Coarse-Grained Fluctuation Probabilities in the Standard Model and Subcritical Bubbles

We compute systematically the probability for fluctuations of the Higgs field, averaged over a given spatial scale, to exceed a specified value, in the Standard Model. For the particular case of interest of averages over one coherence volume we show that, even in the worst possible case of taking the one-loop improved effective potential parameters, the probability for the field to fluctuate from the symmetric to the asymmetric minimum before the latter becomes stable is very small for Higgs masses of the order of those of the $W$ and $Z$ bosons, whereas the converse is more likely. As such, metastability should be satisfied dynamically at the Electroweak phase transition and its dynamics should therefore proceed by the usual mechanism of bubble nucleation with subcritical fluctuations playing no particularly relevant role in it.Comment: Latex file, 13 pages. 7 figures, available in compressed form by anonymous ftp from ftp://euclid.tp.ph.ic.ac.uk/papers/94-5_38.fig Latex and postscript versions also available at http://euclid.tp.ph.ic.ac.uk/Papers/index.htm

Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as {provably} good measurement matrices for compressed sensing under $\ell_1$-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of $\ell_0$-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.Comment: 12 pages, 11 figure