Isomorphism in Wavelets

Journal of Fourier Analysis and Applications - Two scaling functions $$\varphi _A$$ and $$\varphi _B$$ for Parseval frame wavelets are algebraically isomorphic, $$\varphi _A \simeq \varphi _B$$, if...     

Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

Given an inclusion of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets , giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of implies that of the scaling limit of . As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion of local nets with the same canonical field net , we find sufficient conditions which entail the equality of the canonical field nets of . Work supported by MIUR, GNAMPA-INDAM, the EU and SNS. Submitted: August 29, 2008. Accepted: March 23, 2009.     

A sufficient condition for random zero sets of Fock spaces

Archiv der Mathematik - Let $$(r_n)_{n=1}^\infty $$ be a non-decreasing sequence of radii in $$(0, \infty )$$ , and let $$(\theta _n)_{n=1}^\infty $$ be a sequence of independent random arguments...     

Asymptotics of the Number of Threshold Functions and the Singularity Probability of Random {±1}-Matrices

Doklady Mathematics - Two results concerning the number $$P(2,n)$$ of threshold functions and the singularity probability $${{\mathbb{P}}_{n}}$$ of random ( $$n \times n$$ ) $${\text{\{ }} \pm...     

Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation 1 $\frac{{\partial f}}{{\partial t}} = z\frac{{\zeta (t) + z}}{{\zeta (t) - z}}\frac{{\partial f}}{{\partial z}}$ , with boundary valuesf(z,0)=z, in the rangez ∈U= {w ∈ ? : ?w? < 1},t≤0. We choose ζ(t):=B(?2t), where B(t) is Brownian motion on ? $ \mathbb{U} The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the L?wner differential equation

(1)

, with boundary valuesf(z,0)=z, in the rangez ∈U= {w ∈ ℂ : •w• < 1},t≤0. We choose ζ(t):=B(−2t), where B(t) is Brownian motion on ∂ starting at a random-uniform point in ∂ . Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂ has the same law as that of the pathf(ζ(t),t) (wheref(z,t) is extended continuously to ∂ ) ×(−∞,0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters. Research supported by the Sam and Ayala Zacks Professorial Chair.     

On Optimal Bounds in the Local Semicircle Law under Four Moment Condition

Doklady Mathematics - We consider symmetric random matrices $${{{\mathbf{X}}}_{n}} = [{{X}_{{jk}}}]_{{j,k = 1}}^{n},n \geqslant 1$$ , whose upper triangular entries are independent random variables...     

On scaling invariance and type-I singularities for the compressible Navier-Stokes equations

Science China Mathematics - We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-I singularities of solutions with $$\mathop {\lim...     

On norms in some class of exponential type Orlicz spaces of random variables

Positivity - A new characterization of the exponential type Orlicz spaces generated by the functions $$\exp (|x|^p)-1$$ ( $$p\ge 1$$ ) is given. We define norms for centered random variables...     

On the Use of Bivariate Mellin Transform in Bivariate Random Scaling and Some Applications

We discuss here the problem of bivariate random scaling. Both direct and inverse problems of bivariate random scaling are solved by two methods. While the first method is a distributional one, the second method is an indirect one associated with bivariate Mellin transform. Finally, we use bivariate random scaling for some statistical and simulational applications.     

Trigonometric multiplicative chaos and applications to random distributions

Science China Mathematics - The random trigonometric series $$\sum\nolimits_{n = 1}^\infty {{\rho _n}\cos \left( {nt + {\omega _n}} \right)} $$ on the circle $$\mathbb{T}$$ are studied under the...     

Algorithmic randomness of continuous functions

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}We investigate notions of randomness in the space of continuous functions on . A probability measure is given and a version of the Martin-L?f test for randomness is defined. Random continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. Research partially supported by the National Science Foundation grants DMS 0532644 and 0554841 and 00652732. Thanks also to the American Institute of Mathematics for support during 2006 Effective Randomness Workshop; Remmel partially supported by NSF grant 0400307; Weber partially supported by NSF grant 0652326. Preliminary version published in the Third International Conference on Computability and Complexity in Analysis, Springer Electronic Notes in Computer Science, 2006.     

ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

Convergence of the Best Linear Predictor of a Weakly Stationary Random Field

Journal of Fourier Analysis and Applications - Suppose that $$\{X_{m,n}\}_{(m,n)\in \mathbb {Z}^2}$$ is a centered, weakly stationary random field with spectral density function W. Let $$X'$$...     

Geodesics Closed On The Projective Plane

Given a C ^∞ Riemannian metric g on P ² we prove that (, g) has constant curvature iff all geodesics are closed. Therefore is the first non-trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large set of C ^∞ metrics whose geodesics are all closed and have the same period 2π (called Zoll metrics), but no metric of this set can be obtained from another metric of this set via an isometry and scaling. As a corollary we conclude that all two-dimensional P-manifolds are SC-manifolds. Received: April 2007; Revision: September 2007; Accepted: September 2007     

Adjacency and transition matrices related to random walks on graphs

Journal of Algebraic Combinatorics - A pointed graph $$(\Gamma ,v_0)$$ induces a family of transition matrices in Wildberger’s construction of a hermitian hypergroup using a random walk on...     

Randomized consistent statistical inference for random processes and fields

Statistical Inference for Stochastic Processes - We propose a randomized approach to the consistent statistical analysis of random processes and fields on $${\mathbb {R}}^m$$ and $${\mathbb {Z}}^m,...     

On the Convergence of Series of Dependent Random Variables

Journal of Theoretical Probability - Given a sequence $$(X_n)$$ of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under...     

Divergence Criterion for a Class of Random Series Related to the Partial Sums of I.I.D. Random Variables

Journal of Theoretical Probability - Let $$ \{X, X_{n};~n \ge 1 \}$$ be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to...     

Asymptotic Behavior of the Mean Sojourn Time for a Random Walk to be in a Domain of Large Deviations

Siberian Advances in Mathematics - We study the asymptotic behavior of the mean of sojourn time for a homogeneous random walk defined on $$ [0,n]$$ to be above a receding curvilinear boundary in a...     

Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

Foundations of Computational Mathematics - We study the approximation of expectations $${\text {E}}(f(X))$$ for Gaussian random elements X with values in a separable Hilbert space H and Lipschitz...