical behavior for pinning models in a random correlated environment

This article investigates the effect for random pinning models of long range power-law decaying correlations in the environment. For a particular type of environment based on a renewal construction, we are able to sharply describe the phase transition from the delocalized phase to the localized one, giving the critical exponent for the (quenched) free-energy, and proving that at the critical point the trajectories are fully delocalized. These results contrast with what happens both for the pure model (i.e., without disorder) and for the widely studied case of i.i.d. disorder, where the relevance or irrelevance of disorder on the critical properties is decided via the so-called Harris Criterion (Harris, 1974) [21].     

Universality of Chaos and Ultrametricity in Mixed p‐Spin Models

We prove disorder universality of chaos phenomena and ultrametricity in the mixed p‐spin model under mild moment assumptions on the environment. This establishes the longstanding belief among physicists that the solution of mean‐field models with Gaussian disorder also holds for different environments. Our results extend to the mixed p‐spin model as well as to different spin glass models. These include universality of quenched disorder chaos in the Edwards‐Anderson (EA) model and quenched concentration for the magnetization in both EA and mixed p‐spin models under non‐Gaussian environments. In addition, we show quenched self‐averaging for the overlap in the random field Ising model under small perturbation of the external field.© 2015 Wiley Periodicals, Inc.     

Continuous spin models on annealed generalized random graphs

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution.First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case.We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in Dommers et al. (2016). On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.     

The Influence of Hughes Type Action

We call the action of an automorphism α of a finite group G a Hughes type action if it is described by conditions on the orders of elements of G ? α ? ? G. In the present paper we study the structure of finite group G admitting an automorphism α of prime order p so that the orders of elements in G ? α ? ? G are not divisible by p ².     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On Finite p-Groups Whose IA-Automorphisms Are All Class Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism α of G is called a class preserving automorphism if α(x) ∈ x ^G the conjugacy class of x in G, for all x ∈ G. An automorphism α of G is called an IA-automorphism if x ^?1α(x) ∈ G′ for each x ∈ G. In this paper, we give necessary and sufficient conditions on finite p-group G of nilpotency class 2 such that every IA-automorphism is class preserving.     

Directed polymers in a random environment with heavy tails

We study the model of directed polymers in a random environment in 1 + 1 dimensions, where the distribution at a site has a tail that decays regularly polynomially with power α, where α ∈ (0,2). After proper scaling of temperature β⁻¹, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (α, β)‐indexed family of measures on Lipschitz curves lying inside the 45°‐rotated square with unit diagonal. In particular, this shows order‐n transversal fluctuations of the polymer. If, and only if, α is small enough, we find that there exists a random critical temperature below which, but not above which, the effect of the environment is macroscopic. The results carry over to d + 1 dimensions for d > 1 with minor modifications. © 2010 Wiley Periodicals, Inc.     

Continuation of blowup solutions of nonlinear heat equations in several space dimensions

The possible continuation of solutions of the nonlinear heat equation in R^N × R₊ u_t = Δu^m + u^p with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation u_t = Δu^m − u^p, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc.     

Universality in two‐dimensional enhancement percolation

We consider a type of dependent percolation introduced in 2 , where it is shown that certain “enhancements” of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this study we first prove that, for two‐dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two‐dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit 12 , 13 is not affected by any monotonic enhancement that does not shift the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Long-wave instabilities and saturation in thin film equations

Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard-type interface models develop finite-time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term (Phys.˜ D 67, 1993, pp. 113–125). We consider this conjecture for a class of equations, often used to model thin films in a lubrication context, by showing that if the solutions are uniformly bounded above or below (e.g., are nonnegative), then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded. For example, they conjecture that for the long-wave unstable equation m > n leads to blowup. Using a conservation-of-volume constraint for the case m > n > 0, we conjecture a different critical exponent for possible singularities of nonnegative solutions. We prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions. Specifically, for the above equation, solutions are bounded if m < n + 2. The bound is proved using energy methods and is then used to prove the existence of nonnegative weak solutions in the sense of distributions. We present preliminary numerical evidence suggesting that m ≥ n + 2 can allow blowup. © 1998 John Wiley & Sons, Inc.     

