arXiv subjects
Publications and source records attributed to Boris Tsirelson.
We show that two different ideas of uniform spreading of locally finite measures in the d-dimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connecting a given measure with the Lebesgue measure.
Moderate deviations for random complex zeroes are deduced from a new theorem on moderate deviations for random fields.
The divergence of a stationary random vector field at a given point is usually a centered (that is, zero mean) random variable. Strangely enough, it can be equal to 1 almost surely. This fact is another form of a phenomenon disclosed by B. Weiss in 1997.
The best approximation by bounded product functions is calculated for some very simple two-valued functions of two variables.
A moderate deviation principle for nonlinear functions of Gaussian processes is established. The nonlinear functions need not be locally bounded. Especially, the logarithm is allowed. (Thus, small deviations of the process are relevant.) Both discrete and continuous time is treated. An integrable power-like decay of the correlation function is assumed.
Motions of the plane (shifts and rotations) correspond to automorphisms of the type I Arveson system of white noise. I prove that automorphisms corresponding to rotations cannot be extended to the type II Arveson system of Warren's noise.
The coordinate of a harmonic oscillator is measured at a time chosen at random among three equiprobable instants: now, after one third of the period, or after two thirds. The (total) probability that the outcome is positive depends on the state of the oscillator. In the classical case the probability varies between 1/3 and 2/3, but in the quantum case -- between 0.29 and 0.71.
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts, proposed here, includes a wide class of random equivalence classes.
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this fact is now superseded by a general result.
We show that the flat chaotic analytic zero points (i.e. zeroes of a random entire function whose Taylor coefficients are independent complex-valued Gaussian random variables, and the variance of the k-th coefficient is 1/k!) can be regarded as a random perturbation of a lattice in the plane. The distribution of the distances between the zeroes and the lattice points is shift-invariant and has a Gaussian-type decay of the tails.
We compare two examples of random dense countable sets, `Brownian local minima' and `unordered uniform infinite sample'. They appear to be identically distributed. A framework for such notions is proposed. In addition, random elements of other singular spaces (especially, reals modulo rationals) are considered.
The scaling limit of the critical percolation, is it a black noise? The answer depends on stability to perturbations concentrated along a line. This text, containing no proofs, reports experimental results that suggest the affirmative answer.
Contrary to the classical wisdom, processes with independent values (defined properly) are much more diverse than white noise combined with Poisson point processes, and product systems are much more diverse than Fock spaces. This text is a survey of recent progress in constructing and investigating nonclassical stochastic flows and continuous products of probability spaces and Hilbert spaces.
We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes.
By a hole we mean a disc that contains no flat chaotic analytic zero points (i.e. zeroes of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!). A given disc of radius r has a probability of being a hole, - the hole probability. We show that for large r the hole probability decays as exp(-cr^4).
Uncountably many mutually non-isomorphic product systems (that is, continuous tensor products of Hilbert spaces) of types II-0 and III are constructed by probabilistic means (random sets and off-white noises), answering four questions of W. Arveson. Results of math.FA/0001070, math.FA/0006165 are improved, and proofs are more readable.
Linear functions of many independent random variables lead to classical noises (white, Poisson, and their combinations) in the scaling limit. Some singular stochastic flows and some models of oriented percolation involve very nonlinear functions and lead to nonclassical noises. Two examples are examined, Warren's `noise made by a Poisson snake' and the author's `Brownian web as a black noise'. Classical noises are stable, nonclassical are not. A new framework for the scaling limit is proposed. Old and new results are presented about noises, stability, and spectral measures.
Recently T. Kieu (arXiv:quant-ph/0110136) claimed a quantum algorithm computing some functions beyond the Church-Turing class. He notes that "it is in fact widely believed that quantum computation cannot offer anything new about computability" and claims the opposite. However, his quantum algorithm does not work, which is the point of my short note. I still believe that quantum computation leads to new complexity but retains the old computability.