arXiv subjects
Publications and source records attributed to Pavel Zakharov.
This paper studies the giant component of the sparse random bipartite graph $G(n, n, p)$, where $p = c/n$ for a fixed constant $c > 1$. We prove that its size is asymptotically normal.
We investigate the ferromagnetic Ising model on the Erdős-Rényi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(β,B)$, where $Z_{\mathbb{G}(n,m)}(β,B)$ is the partition function at inverse temperature $β>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $β>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $Θ(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $β<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.
We prove that for any $k\geq3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the `replica symmetric solution' predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)].The improvement is significant particularly for small $k$.
In an influential article Papadimitriou [FOCS 1991] proved that a local search algorithm called WalkSAT finds a satisfying assignment of a satisfiable 2-CNF with $n$ variables in $O(n^2)$ expected time. Variants of the WalkSAT algorithm have become a mainstay of practical SAT solving (e.g., [Hoos and Stützle 2000]). In the present article we analyse the expected running time of WalkSAT on random 2-SAT instances. Answering a question raised by Alekhnovich and Ben-Sasson [SICOMP 2007], we show that WalkSAT runs in linear expected time for all clause/variable densities up to the random 2-SAT satisfiability threshold.
Fluorescence telescopes are among the key instruments used for studying ultra-high energy cosmic rays in all modern experiments. We use model data for a small ground-based telescope EUSO-TA to try some methods of machine learning and neural networks for recognizing tracks of extensive air showers in its data and for reconstruction of energy and arrival directions of primary particles. We also comment on the opportunities to use this approach for other fluorescence telescopes and outline possible ways of improving the performance of the suggested methods.
We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order $\sqrt n$, with $n$ the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.
We study the noise of the intensity variance and of the intensity correlation and structure functions measured in light scattering from a random medium in the case when these quantities are obtained by averaging over a finite number N of pixels of a digital camera. We show that the noise scales as 1/N in all cases and that it is sensitive to correlations of signals corresponding to adjacent pixels as well as to the effective time averaging (due to the finite sampling time) and spatial averaging (due to the finite pixel size). Our results provide a guide to estimation of noise level in such applications as the multi-speckle dynamic light scattering, time-resolved correlation spectroscopy, speckle visibility spectroscopy, laser speckle imaging etc.
The thermodynamic approach of concentration mapping is used to discover communities in the directional friendship network of LiveJournal users. We show that this Internet-based social network has a power-law region in degree distribution with exponent γ= 3.45. It is also a small-world network with high clustering of nodes. To study the community structure we simulate diffusion of a virtual substance immersed in such a network as in a multi-dimensional porous system. By analyzing concentration profiles at intermediate stage of the diffusion process the well-interconnected cliques of users can be identified as nodes with equal values of concentration.
We introduce a charge coupled device (CCD) camera based detection scheme in dynamic light scattering that provides information on the single-scattered auto-correlation function even for fairly turbid samples. It is based on the single focused laser beam geometry combined with the selective cross correlation analysis of the scattered light intensity. Using a CCD camera as a multispeckle detector we show how spatial correlations in the intensity pattern can be linked to single and multiple scattering processes. Multiple scattering suppression is then achieved by an efficient cross correlation algorithm working in real time with a temporal resolution down to 0.2 seconds. Our approach allows access to the extensive range of systems that show low-order scattering by selective detection of the singly scattered light. Model experiments on slowly relaxing suspensions of titanium dioxide in glycerol were carried out to establish the validity range of our approach. Successful application of the method is demonstrated up to a scattering coefficient of more than $μ_s = 5$ cm$^{-1}$ for the sample size of L=1 cm.