Restricted Percolation Critical Exponents in High Dimensions

Despite great progress in the study of critical percolation on ℤ^d for d large, properties of critical clusters in high-dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high-dimensional critical exponents; see in particular the conjecture by Kozma-Nachmias that the probability that 0 and (n, n, n, …) are connected within [−n, n]^d scales as n^{−2 − 2d} . In this paper, we study the properties of critical clusters in high-dimensional half-spaces and boxes. In half-spaces, we show that the probability of an open connection (“arm”) from 0 to the boundary of a sidelength n box scales as n⁻³ . We also find the scaling of the half-space two-point function (the probability of an open connection between two vertices) and the tail of the cluster size distribution. In boxes, we obtain the scaling of the two-point function between vertices which are any macroscopic distance away from the boundary. Our argument involves a new application of the “mass transport" principle which we expect will be useful to obtain quantitative estimates for a range of other problems. © 2020 Wiley Periodicals LLC     

Lyapunov exponents of Green��s functions for random potentials tending to zero

We consider quenched and annealed Lyapunov exponents for the Green??s function of ????+? ??V, where the potentials ${V(x),\ x\in\mathbb {Z}^d}$ , are i.i.d.? nonnegative random variables and ?? > 0 is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like ${c\sqrt{\gamma}}$ as ?? tends to 0. Here the constant c is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.     

A Quenched CLT for Super-Brownian Motion with Random Immigration

A quenched central limit theorem is derived for the super-Brownian motion with super-Brownian immigration, in dimension d≥4. At the critical dimension d=4, the quenched and annealed fluctuations are of the same order but are not equal. W. Hong was supported by the Program for New Century Excellent Talents in University (NCET) and NSFC (Grant No. 10121101). O. Zeitouni was partially supported by NSF grant DMS-0503775.     

On the repulsion of an interface above a correlated substrate

We analyze a model of an interface fluctuating above a rough substrate. It is based on harmonic crystals, or lattice free fields, indexed by ^d , d 3. The phenomenon for which we want to get precise quantitative estimates is the repulsion effect of the substrate on the interface: the substrate is itself a random field, but its randomness is quenched (this generalizes the widely considered case of a flat deterministic substrate). With respect to [2] in which the substrate has been taken to be an IID field, here the substrate is an harmonic crystal, as the interface, and as such it is strongly correlated. We obtain the leading asymptotic behavior of the model in the limit of a very extended substrate: we show in particular that, to leading order, the effect of an IID substrate cannot be distinguished from the effect of an harmonic crystal substrate. We observe however that, unlike in the IID substrate case, annealed and quenched models display sharply different features.     

Renormalization group and the <Emphasis Type="Italic">ɛ</Emphasis>-expansion: Representation of the <Emphasis Type="Italic">β</Emphasis>-function and anomalous dimensions by nonsingular integrals

In the framework of the renormalization group and the ɛ-expansion, we propose expressions for the β-function and anomalous dimensions in terms of renormalized one-irreducible functions. These expressions are convenient for numerical calculations. We choose the renormalization scheme in which the quantities calculated using R operations are represented by integrals that do not contain singularities in ɛ. We develop a completely automated calculation system starting from constructing diagrams, determining relevant subgraphs, combinatorial coefficients, etc., up to determining critical exponents. As an example, we calculate the critical exponents of the φ ³ model in the order ɛ ⁴ .     

Half-transitive graphs of order a product of two distinct primes *

A simple undirected graph X is said to be ½-transitive if the automorphism group AutX of X acts transitively onthe vertices and edges, but not on the arcs of X. In this pape we determine all ½-transitive graphs of order a product of two distinct primes     

On a Hardy Type General Weighted Inequality in Spaces L p(·)

A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L ^p(·). Equivalent necessary and sufficient conditions are found for the ${L^{p(\cdot)} \longrightarrow L^{q(\cdot)}}A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L ^p(·). Equivalent necessary and sufficient conditions are found for the L^p(·) ? L^q(·){L^{p(\cdot)} \longrightarrow L^{q(\cdot)}} boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.     

On the Marginal Automorphisms of a Group

Let W be a nonempty subset of a free group. We call an automorphism α of a group G a marginal automorphism if x ^?1α(x) ∈ W*(G) for each x ∈ G, where W*(G) is the marginal subgroup of G. In this article, we give some results on marginal automorphisms of a group.