Band 17, Heft 4

We study the spectral properties of matrices of long-range percolation model. These are N × N random real symmetric matrices H = { H ( i, j )} i,j whose elements are independent random variables taking zero value with probability , where ψ is an even positive function with ψ ( t ) ≤ 1 and vanishing at infinity. We study the resolvent G ( z ) = ( H – z ) –1 , Im z ≠ 0, in the limit N, b → ∞, b = O ( N α ), 1/3 < α < 1, and obtain the explicit expression T ( z 1 , z 2 ) for the leading term of the correlation function of the normalized trace of the resolvent g N,b ( z ) = N –1 Tr G ( z ). We show that in the scaling limit of local correlations, this term leads to the expression found earlier by other authors for band random matrix ensembles. This shows that the ratio b 2 / N is the correct scale for the eigenvalue density correlation function and that the ensemble we study and that of band random matrices belong to the same class of spectral universality.

In this paper we introduce the notion of T-convolution, which is a generalization of convolution to higher dimensions. By using T-convolution we construct n -dimensional distributions having n + 1 axes of symmetry. In addition, we can generalize well-known symmetric probability distributions in one dimension to higher dimensions. In particular, we consider generalizations of Laplace and triangle continuous distributions and we show their plots in the two-dimensional case. As an example of discrete distributions, we study the T-convolution of Poisson distributions in the plane.

Let X and Y be separable Banach spaces and E be a subset of X . By a random operator from X into Y we mean a rule Φ that assigns to each element x ∈ X a unique Y - valued random variable Φ x . Taking into account many circumstances in which the inputs are also subject to the influence of a random environment there arises the need to define the action of Φ on some random inputs. In this paper, some procedures for extending random mappings will be proposed. Some conditions under which a random mapping can be extended to apply to all X -valued random variables will be presented.

In this paper, we establish the existence of common random fixed points for multivalued random operators. In our results we may not assume that ƒ is an S -weakly commuting random operator and g is a T -weakly commuting random operator. We improve and extend the recent results of Naseer Shahzad and Nawab Hussain [J. Math. Anal. Appl. 323: 1038–1046, 2006] and many authors.

We study in this paper a class of fractional stochastic partial differential equations with non-Lipschitz coefficients in the whole real line. Existence, uniqueness and regularity of the solution are proved.