Critical behavior of a metallic diffuse-fringe film percolation system

Preparation of a metallic diffuse-fringe film system and its electrical properties are reported. The anomalousR-I characteristics, third-harmonic coefficient and critical behavior are observed in the film system. Analysis shows that this behavior is caused by the location-dependent tunneling and hopping effects, which result from. the continuous variation of the diffuse fringe structure of the film due to the increase of the current. Project supported by the National Natural Science Foundation of China and Zhejiang Provincial Natural Science Foundation of China.     

Asymptotic behavior of M-estimator and related random field for diffusion process

The M-estimate which maximizes a positive stochastic process Q is treated for multidimensional diffusion models. The convergence in distribution of the process of ratio of Q's after normalizing is proved. The asymptotic behavior of M-estimates is stated. We present the asymptotic variance in general cases and in estimation by misspecified models.     

Dispersion of Lagrangian trajectories in a random large-scale velocity field

We study the distribution of the distance R(t) between two Lagrangian trajectories in a spatially smooth turbulent velocity field with an arbitrary correlation time and a non-Gaussian distribution. There are two dimensionless parameters, the degree of deviation from the Gaussian distribution α and β=τD, where τ is the velocity correlation time and D is a characteristic velocity gradient. Asymptotically, R(t) has a lognormal distribution characterized by the mean runaway velocity and the dispersion Δ. We use the method of higher space dimensions d to estimate and Δ for different values of α and β. It was shown previously that for β≪ 1 and for β≫ 1. The estimate of Δ is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contribution to Δ estimated as αD²τ for β≪1 and αβ/τ for β≫1. For α above some critical value σ_cr, the values of and Δ are determined by higher irreducible correlators of the velocity gradient, and our approach loses its applicability. This critical value can be estimated as for β≪1 and for β≫1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 456–467, March, 2000.     

Extremal behavior of the heat random field

We study the exponential type inequalities for the distribution of the supremum of a random field that arises as the solution of the heat equation with a random initial condition that is a strictly sub-Gaussian random field. AMS 2000 Subject Classifications Primary—60G60; Secondary—60G17, 60F05     

Critical percolation: the expected number of clusters in a rectangle

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.     

Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation

We prove that the Fourier transform of the properly scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α > 0 converges to ${e^{-C|k|^{\alpha\wedge2}}}$ for some ${C\in(0,\infty)}$ above the upper-critical dimension ${{{dc \equiv 2(\alpha \wedge 2)}}}$ . This answers the open question remained in the previous paper (Chen and Sakai in Probab Theory Relat Fields 142:151–188, 2008). Moreover, we show that the constant C exhibits crossover at α = 2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.     

Critical behavior of two tensor models in correlated random fields

The renormalization-group method is used to investigate the critical behavior of two models with tensor order parameter in correlated random fields. The critical exponents are calculated for both short- and long-range interaction potentials. A scaling relation that differs from the analogous one in the vector model is obtained.Kazan State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 1, pp. 40–46, January, 1995.     

Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

Dynamic approximation of a random information functional

Using the probabilistic evaluation of the Marcovian diffusion process by a functional of action, the paper introduces a dynamic approximation of a random information functional defined on the process trajectories and determined by the parameters of a controlled stochastic equation. The developed mathematical formalism is aimed toward a dynamic modeling of a random object.     

Dynamic approximation of a random information functional

Using the probabilistic evaluation of the Marcovian diffusion process by a functional of action, the paper introduces a dynamic approximation of a random information functional defined on the process trajectories and determined by the parameters of a controlled stochastic equation. The developed mathematical formalism is aimed toward a dynamic modeling of a random object.     

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

An approximation of the thermal field in a continuous casting process of a thin metal layer

We consider the problem of a phase change in a continuous casting process: molten bronze is poured on to a moving steel strip cooled from below, in order to solidify the bronze. An estimate of the width of the solidification zone, depending on the thickness of the strip and on the casting velocity, is obtained, neglecting conduction in the direction of the strip motion.     

Weak approximation of CIR equation by discrete random variables

For the CIR equation \( {\text{d}}{X_t} = \left( {\theta - k\,{X_t}} \right){\text{d}}t + \sigma \sqrt {{{X_t}}} {\text{d}}{B_t} \), we propose positive weak first- and second-order approximations that use, at each step, generation of discrete (respectively two- and three-valued) random variables (Theorems 3 and 4). The equation is split into deterministic part \( {\text{d}}{D_t} = \left( {\theta - k{D_t}} \right){\text{d}}t \), which is solved exactly, and stochastic part \( {\text{d}}{S_t} = \sigma \sqrt {{{S_t}}} {\text{d}}{B_t} \), which is actually approximated in distribution.     

On the normal approximation of a sum of a random number of independent random variables

In this paper, we extend the results obtained in [J. Sunklodas, Some estimates of normal approximation for the distribution of a sum of a random number of independent random variables, Lith. Math. J., 52(3):326?C333, 2012] for a thrice-differentiable function h : ? ?? ? to the case of h ?? BL(?); namely, we estimate the quantity | E h(Z _N )? E h(Y)| where h is a real bounded Lipschitz function, $ {Z_N}={{{\left( {{S_N}-\mathrm{E}{S_N}} \right)}} \left/ {{\sqrt{{\mathrm{D}{S_N}}}}} \right.} $ , S _N = X ₁ + · · · + X _N , X ₁ , X ₂ , . . . are independent, not necessarily identically distributed, real random variables, N is a positive integer-valued r.v. independent of X ₁ , X ₂ , . . . , and Y is a standard normal random variable.     

Gravitational field of an expanding spherically symmetric shell in RTG. Linear velocity approximation

The appearance of nonstatic behavior of the gravitational fieldof an expanding, spherically symmetric shell is shown in RTG under the linear velocity approximation for a gravitational field of arbitrary magnitude.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 79–83. July, 1996.     

Inverse problems in underwater acoustics and approximation of the velocity field for signal propagation

A new approach is suggested for solving the inverse problem in underwater acoustics — the determination of the velocity of signal propagation in an inhomogeneous medium. The unknown velocity u(x(t), t) is assumed to be independent of the time t and is sought in the form of a polynomial with respect to the degree of the space coordinate x ₁, x₂, x₃.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 102–105, 1989.     

Longtime behavior for the occupation time process of a super-Brownian motion with random immigration

Longtime behavior for the occupation time of a super-Brownian motion with immigration governed by the trajectory of another super-Brownian motion is considered. Central limit theorems are obtained for dimensions d⩾3 that lead to some Gaussian random fields: for 3⩽d⩽5, the field is spatially uniform, which is caused by the randomness of the immigration branching; for d⩾7, the covariance of the limit field is given by the potential operator of the Brownian motion, which is caused by the randomness of the underlying branching; and for d=6, the limit field involves a mixture of the two kinds of fluctuations. Some extensions are made in higher dimensions. An ergodic theorem is proved as well for dimension d=2, which is characterized by an evolution equation.