We prove that there exist (infinitely many) values of the real parameters a and b for which the matrices a 1 1 0 1 and b 1 0 1 1 have the following property: all infinite periodic products of the two matrices converge to zero, but there exists a nonperiodic product that doesn't. Our proof is self-contained and fairly elementary; it uses only elementary facts from the theory of formal languages and from linear algebra. It is not constructive in that we do not exhibit any explicit values of a and b with the stated property; the problem of finding explicit matrices with this property remains open.
The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations?The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span d(t) ∼ t γ of the trajectory of t steps above the top of the semicircle or the triangle. We show that γ = 1 3 , i.e. d(t) shares the KPZ statistics for the semicircle, while γ = 0 for the triangle. We propose heuristic derivations of scaling exponents γ for different geometries, justify them by explicit analytic computations and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.
We study large fluctuations of emitted radiation in the system of N non-interacting twolevel atoms. Two methods are used to calculate the probability of large fluctuations and the time dependence of excitation and emission. The first method is based on the large deviation principle for Markov processes. The second one uses an analogue of the quantum formalism for classical probability problems.Particularly we prove that in a large fluctuation limit approximately half of the atoms are excited. This fact is independent on the fraction of the excited atoms in equilibrium.
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