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We discuss finite-size effects in one disordered $\lambda {\phi }^4$ model defined in a $d$-dimensional Euclidean space. We consider that the scalar field satisfies periodic boundary conditions in one dimension and it is coupled with a quenched random field. In order to obtain the average value of the free energy of the system, we use the replica method. We first discuss finite-size effects in the one-loop approximation in $d=3$ and $d=4$. We show that in both cases, there is a critical length where the system develop a second-order phase transition, when the system presents long-range correlations with power-law decay. Next, we improve the above result studying the gap equation for the size-dependent squared mass, using the composite field operator method. We obtain again that the system present a second-order phase transition with long-range correlation with power-law decay.

Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.

I analyze empirical correlations of electricity forward returns from the perspective of a random field model that specifies the correlations in terms of the temporal separation between forward maturities. It turns out that temporal separation cannot fully account for the empirical forward return correlations. Specifically, the relation between correlations and temporal separation does not seem to be invariant across segments of the electricity forward market or trading periods.

We provide a new projective condition for a stationary real random field indexed by the lattice ${\mathbb{Z}}^d$ to be well approximated by an orthomartingale in the sense of Cairoli (1969). Our main result can be viewed as a multidimensional version of the martingale-coboundary decomposition method which the idea goes back to Gordin (1969). It is a powerful tool for proving limit theorems or large deviations inequalities for stationary random fields when the corresponding result is valid for orthomartingales.

We prove a martingale-coboundary representation for random fields with a completely commuting filtration. For random variables in $L^2$, we present a necessary and sufficient condition which is a generalization of Heyde’s condition for one-dimensional processes from 1975. For $L^p$ spaces with $2\le p<\infty$ we give a necessary and sufficient condition which extends Volný’s result from 1993 to random fields and improves condition of El Machkouri and Giraudo from 2016. A new sufficient condition is presented which for dimension one improves Gordin’s condition from 1969. In application, new weak invariance principle and estimates of large deviations are found.

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

The rapid development of additive manufacturing has made coated structures an innovative configuration with high design flexibility. However, poor forming accuracy and surface roughness during manufacturing will cause uncertainty in surface layer thickness, which results in structure performance deviation and failure to achieve the expected goals. This paper proposes a robust topology optimization method for coated structures considering the surface layer thickness uncertainty to obtain high-quality designs that can resist disturbance by uncertainties. First, an erosion-based approach is used to establish the model of the coated structure surface layer. Second, modeling the surface layer thickness uncertainty applies a random field whose dimensionality of the random fields is reduced by the Expansion Optimal Linear Estimation (EOLE) method. Then, minimizing the weighted sum of the mean and standard deviation of structural compliance is taken as the optimization objective, and robust topology optimization considering uncertainty is established. Finally, estimate the stochastic response by the perturbation technique, then the sensitivity of the objective function with respect to the design variables is derived. Numerical examples show that the structural design obtained with the proposed method has a stronger resistance to uncertainty than the deterministic topology optimization method, proving the method’s effectiveness in this paper.

The objective of this study was to investigate the liquefaction response of soil using the spatial variability of the shear modulus by considering different values of the coefficient of variation (COV) and the horizontal scale of fluctuation (SOF). For this purpose, a Monte Carlo simulation, combining the digital generation of a non-Gaussian random field with finite difference analyses, was utilized. Parametric studies were performed from the perspectives of the liquefaction area, excess pore water pressure (EPWP), and displacement at the ground surface. We found that a larger COV of the soil shear modulus was correlated with a slower reduction of liquefaction area and a lower EPWP ratio. To explain the influence from the perspective of the spatial distribution characteristics of the shear modulus, a deterministic model test was carried out. Additionally, it was found that the displacement history and the differential settlement at the end of shaking were regularly affected by the COV and the horizontal SOF, especially for large COV and horizontal SOF.

In this paper, empirical mode decomposition technique is used to analyze the spatial slip distribution of five past earthquakes. It is shown that the finite fault slip models exhibit five empirical modes of oscillation. The last intrinsic mode is positive and characterizes the non-stationary mean of the slip distribution. This helps in splitting the spatial variability of slip into trend and the remaining modes sum as the fluctuation in the data. The fluctuation component indicates that it can be modeled as an anisotropic random field. Important parameters of this random field have been estimated. The effect of these modes on ground motion is presented by simulating both acceleration and displacement time histories.

Mathematical modeling and development of statistical algorithms for multidimensional image processing are examined. Optimal and quasioptimal solutions of the main image processing tasks (e.g. filtration, detection, recognition, and alignment) are also described. The authors suggest adaptive processing algorithms with pseudo-gradient procedures.

The problem of interest is to estimate an integral (or the total) of a random field from observations at a finite number of sampling points. The quality of an estimator is measured by mean square error (MSE). Sampling points are appropriately selected so that the resulting MSE is as close to zero as possible. In this article, we begin by introducing the basic elements of estimation of integrals of random fields, including a review of the number-theoretic method. We then show how the number-theoretic method is related to the design of sampling points along with the sample mean estimator.

We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^d-σ. Using FRG we obtained the phase diagram in (d, σ, N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4 + σ, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.