Universality of the limit shape of convex lattice polygonal lines

Let ${\varPi}_n$ be the set of convex polygonal lines $\varGamma$ with vertices on $\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as $n\to\infty$, of "typical" $\varGamma\in {\varPi}_n$ with respect to a parametric family of probability measures $\{P_n^r,0<r<\infty\}$ on ${\varPi}_n$, including the uniform distribution ($r=1$) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. B\'ar\'any and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class $\{P^r_n\}$, even though $P^r_n$ ($r\ne1$) and $P^1_n$ are asymptotically singular. Measures $P^r_n$ are constructed, following Sinai's approach, as conditional distributions $Q_z^r(\cdot |{\varPi}_n)$, where $Q_z^r$ are suitable product measures on the space ${\varPi}=\bigcup_n{\varPi}_n$, depending on an auxiliary "free" parameter $z=(z_1,z_2)$. The transition from $({\varPi},Q_z^r)$ to $({\varPi}_n,P_n^r)$ is based on the asymptotics of the probability $Q_z^r({\varPi}_n)$, furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the M\"obius inversion formula and properties of zeroes of the Riemann zeta function.Comment: Published in at http://dx.doi.org/10.1214/10-AOP607 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gibbs cluster measures on configuration spaces

The probability distribution g_cl of a Gibbs cluster point process in X = R^d (with i.i.d. random clusters attached to points of a Gibbs configuration with distribution g) is studied via the projection of an auxiliary Gibbs measure ĝ in the space of configurations ^γ={(x,\bar{y})}, where x∈X indicates a cluster "center" and y∈\mathfrak{X}=\sqcup_{n} X^n represents a corresponding cluster relative to x. We show that the measure g_cl is quasi-invariant with respect to the group Diff_0(X) of compactly supported diffeomorphisms of X, and prove an integration-by-parts formula for g_cl. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms

Poisson cluster measures : Quasi-invariance, integration by parts and equilibrium stochastic dynamics

The distribution µcl of a Poisson cluster process in X = Rd (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X = FnXn, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure µcl is quasiinvariant with respect to the group of compactly supported diffeomorphisms ofX and prove an integration-by-parts formula for µcl. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet form

Limit theorems for sums of random exponentials

We study limiting distributions of exponential sums $S_N(t)=\sum_{i=1}^N e^{tX_i}$ as $t\to\infty$, $N\to\infty$, where $(X_i)$ are i.i.d.\ random variables. Two cases are considered: (A) \esssup X_i=0 and (B) \esssup X_i=\infty. We assume that the function $h(x)=-\log P(X_i>x)$ (case B) or $h(x)=-\log P(X_i>-1/x)$ (case A) is regularly varying at $\infty$ with index $1<\varrho<\infty$ (case B) or $0<\varrho<\infty$ (case A). The appropriate growth scale of $N$ relative to $t$ is of the form $e^{\lambda H_0(t)}$ ($0<\lambda<\infty$), where the rate function $H_0(t)$ is a certain asymptotic version of the function $H(t)=\log E [e^{tX_i}]$ (case B) or $H(t)=-\log E [e^{tX_i}]$ (case A). We have found two critical points, $\lambda_{1}<\lambda_{2}$, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For $0<\lambda<\lambda_{2}$, under the slightly stronger condition of normalized regular variation of $h$ we prove that the limit laws are stable, with characteristic exponent $\alpha=\alpha(\varrho,\lambda)\in(0,2)$ and skewness parameter $\beta\equiv1$

Occupation time distributions for the telegraph process

For the one-dimensional telegraph process, we obtain explicit distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of sub-normal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals.Comment: 23 pages, 4 figure

Qualitative and quantitative characteristics of the extracellular DNA delivered to the nucleus of a living cell

BACKGROUND: The blood plasma and other intertissue fluids usually contain a certain amount of DNA, getting there due to a natural cell death in the organism. Cells of this organism can capture the extracellular DNA, whereupon it is delivered to various cell compartments. It is hypothesized that the extracellular DNA is involved in the transfer of genetic information and its fixation in the genome of recipient cell. RESULTS: The existence of an active flow of extracellular DNA into the cell is demonstrated using human breast adenocarcinoma (MCF-7) cells as a recipient culture. The qualitative state of the DNA fragments delivered to the main cell compartments (cytoplasm and interchromosomal fraction) was assessed. The extracellular DNA delivered to the cell is characterized quantitatively. CONCLUSION: It is demonstrated that the extracellular DNA fragments in several minutes reach the nuclear space, where they are processed so that their linear size increases from about 500 bp to 10,000 bp. The amount of free extracellular DNA fragments simultaneously present in the nuclear space may reach up to 2% of the haploid genome. Using individual DNA fragments with a known molecular weight and sequence as an extracellular DNA, it is found that these fragments degrade instantly in the culture liquid in the absence of a competitor DNA and are delivered into the cell as degradants. When adding a sufficient amount of competitor DNA, the initial undegraded molecules of the DNA fragments with the known molecular weight and sequence are detectable both in the cytoplasm and nuclear space only at the zero point of experiments. The labeled precursor α-dNTP*, added to culture medium, was undetectable inside the cell in all the experiments